Projective Varieties

with Bad Semi-stable Reduction at 3 Only

To I. R. Shafarevich, on the occasion of his 90th Birthday

Victor Abrashkin

Received: April 30, 2013

Communicated by Peter Schneider

Abstract. Suppose F = W (k)[1/p] where W (k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p ≠ 2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0, p). This modification of Breuil’s theory results in the following application in the spirit of the Shafarevich Conjecture. If Y is a projective algebraic variety over ℚ with good reduction modulo all primes l ≠ 3 and semi-stable reduction modulo 3 then for the Hodge numbers of Y_{ℂ} = Y ⊗_{ℚ} ℂ, one has h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ}).

2010 Mathematics Subject Classification: 11S20, 11G35, 14K15 Keywords and Phrases: p-adic semi-stable representations, Shafarevich Conjecture

Introduction

Everywhere in the paper p is a fixed prime number, p ≠ 2, k is algebraically closed field of charactersitic p, F is the fraction field of the ring of Witt vectors W (k), F̄ is a fixed algebraic closure of F and Γ_{F} = Gal(F̄/F) is the absolute Galois group of F. Suppose Y is a projective algebraic variety over ℚ. Denote by Y_{ℂ} the corresponding complex variety Y ⊗_{ℚ} ℂ. For integers n, m ⩾ 0, set h^{n}(Y_{ℂ}) = dim_{ℂ} H^{n}(Y_{ℂ}, ℂ) and h^{n,m}(Y_{ℂ}) = dim_{ℂ} H^{n}(Ω_{Y_{ℂ}}^{m}). The main result of this paper can be stated as follows.

Theorem 0.1. If Y has semi-stable reduction modulo 3 and good reduction modulo all primes l ≠ 3 then h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ}).

Remind that a generalization of the Shafarevich Conjecture about the nonexistence of non-trivial abelian varieties over ℚ with everywhere good reduction was proved by Fontaine [16] and the author [2], and states that

(0.1)  h^{1}(Y_{ℂ}) = h^{3}(Y_{ℂ}) = 0, h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ})

if Y has everywhere good reduction. (The Shafarevich Conjecture appears then as the equality h^{1}(Y_{ℂ}) = 0.) This result became possible due to the following two important achievements of Fontaine’s theory of p-adic crystalline representations:

— the Fontaine-Messing theorem relating etale and de Rham cohomology of smooth proper schemes over W (k) in dimensions [0, p), [15] (it was later proved by Faltings in full generality, [12]);

— the Fontaine-Laffaille integral theory of crystalline representations of Γ_{F} with Hodge-Tate weights from [0, p − 2], [13]. Note that the Fontaine-Laffaille theory works essentially for Hodge-Tate weights from [0, p) but does not give all Galois invariant lattices in the corresponding crystalline representations. Nevertheless, this theory admits improvement developed by the author in [1]. As a result, there was obtained a suitable integral theory for the case of Hodge-Tate weights from [0, p), which allowed us to prove some extras to statements (0.1), in particular, that modulo the Generalized Riemann Hypothesis one has h^{4}(Y_{ℂ}) = h^{2,2}(Y_{ℂ}). Since that time there was a huge progress in the study of semi-stable p-adic representations. Tsuji [23] proved a semi-stable case of the relation between etale and crystalline cohomology and Breuil [5, 6] developed an analogue of the Fontaine-Laffaille theory in the context of semi-stable representations (even for ramified basic fields). The papers [4] and [21] studied the problem of the existence of abelian varieties over ℚ with only one prime of bad semi-stable reduction. Note that the progress in this direction is quite restrictive because our knowledge of algebraic number fields with prescribed ramification at a given prime number p (and unramified outside p) is very far from to be complete. Theorem 0.1 represents an exceptional situation where the standard tools: the Odlyzko estimates of the minimal discriminants of algebraic number fields and the modern computing facilities (SAGE) are sufficient to resolve upcoming problems. In addition, the proof of this theorem requires a modification of Breuil’s theory to work with semi-stable representations of Γ_{F} with Hodge-Tate weights from [0, p). The structure of this paper can be described as follows. In Section 1 we introduce the category \underline{L}^{∗} of filtered (φ, N)-modules over W_{1} := k[[u]]. This is a special pre-abelian category, that is an additive category with kernels, cokernels and sufficiently nice behaving short exact sequences. Note that such categories play quite appreciable role in all our constructions. In Section 2 we construct the functor V^{∗} from \underline{L}^{∗} to the category of 𝔽_{p}[Γ_{F}]- modules \underline{MΓ}_{F} by introducing a “truncated” version of Fontaine’s ring of semistable periods Â_{st}. The functor V^{∗} is not fully faithful but by taking into account the maximal etale subobjects of filtered modules from \underline{L}^{∗} we define a modification CV^{∗} of V^{∗}. This functor gives already a fully faithful functor from \underline{L}^{∗} to the category of cofiltered Γ_{F} -modules \underline{CMΓ}_{F}. In Section 3 we give an interpretation of Breuil’s theory in terms of W := W (k)[[u]]-modules (Breuil worked with modules over the divided powers envelope of W) by introducing the category of filtered (φ, N)-modules \underline{L}^{ft} over W. The advantage of this construction is that the objects of this category appear as strict subquotients of p-divisible groups in suitable pre-abelian category. This allows us to use devissage despite that all involved categories are not abelian. We also introduce the subcategories \underline{L}^{u,ft} and, resp., \underline{L}^{m,ft} of unipotent and, resp., multiplicative objects in \underline{L}^{ft} and prove that any L ∈ \underline{L}^{ft} is a canonical extension

(0.2)  0 ⟶ L^{u} ⟶ L ⟶ L^{m} ⟶ 0

of a multiplicative object L^{m} by a unipotent object L^{u}. In Section 4 we study Breuil’s functor V^{ft} : \underline{L}^{ft} ⟶ \underline{MΓ}_{F} in the situation of Hodge-Tate weights from [0, p). We show that on the subcategory \underline{L}^{u,ft} this functor is still fully faithful by proving that on the subcategory of killed by p unipotent objects the functors V^{ft} and V^{∗} coincide. Then we show that for any killed by p object L of \underline{L}^{ft}, the functor V^{ft} transforms the standard short exact sequence (0.2) into a short exact sequence in \underline{MΓ}_{F}, which admits a functorial splitting. This

splitting is used then to construct a modified version C̃V^{ft} : \underline{L}^{ft} ⟶ \underline{CMΓ}_{F} of V^{ft}, which is already fully faithful. This gives us an efficient control on all Galois invariant lattices of semi-stable representations with weights from [0, p). Especially, we have an explicit description of all killed by p subquotients of such lattices and the corresponding ramification estimates. Finally, in Section 5 we give a proof of Theorem 0.1 following the strategy from [2]. Essentially, we obtain the following result: if V is a 3-adic representation of Γ_{ℚ} = Gal(ℚ̄/ℚ) which is unramified outside 3 and is semi-stable at 3 then there is a Γ_{ℚ}-equivariant filtration by ℚ_{3}-subspaces V = V_{0} ⊃ V_{1} ⊃ V_{2} ⊃ V_{3} = 0 such that for 0 ⩽ i ⩽ 2, the Γ_{ℚ}-module V_{i}/V_{i+1} is isomorphic to the product of finitely many copies of the Tate twist ℚ_{3}(i). If V = H_{et}^{2}(Y_{F} , ℚ_{3}) then looking at the eigenvalues of the Frobenius morphisms of reductions modulo l ≠ 3, we obtain that V = V_{1} and V_{2} = 0, and this implies that h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ}). Note that our construction of the modification of Breuil’s functor gives automatically the modification of the Fontaine-Laffaille functor, which essentially coincides with the modification constructed in [1]. It is worth mentioning that switching from Breuil’s S-modules to W-modules means moving in the direction of Kisin’s approach [18] and recent approach to integral theory of p-adic representations by Liu [19, 20]. It would be also very interesting to study the opportunity to modify Breuil’s functor over ramified base [8, 9] to the case of Hodge-Tate weights from [0, p). Finally, mention quite surprising matching of the ramification estimates for semi-stable representations and the Leopoldt conjecture for the field ℚ(\sqrt[3]{3}, ζ_{9}), cf. Section 5. Acknowledgements. The author is very grateful to Shin Hattori for numerous and helpful discussions.

1. The categories ˜ \underline{L}^{∗}, ˜ \underline{L}_{0}^{∗}, \underline{L}^{∗}, \underline{L}_{0}^{∗}

Remind that k is algebraically closed field of characteristic p > 2. Let W = W (k)[[u]], where W (k) is the ring of Witt vectors with coefficients in k and u is an indeterminate. Denote by σ the automorphism of W (k) induced by the p-th power map on k and agree to use the same symbol for its continuous extension to W such that σ(u) = u^{p}. Denote by N : W ⟶ W the continuous W (k)-linear derivation such that N(u) = −u. We shall often use below the following statement.

Lemma 1.1. Suppose L is a finitely generated W-module and A is a σ-linear operator on L. Then the operator id_{L} − A is epimorphic. If, in addition, A is nilpotent then id_{L} − A is bijective.

Proof. Part b) is obvious. In order to prove a) notice first that we can replace L by L/uL and, therefore, assume that L is a finitely generated W (k)-module. Clearly, it will be enough to consider the case pL = 0. Then there is a decomposition of k-vector spaces L = L_{1} ⊕L_{2}, where A is invertible on L_{1} and nilpotent on L_{2}. It remains to note that L_{1} = L_{0} ⊗_{𝔽_{p}} k, where L_{0} is a finite dimensional 𝔽_{p}-vector space such that A|_{L_{0}} = id. The existence of L_{0} is a standard fact of σ-linear algebra: if s = dim_{k} L_{1} and A ∈ M_{s}(k) is a matrix of A|_{L_{1}} in some k-basis of L_{1} then L_{0} = {(x_{1}, …, x_{s}) ∈ k^{s} | (x_{1}^{p}, …, x_{s}^{p})A = (x_{1}, …, x_{s})}; the 𝔽_{p}-linear space L_{0} has dimension s because the corresponding equations determine an etale algebra of rank p^{s} over algebraically closed field k.

□

Remark. In above Lemma and everywhere below we use the following agreement: A is nilpotent on L iff it is “topologically nilpotent”, i.e. ⋂_{n} A^{n}(L) = 0.

1.1. Definitions and general properties. Let W_{1} = W/pW with induced σ, φ and N.

Definition. The objects of the category ˜ \underline{L}_{0}^{∗} are the triples L = (L, F (L), φ), where

• L and F (L) are W_{1}-modules such that L ⊃ F (L);

• φ : F (L) ⟶ L is a σ-linear morphism of W_{1}-modules; (Note that φ(F (L)) is a σ(W_{1})-submodule in L.) If L_{1} = (L_{1}, F (L_{1}), φ) is also an object of ˜ \underline{L}_{0}^{∗} then the morphisms f ∈ Hom˜\underline{L}_{0}^{∗}(L_{1}, L) are given by W_{1}-linear maps f : L_{1} ⟶ L such that f(F (L_{1})) ⊂ F (L) and fφ = φf.

Definition. The objects of the category ˜ \underline{L}^{∗} are the quadruples L = (L, F (L), φ, N), where

• (L, F (L), φ) is an object of the category ˜ \underline{L}_{0}^{∗};

• N : L ⟶ L/u^{2p}L is a W_{1}-differentiation, i.e. for all w ∈ W_{1} and l ∈ L, N(wl) = N(w)(l mod u^{2p}L) + wN(l); • if L_{1} = (L_{1}, F (L_{1}), φ, N) is another object of ˜ \underline{L}^{∗} then the morphisms Hom˜\underline{L}^{∗}(L_{1}, L) are given by f : (L_{1}, F (L_{1}), φ) ⟶ (L, F (L), φ) from ˜ \underline{L}_{0}^{∗} such that fN = Nf. (We use the same notation f for the reduction of f modulo u^{2p}L.)

The categories ˜ \underline{L}^{∗} and ˜ \underline{L}_{0}^{∗} are additive.

Definition. The category \underline{L}_{0}^{∗} is a full subcategory of ˜ \underline{L}_{0}^{∗} consisting of the objects L = (L, F (L), φ) such that

• L is a free W_{1}-module of finite rank;

• F (L) ⊃ u^{p−1}L;

• the natural embedding φ(F (L)) ⊂ L induces the identification φ(F (L)) ⊗_{σ(W_{1})} W_{1} = L.

Note that φ induces a map F (L)/u^{2p}L ⟶ L/u^{2p}L: use that u^{2p}L ⊂ u^{p+1}F (L) ⊂ u^{2}F (L) and φ(u^{2}F (L)) ⊂ u^{2p}L. We shall denote this map by the same symbol φ.

Definition. The category \underline{L}^{∗} is a full subcategory of ˜ \underline{L}^{∗} consisting of the objects L = (L, F (L), φ, N) such that

• (L, F (L), φ) ∈ \underline{L}_{0}^{∗};

• for all l ∈ F (L), uN(l) ∈ F (L) mod u^{2p}L and N(φ(l)) = φ(uN(l)).

The categories \underline{L}_{0}^{∗} and \underline{L}^{∗} are additive. In the case of objects (L, F (L), φ, N) of \underline{L}^{∗} the morphism N can be uniquely recovered from the W_{1}-differentiation N_{1} = N mod u^{p}L due to the following property.

Proposition 1.2. Suppose (L, F (L), φ) ∈ \underline{L}_{0}^{∗} and N_{1} : L ↦ L/u^{p}L is a W_{1}- differentiation such that for any m ∈ F (L), uN_{1}(m) ∈ F (L) mod u^{p}L and N_{1}(φ(l)) = φ(uN_{1}(l)). Then there is a unique W_{1}-differentiation N : L ⟶ L/u^{2p}L such that N mod u^{p} = N_{1} and (L, F (L), φ, N) ∈ \underline{L}^{∗}.

Proof. Choose a W_{1}-basis m_{1}, …, m_{s} of F (L). Then l_{1} = φ(m_{1}), …, l_{s} = φ(m_{s}) is a W_{1}-basis of L and a σ(W_{1})-basis of φ(F (L)). Let N(l_{i}) := φ(uN_{1}(m_{i})′) ∈ L/u^{2p}L, where N_{1}(m_{i})′ are some lifts of N_{1}(m_{i}) to L/u^{2p}L. Clearly, the elements N(l_{i}) ∈ φ(F (L)) ⊂ L/u^{2p}L are well-defined (use that φ(u^{p+1}L) ⊂ u^{2p}L). For any l = ∑_{i} w_{i}l_{i} ∈ L, let N(l) := ∑_{i} N(w_{i})l_{i} + ∑_{i} w_{i}N(l_{i}). Then N : L ⟶ L/u^{2p} is a W_{1}-differentiation and N mod u^{p} = N_{1}. Clearly, N is the only candidate to satisfy the requirements of our Proposition. Now suppose m = ∑_{i} w_{i}m_{i} ∈ F (L) with all w_{i} ∈ W_{1}. Then N(φ(m)) = ∑_{i} σ(w_{i})l_{i} mod u^{2p}. On the other hand, φ(uN(m)) equals

∑_{i} u^{p}σ(N(w_{i}))l_{i} + ∑_{i} φ(w_{i}uN(m_{i})) = ∑_{i} σ(w_{i})l_{i} mod u^{2p}

because all σ(N(w_{i})) ∈ u^{p}σ(W_{1}). The proposition is proved. □

Remark. By above Proposition in the definition of objects of \underline{L}^{∗} one can replace N : L ⟶ L/u^{2p}L by N_{1} = N mod u^{p}L and use N as a unique extension of N_{1} if neccessary. An example of the situation where we do need to extend N_{1} is described in Proposition 1.3 below. Another situation is related to the definition of the truncated version R_{st}^{0} of Â_{st} in Subsection 2. Here we need N to be defined modulo some smaller module than u^{p}L, e.g. u^{p+1}L. Our choice was done in favour of the module u^{2p}L because it is the smallest possible module where the definition of N makes sense.

Proposition 1.3. \underline{L}_{0}^{∗} and \underline{L}^{∗} are pre-abelian categories (cf. Appendix A for the concept of pre-abelian category).

Proof. Suppose S is an additive category and f ∈ Hom_{S}(A, B), A, B ∈ S. Then i ∈ Hom_{S}(K, A) is a kernel of f if for any D ∈ S, the sequence of abelian groups

0 ⟶ Hom_{S}(D, K) ⟶^{i_{∗}} Hom_{S}(D, A) ⟶^{f_{∗}} Hom_{S}(D, B)

is exact. Similarly, j ∈ Hom_{S}(B, C), B, C ∈ S, is a cokernel of f if for any D ∈ S, the sequence

0 ⟶ Hom_{S}(C, D) ⟶^{j^{∗}} Hom_{S}(B, D) ⟶^{f^{∗}} Hom_{S}(A, D)

is exact. Let F F_{W_{1}} be the category of free W_{1}-modules with filtration. This category is pre-abelian. More precisely, consider the objects L = (L, F (L)) and M = (M, F (M)) in F F_{W_{1}} and let f ∈ Hom_{F F_{W_{1}}} (L, M). Then Ker_{F F_{W_{1}}} f is a natural embedding i_{L} : K = (K, F (K)) ⟶ L, where K = Ker(f : L ⟶ M) and F (K) = K ∩ F (L). The coimage Coim_{F F_{W_{1}}} f = Coker_{F F_{W_{1}}} (Ker_{F F_{W_{1}}} f) appears as a natural projection j_{L} : L ⟶ L′ = (L′, F (L′)), where L′ = f(L) and F (L′) = f(F (L)). Similarly, Cokerf is a natural projection j_{M} : M ⟶ C = (C, F (C)), where C = (M/L′)/(M/L′)_{tor} and F (C) = j_{M}(F (M)). Then the image Im_{F F_{W_{1}}} f = Ker_{F F_{W_{1}}} (Coker_{F F_{W_{1}}} f) is a natural embedding M′ = (M′, F (M′)) ⟶ M, where M′ is the kernel of j_{M} and F (M′) = F (M) ∩ M′. As usually, there is a natural map L′ ⟶ M′ induced by L′ ⊂ M′. Note that M/M′ = C is free and M′/L′ is torsion W_{1}-modules and these properties completely characterize M′ as a W_{1}-submodule of M. Now suppose L = (L, F (L), φ), M = (M, F (M), φ) are objects of \underline{L}_{0}^{∗} and f ∈ Hom\underline{L}_{0}^{∗}(L, M). Use the obvious forgetful functor \underline{L}_{0}^{∗} ⟶ F F_{W_{1}} and the same notation for the corresponding images of L, M and f. Show that K = Ker_{F F_{W_{1}}} f and C = Coker_{F F_{W_{1}}} f have the natural structures of objects of \underline{L}_{0}^{∗} and with respect to this structure they become the kernel and, resp, cokernel of f in \underline{L}_{0}^{∗}. Indeed, u^{p−1}K = u^{p−1}L ∩ K ⊂ F (L) ∩ K = F (K) = Ker(f : F (L) ⟶ F (M)). Therefore, φ(F (K)) ⊂ K ∩ φ(F (L)) and there is a natural embedding ι : φ(F (K)) ⊗_{σW_{1}} W_{1} ⊂ K. On the one hand,

rk_{σW_{1}} φ(F (K)) = rk_{W_{1}} F (K) = rk_{W_{1}} K.

On the other hand, F (L)/F (K) ⊂ L/K = L′ have no W_{1}-torsion. This implies that the quotient φ(F (L))/φ(F (K)) has no σW_{1}-torsion and the factor of L = φ(F (L)) ⊗_{σW_{1}} W_{1} by φ(F (K)) ⊗_{σW_{1}} W_{1} also has no W_{1}-torsion. So, ι becomes the equality φ(F (K))⊗_{σW_{1}} W_{1} = K and K = (K, F (K), φ) = Ker\underline{L}_{0}^{∗}f. The above description of Ker\underline{L}_{0}^{∗} implies that u^{p−1}L′ ⊂ F (L′), φ(F (L′)) = φ(F (M))/φ(F (K)) and L′ = φ(F (L′)) ⊗_{σW_{1}} W_{1}. In other words, L′ = (L′, F (L′), φ) ∈ \underline{L}_{0}^{∗}. Now note that for M′ = (M′, F (M′)), we have

u^{p−1}M′ = (u^{p−1}M) ∩ M′ ⊂ F (M) ∩ M′ = F (M′)

and, therefore, F (M′)/F (L′) is torsion W_{1}-module and

• (φ(F (M′)) ⊗_{σW_{1}} W_{1})/L′ is torsion W_{1}-module;

On the other hand, F (M)/F (M′) = F (C) is torsion free W_{1}-module. This implies that φ(F (M))/φ(F (M′)) is torsion free σW_{1}-module and, therefore,

• M/(φ(F (M′)) ⊗_{σW_{1}} W_{1}) is torsion free W_{1}-module.

The above two conditions completely characterize M′ as a submodule of M. Therefore, φ(F (M′)) ⊗_{σW_{1}} W_{1} = M′, M′ = (M′, F (M′), φ) ∈ \underline{L}_{0}^{∗} and (M/M′, F (M)/F (M′), φ) = (C, F (C), φ) = C ∈ \underline{L}_{0}^{∗}. Now a formal verification shows that C = Coker\underline{L}_{0}^{∗}f. Again Coim\underline{L}_{0}^{∗} f = (L′, F (L′), φ) and Im\underline{L}_{0}^{∗} f = (M′, F (M′), φ) together with their natural embedding Coim\underline{L}_{0}^{∗} f ⟶ Im\underline{L}_{0}^{∗} f in \underline{L}_{0}^{∗}. As a matter of fact, these two objects of the category \underline{L}_{0}^{∗} do not differ very much due to the following Lemma.

Lemma 1.4. φ(F (L′)) ⊃ u^{p}φ(F (M′)) (and, therefore, L′ ⊃ u^{p}M′).

Proof of Lemma. Otherwise, there is an l ∈ φ(F (L′)) \ u^{p}φ(F (L′)) such that l ∈ u^{2p}φ(F (M′)). Form the sequence l_{n} ∈ L′ such that l_{1} = l and for all n ⩾ 2, l_{n+1} = φ(u^{a_{n}}l_{n}), where a_{n} ⩾ 0 is such that u^{a_{n}}l_{n} ∈ F (L′) \ uF (L′). Clearly, all l_{n} ∉ uF (L′) ⊃ u^{p}L′. On the other hand, l ∈ u^{2p}φ(F (M′)) ⊂ u^{p+1}F (M′) and, therefore, for all n ⩾ 1, l_{n} ∈ φ^{n}(u^{2p}M′) ⊂ u^{p^{n}+p}M′. So, for n ≫ 0, l_{n} ∈ u^{p}L′. The contradiction. □

Now suppose L = (L, F (L), φ, N) and M = (M, F (M), φ, N) are objects of \underline{L}^{∗} and f ∈ Hom\underline{L}^{∗}(L, M). Prove that the kernel (K, F (K), φ) and the cokernel (C, F (C), φ) of f in the category \underline{L}_{0}^{∗} have a natural structure of objects of the category \underline{L}^{∗}. Clearly, N(K) ⊂ Ker (f mod u^{2p} : L/u^{2p}L ⟶ M/u^{2p}M). The above Lemma 1.4 implies that u^{p}L′ ⊃ u^{2p}M′ and we obtain the following natural maps

L′/u^{p}L′ ⟵^{α} L′/u^{2p}M′ ⟶^{β} M′/u^{2p}M′ ⟶^{γ} M/u^{2p}M.

Note that α is epimorphic but β and γ are monomorphic. This implies that N(K) ⊂ Ker(L/u^{2p}L ⟶ L′/u^{p}L′) and

N(K) mod u^{p}L ⊂ Ker(L/u^{p}L ⟶ L′/u^{p}L′) = K/u^{p}K.

Therefore, by Proposition 1.2, N (as a unique lift of N_{1} = N mod u^{p}) maps K to K/u^{2p}K and (K, F (K), φ, N) ∈ \underline{L}^{∗}. The above property of Ker\underline{L}^{∗}f implies that N(L′) ⊂ L′/u^{2p}L′. Now use that u^{p}M′ ⊂ L′, u^{2p}L′ ⊂ u^{2p}M′ and N(u^{p}M′) ⊂ u^{p}M/u^{2p}M to deduce that

N(u^{p}M′) ⊂ (L′/u^{2p}M′) ∩ (u^{p}M/u^{2p}M) = u^{p}M′/u^{2p}M′.

So, N mod u^{p} maps M′ to M′/u^{p}M′ and again by Proposition 1.2 N(M′) ⊂ M′/u^{2p}M′. This means that the kernel of the above constructed Cokerf : (M, F (M), φ) ⟶ (C, F (C), φ) is provided with the structure of object of the category \underline{L}^{∗}. Therefore, N induces the map N : C ⟶ C/u^{2p}C and (C, F (C), φ, N) ∈ \underline{L}^{∗}. The proposition is proved. □

The above proof shows that the kernels and cokernels in the category \underline{L}^{∗} appear on the level of filtered modules as the kernel and cokernel of the corresponding map of filtered modules (L_{1}, F (L_{1})) to (L, F (L)) in the category of filtered W_{1}-modules. Therefore, the category \underline{L}^{∗} is special, cf. Appendix A, and we can apply the corresponding formalism of short exact sequences. In particular, if we take another object L_{2} = (L_{2}, F (L_{2}), φ, N) ∈ \underline{L}^{∗} then

• i ∈ Hom\underline{L}^{∗}(L_{1}, L) is strict monomorphism iff i : L_{1} ⟶ L is injective and i(L_{1}) ∩ F (L) = i(F (L_{1}));

• j ∈ Hom\underline{L}^{∗}(L, L_{2}) is strict epimorphism iff j : L ⟶ L_{2} is epimorphic and j(F (L)) = F (L_{2}).

As usually, cf. Appendix A, if a monomorphism i is strict then the monomorphism j = Coker i is strict, and if an epimorphism j is strict then the monomorphism i = Ker j is strict, and under these assumptions 0 ⟶ L_{1} ⟶^{i} L ⟶^{j} L_{2} ⟶ 0 is short exact sequence.

With relation to the above result that the categories \underline{L}_{0}^{∗} and \underline{L}^{∗} are preabelian, note that the situation with the categories ˜ \underline{L}_{0}^{∗} and ˜ \underline{L}^{∗} is different. Indeed, let F M_{W_{1}} be the category of filtered (not necessarily free) modules over W_{1}. This category is pre-abelian: for M_{i} = (M_{i}, F (M_{i})), i = 1, 2, and f ∈ Hom_{F M_{W_{1}}} (M_{1}, M_{2}), we have the following equalities Ker_{F M_{W_{1}}} f = (Kerf, Kerf ∩ F (M_{1})) (together with its natural embedding into M_{1}) and Coker_{F M_{W_{1}}} f = (Cokerf, F (M_{2})/(f(M_{1})∩F (M_{2})) (together with the natural projection from M_{2}). Now suppose that M_{i} = (M_{i}, F (M_{i}), φ) ∈ ˜ \underline{L}_{0}^{∗}, i = 1, 2, and f ∈ Hom˜\underline{L}_{0}^{∗}(M_{1}, M_{2}). Then Ker˜\underline{L}_{0}^{∗}f exists (and coincides on the level of filtered modules with Ker_{F M_{W_{1}}} f) but Coker˜\underline{L}_{0}^{∗}f exists (and coincides on the level of filtered modules with Coker_{F M_{W_{1}}} f) only if we have f(F (M_{1})) = f(M_{1}) ∩ F (M_{2}). In particular, on the level of filtered modules the composition Coker_{F M_{W_{1}}} (Ker_{F M_{W_{1}}} f) always makes sense and coincides with the natural projection M_{1} ⟶ (f(M_{1}), f(F (M_{1})). Therefore, one can introduce the concept of strict epimorphism in ˜ \underline{L}_{0}^{∗}: f is strict epimorphism iff f(M_{1}) = M_{2} and f(F (M_{1})) = F (M_{2}).

The following situation will appear several times below.

Lemma 1.5. Suppose M_{1}, M_{2} ∈ ˜ \underline{L}_{0}^{∗}, ι ∈ Hom˜\underline{L}_{0}^{∗}(M_{1}, M_{2}) is a strict epimorphism and Ker˜\underline{L}_{0}^{∗}ι = (K, K, φ). Then for any L ∈ \underline{L}_{0}^{∗},

ι^{∗} : Hom˜\underline{L}_{0}^{∗}(L, M_{1}) ⟶ Hom˜\underline{L}_{0}^{∗}(L, M_{2})

is epimorphic. In addition, if φ|_{K} is nilpotent then ι^{∗} is bijective.

Proof. The structure of L = (L, F (L), φ) can be given by a vector l̄ = (l_{1}, …, l_{s}) and a matrix C ∈ M_{s}(W_{1}) such that

— l_{1}, …, l_{s} is a W_{1}-basis of L;

— if l̄C = m̄ = (m_{1}, …, m_{s}) then m_{1}, …, m_{s} is a W_{1}-basis of F (L);

— l̄= φ(m̄) := (φ(m_{1}), …, φ(m_{s})).

Suppose M_{1} = (M_{1}, F (M_{1}), φ) and M_{2} = (M_{2}, F (M_{2}), φ). Any f ∈ Hom˜\underline{L}_{0}^{∗}(L, M_{2}) is given by f(l̄) ∈ M_{2}^{s} such that f(l̄)C ∈ F (M_{2})^{s} and φ(f(l̄)C) = f(l̄). Choose an f̂(l̄) ∈ M_{1}^{s} such that f̂(l̄) mod K = f(l̄). Then f̂(l̄)C modulo K belongs to F (M_{2})^{s} and, therefore, f̂(l̄)C ∈ F (M_{1})^{s}. Clearly, we have that k̄_{0} := φ(f̂(l̄)C) − f̂(l̄) ∈ K^{s}. We must prove the existence of k̄_{1} ∈ K^{s} such that φ((f̂(l̄) + k̄_{1})C) = f̂(l̄) + k̄_{1}. This is equivalent to

k̄_{1} − φ(k̄_{1}C) = k̄_{0}

and the existence of k̄_{1} follows from Lemma 1.1. This proves that ι^{∗} is surjective. If φ|_{K} is nilpotent then the bijectivity of ι^{∗} follows in a similar way from part b) of Lemma 1.1. □

1.2. Standard exact sequences. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗}. Introduce a σ-linear map ϕ : L ⟶ L by ϕ : l ↦ φ(u^{p−1}l).

Definition. The object L is etale (resp., connected) if ϕ mod u is invertible (resp., nilpotent) on L/uL.

Let L(0) = (W_{1}, W_{1}u^{p−1}, φ, N) ∈ \underline{L}^{∗}, where φ(u^{p−1}) = 1 and N(1) = u^{p} mod u^{2p}. Then L(0) is etale. As a matter of fact, it is the simplest etale object of \underline{L}^{∗} due to the following Lemma.

Lemma 1.6. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗} is etale. Then L is a product of finitely many copies of L(0).

Proof. If L̃_{0} = {l ∈ L/uL | ϕ(l) = l} then L/uL = L̃_{0} ⊗_{𝔽_{p}} k. Then there is a unique 𝔽_{p}-submodule L_{0} of L such that ϕ|_{L_{0}} = id and L = L_{0} ⊗_{𝔽_{p}} W_{1}. Suppose l ∈ L_{0}. Then φ(u^{p−1}l) = l and N(l) = N(φ(u^{p−1}l)) = φ(uN(u^{p−1}l)) = φ(u^{p}(l mod u^{2p}) + u^{p}N(l)) = u^{p}l mod u^{2p}L. Therefore, if e_{1}, …, e_{s} is an 𝔽_{p}-basis of L_{0} then all (W_{1}e_{i}, W_{1}u^{p−1}e_{i}) determine the subobjects L_{i} ≃ L(0) of L and L ≃ L_{1} × ⋯ × L_{s}. □

Proposition 1.7. Any L ∈ \underline{L}^{∗} contains a unique maximal etale subobject (L^{et}, i^{et}) and a unique maximal connected quotient object (L^{c}, j^{c}) and the se-

quence 0 ⟶ L^{et} ⟶^{i^{et}} L ⟶^{j^{c}} L^{c} ⟶ 0 is short exact.

Proof. Let L = (L, F (L), φ, N) and, as earlier, let ϕ : L ⟶ L be such that for any l ∈ L, ϕ(l) = φ(u^{p−1}l). Then for L̃ = L/uL, we have the k-linear subspaces L̃^{et} and L̃^{c} in L̃ such that ϕ̃ := ϕ mod u is invertible on L̃^{et} and nilpotent on L̃^{c} and L̃ = L̃^{et} ⊕ L̃^{c}. Then there is a unique W_{1}-submodule L^{et} of L such that ϕ|_{L^{et}} is invertible and L^{et}/uL^{et} = L̃^{et}. The filtered submodule (L^{et}, u^{p−1}L^{et}) determines an etale subobject ι^{et} : L^{et} ⟶ L. Clearly, u^{p−1}L^{et} ⊂ L^{et} ∩ F (L). If the inverse embedding does not take place then there is an l ∈ L^{et} \uL^{et} such that u^{p−1}l ∈ uF (L). Therefore, ϕ(l) = φ(u^{p−1}l) ∈ u^{p}L but ϕ|_{L^{et}} is invertible. So, ι^{et} is strict monomorphism and we can consider Coker ι^{et} = j^{c} : L ⟶ L^{c}. Clearly, L^{c} is connected. The maximality properties of L^{et} and L^{c} are formally implied by the following easy statement:

if L_{1} ∈ \underline{L}^{∗} is etale and L_{2} ∈ \underline{L}^{∗} is connected then Hom\underline{L}^{∗}(L_{1}, L_{2}) = 0. □

Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗}. Then φ(F (L)) is a σ(W_{1})-module and L = φ(F (L)) ⊗_{σ(W_{1})} W_{1}. If l ∈ L and for 0 ⩽ i < p, l^{(i)} ∈ F (L) are such that l = ∑_{0⩽i<p} φ(l^{(i)}) ⊗ u^{i}, set V (l) = l^{(0)}. Then V mod u is a σ^{−1}-linear endomorphism of the k-vector space L/uL.

Definition. The module L is multiplicative (resp., unipotent) if Ṽ := V mod u is invertible (resp., nilpotent) on L̃ := L/uL.

Let L(1) = (W_{1}, W_{1}, φ, N) ∈ \underline{L}^{∗}, where φ(1) = 1 and N(1) = 0. Then L(1) is multiplicative. As a matter of fact, it is the simplest multiplicative object of \underline{L}^{∗} due to the following Lemma.

Lemma 1.8. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗} is multiplicative, then L is the product of finitely many copies of L(1).

Proof. Clearly, the embedding F (L) ⟶ L induces the identification F (L)/uF (L) = L/uL and, therefore, F (L) = L. Let L̃_{0} ⊂ L̃ be such that Ṽ|L̃_{0} = id. If l ∈ L is such that l mod uL ∈ L̃_{0} then φ(l) ≡ l mod uL. This implies the existence of a unique l′ ∈ L such that l′ ≡ l mod uL and φ(l′) = l′. In other words, there is an 𝔽_{p}-submodule L_{0} in L such that L = L_{0} ⊗_{𝔽_{p}} W_{1} and φ|_{L_{0}} = id. If l ∈ L_{0} then N(l) = N(φ(l)) = φ(uN(l)) = u^{p}φ(N(l)) = 0. So, if e_{1}, …, e_{s} is an 𝔽_{p}-basis of L_{0} then the filtered modules (W_{i}e_{i}, W_{1}e_{i}) determine the subobjects L_{i} ≃ L(1) of L and L ≃ L_{1} × ⋯ × L_{s}. □

Proposition 1.9. Any L = (L, M, φ, N) ∈ \underline{L}^{∗} contains a unique maximal multiplicative quotient object (L^{m}, j^{m}) and a unique maximal unipotent subobject (L^{u}, i^{u}) and the sequence

0 ⟶ L^{u} ⟶^{i^{u}} L ⟶^{j^{m}} L^{m} ⟶ 0

is exact.

Proof. Let L̃ = L/uL, M̃ = M/uM and L̃ = L̃^{m} ⊕ L̃^{u}, where Ṽ := V mod u is invertible on L̃^{m} and nilpotent on L̃^{u}. Note that φ induces a σ-linear isomorphism φ̃ : M̃ ⟶ L̃. Denote by ι̃ : M̃ ⟶ L̃ the k-linear morphism induced by the embedding M ⊂ L. With this notation, for any l ∈ L̃, Ṽ(l) = ι̃(φ̃^{−1}(l)). Consider the filtration L̃^{u} ⊃ ṼL̃^{u} ⊃ ⋯ ⊃ Ṽ^{s}L̃^{u} = {0} and set for 1 ⩽ i ⩽ s+1, M̃_{i} = φ̃^{−1}(Ṽ^{i−1}L̃^{u}). Then M̃_{1} ⊃ M̃_{2} ⊃ ⋯ ⊃ M̃_{s} ⊃ M̃_{s+1} = {0} and for 1 ⩽ i ⩽ s,

(1.1)  ι̃(M̃_{i}) = Ṽ^{i}L̃^{u} = φ̃(M̃_{i+1}).

For 1 ⩽ i ⩽ s + 1, introduce the W_{1}-submodules M_{i}^{(0)} of M such that M_{1}^{(0)} ⊃ M_{2}^{(0)} ⊃ ⋯ ⊃ M_{s}^{(0)} ⊃ M_{s+1}^{(0)} = 0 and M_{i}^{(0)}/uM_{i}^{(0)} = M̃_{i} with respect to the natural projection M ⟶ M̃. Then conditions (1.1) imply that for all i, M_{i}^{(0)} ⊂ φ(M_{i+1}^{(0)}) ⊗_{σW_{1}} W_{1} + uL. Let M̃^{m} = φ̃^{−1}(L̃^{m}) and let M^{m} ⊂ M be a W_{1}-submodule such that M^{m}/uM^{m} = M̃^{m} with respect to the natural projection M ⟶ M̃. Then

(1.2)  M^{m} + uL = φ(M^{m}) ⊗_{σW_{1}} W_{1} + uL

and M = M^{m} ⊕ M_{1}^{(0)}. Prove the existence of “more precise” lifts M_{i}^{(n)} of M̃_{i}, where 0 ⩽ i ⩽ s + 1 and n ⩾ 1.

Lemma 1.10. For all n ⩾ 1 and 0 ⩽ i ⩽ s + 1, there are W_{1}-modules M_{i}^{(n)} such that

a) M_{1}^{(n)} ⊃ M_{2}^{(n)} ⊃ ⋯ ⊃ M_{s}^{(n)} ⊃ M_{s+1}^{(n)} = {0} and M_{i}^{(n)}/uM_{i}^{(n)} = M̃_{i} with respect to the natural projection M ⟶ M̃;

b) M_{i}^{(n)} ⊂ φ(M_{i+1}^{(n)}) ⊗_{σW_{1}} W_{1} + uφ(M_{1}^{(n)}) ⊗_{σW_{1}} W_{1} + u^{n+1}L;

c) M_{i}^{(n−1)} + u^{n}M = M_{i}^{(n)} + u^{n}M.

Proof of Lemma. The modules M_{i}^{(0)}, 0 ⩽ i ⩽ s + 1, satisfy the requirements a) and b) of our Lemma. Therefore, we can assume that the modules M_{i}^{(n)} satisfying the requirements a)-c) have been already constructed for n = N − 1, where N ⩾ 1. Note that M = M^{m} ⊕ M_{1}^{(N−1)} (it is known for N = 1 and follows from c) for N > 1). Therefore, (1.2) implies that

L = φ(M^{m}) ⊗_{σW_{1}} W_{1} + φ(M_{1}^{(N−1)}) ⊗_{σW_{1}} W_{1}
⊂ M^{m} + φ(M_{1}^{(N−1)}) ⊗_{σW_{1}} W_{1} + uL.

Therefore, for 1 ⩽ i ⩽ s (use b) for n = N − 1),

M_{i}^{(N−1)} ⊂ φ(M_{i+1}^{(N−1)}) ⊗_{σW_{1}} W_{1} + uφ(M_{1}^{(N−1)}) ⊗_{σW_{1}} W_{1} + u^{N}M^{m} + u^{N+1}L

and we can define the submodules M_{i}^{(N)} in such a way that the property c) holds for n = N

(1.3)  M_{i}^{(N)} + u^{N}M^{m} = M_{i}^{(N−1)} + u^{N}M^{m}

and

(1.4) M_{i}^{(N)} ⊂ φ(M_{i+1}^{(N−1)}) ⊗_{σW_{1}} W_{1} + uφ(M_{1}^{(N−1)}) ⊗_{σW_{1}} W_{1} + u^{N+1}L.

Note that (1.3) implies that φ(M_{i}^{(N)}) + u^{Np}L = φ(M_{i}^{(N−1)}) + u^{Np}L and, therefore, we can replace φ(M_{i}^{(N−1)}) and φ(M_{1}^{(N−1)}) by φ(M_{i}^{(N)}) and, resp. φ(M_{1}^{(N)}) in (1.4). The lemma is proved. □

Let M^{u} = ⋂_{n⩾0}(M_{1}^{(n)} + u^{n+1}M). Then M^{u}/uM^{u} = M̃^{u} with respect to the natural projection M ⟶ M̃ and M = M^{m} ⊕ M^{u}. Let L^{u} = φ(M^{u}) ⊗_{σW_{1}} W_{1}. Then rk_{W_{1}} L^{u} = rk_{W_{1}} M^{u} and

L^{u} = ⋂_{n⩾0} (φ(M_{1}^{(n)}) ⊗_{σW_{1}} W_{1} + u^{(n+1)p}L) ⊃ M^{u}

(use Lemma 1.10b)). On the other hand,

L = φ(M^{m} ⊕ M^{u}) ⊗_{σW_{1}} W_{1} = M^{m} ⊕ L^{u}

implies that M^{u} ⊃ u^{p−1}L^{u} and L^{u} ∩ M = M^{u}. Therefore, the filtered module (L^{u}, M^{u}) defines a unipotent subobject L^{u} of L in the category \underline{L}_{0}^{∗} and the natural embedding L^{u} ⟶ L is strict. Suppose l ∈ M^{u} and N(l) ≡ l_{0} + l_{1} mod u^{p}L, where l_{0} ∈ M^{m} and l_{1} ∈ L^{u}. Then uN(l) ≡ ul_{0} + ul_{1} ∈ (M^{m} ⊕ M^{u}) mod u^{p}L and N(φ(l)) = φ(uN(l)) ≡ φ(ul_{1}) mod u^{p}L implies that N(L^{u}) ⊂ L^{u} mod u^{p}L. Then from Proposition 1.2 it follows that L^{u} is a subobject of L in the category \underline{L}^{∗}. Clearly, the quotient L/L^{u} := L^{m} is multiplicative. The maximality of L^{u} and L^{m} are formally implied by the following easy property of objects L_{1}, L_{2} ∈ \underline{L}^{∗}: if L_{1} is unipotent and L_{2} is multiplicative then Hom\underline{L}^{∗}(L_{1}, L_{2}) = 0. □

Using the above results we can introduce the subcategories \underline{L}^{∗et}, \underline{L}^{∗c}, \underline{L}^{∗m}, \underline{L}^{∗u} in \underline{L}^{∗}. They consist of, resp., etale, connected, multiplicative and unipotent objects of the cattegory \underline{L}^{∗}. The correspondences L ↦ L^{et}, L ↦ L^{c}, L ↦ L^{m}, L ↦ L^{u} determine the natural exact functors from \underline{L}^{∗} to, resp., \underline{L}^{∗et}, \underline{L}^{∗c}, \underline{L}^{∗m} and \underline{L}^{∗u}.

1.3. The category \underline{L}_{cr}^{∗}.

Proposition 1.11. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗}. Then the following conditions are equivalent:

(a) N(F (L)) ⊂ F (L) mod u^{2p}L;

(b) N(φ(F (L))) ⊂ u^{p}L mod u^{2p}L.

Proof. (a) ⇒ (b): if for any l ∈ F (L), N(l) ∈ F (L) mod u^{2p}L then N(φ(l)) = φ(uN(l)) = u^{p}φ(N(l)) ∈ u^{p}L mod u^{2p}L.

(b) ⇒ (a): for any l ∈ F (L), φ(uN(l)) = N(φ(l)) ∈ u^{p}L mod u^{2p}L; now use that φ induces embedding of F (L)/uF (L) into L/u^{p}L to deduce that uN(l) ∈ uF (L) mod u^{2p}L, i.e. N(l) ∈ F (L) mod u^{2p}L (use that u^{p−1}L ⊂ F (L)). □

Definition. The category \underline{L}_{cr}^{∗} is a full subcategory of \underline{L}^{∗} consisting of (L, F (L), φ, N) such that N : L ⟶ L satisfies the equivalent conditions from Proposition 1.11.

Remark. a) If L = (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗} then N_{1} = N mod u^{p} is a unique W_{1}-differentiation N_{1} : L ⟶ L/u^{p} whose restriction to φ(F (L)) is the zero map. Therefore, any L ∈ \underline{L}_{0}^{∗} has at most one structure of object of the category \underline{L}^{∗}.

b) Any etale or multiplicative object from \underline{L}^{∗} belongs to \underline{L}_{cr}^{∗}.

c) If f is a morphism in \underline{L}_{cr}^{∗} then Ker\underline{L}^{∗}f = Ker\underline{L}_{cr}^{∗}f and Coker\underline{L}^{∗}f = Coker\underline{L}_{cr}^{∗}f. In particular, we can introduce the full subcategories \underline{L}_{cr}^{∗et}, \underline{L}_{cr}^{∗c}, \underline{L}_{cr}^{∗m}, \underline{L}_{cr}^{∗u} of, resp., etale, connected, multiplicative and unipotent objects of \underline{L}_{cr}^{∗}.

Proposition 1.12. Suppose L = (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗}. Then there is a σ(W_{1})-basis l_{1}, …, l_{s} of φ(F (L)) and integers 0 ⩽ c_{i} < p, where 1 ⩽ i ⩽ s, such that u^{c_{1}}l_{1}, …, u^{c_{s}}l_{s} is a W_{1}-basis of F (L).

Proof. Choose a W_{1}-basis m_{1}, …, m_{s} of L such that for suitable integers c_{1}, …, c_{s}, the elements u^{c_{1}}m_{1}, …, u^{c_{s}}m_{s} form a W_{1}-basis of F (L). Clearly all 0 ⩽ c_{i} < p. For 1 ⩽ i ⩽ s and j ⩾ 0, let l_{ij} ∈ φ(F (L)) be such that m_{i} = ∑_{j⩾0} u^{j}l_{ij}. Note that {l_{i0} | 1 ⩽ i ⩽ s} is a σ(W_{1})-basis of φ(F (L)) and it will be sufficient to prove that all u^{c_{i}}l_{i0} ∈ F (L) because then the elements l_{i} := l_{i0} will satisfy the requirements of our proposition. For all 1 ⩽ i ⩽ s, the element

N(u^{c_{i}}m_{i}) = − ∑_{j}(j + c_{i})u^{j+c_{i}}(l_{ij} mod u^{2p}L) + ∑_{j} u^{j+c_{i}}N(l_{ij})

belongs to F (L) mod u^{2p}L if and only if ∑_{j}(j + c_{i})u^{j+c_{i}}l_{ij} ∈ F (L). (Use that u^{p}L ⊂ uF (L).) This implies that for all integers k ⩾ 0, ∑_{j}(j + c_{i})^{k}u^{j+c_{i}}l_{ij} ∈ F (L). Therefore, for any α ∈ ℤ/pℤ,

∑_{(j+c_{i})modp=α} u^{j+c_{i}}l_{ij} ∈ F (L).

In particular, taking α = c_{i}mod p and using that u^{p}l_{ij} ∈ F (L), we obtain that u^{c_{i}}l_{i0} ∈ F (L). □

Remark. a) Suppose L = (L, F (L), φ) ∈ \underline{L}_{0}^{∗} and satisfies the conclusion of Proposition 1.12. Define the W_{1}-differentiation N_{1} : L ⟶ L/u^{p}L by setting N_{1}(l_{1}) = ⋯ = N(l_{s}) = 0. If N : L ⟶ L/u^{2p}L is the extension of N_{1} given by Propostion 1.2 then (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗}. In other words, Proposition 1.12 characterizes the objects of \underline{L}_{0}^{∗} coming from \underline{L}_{cr}^{∗}.

b) For an object (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗}, Proposition 1.12 implies that if ∑_{0⩽i<p} u^{i}l_{i} ∈ F (L), where all l_{i} ∈ φ(F (L)), then all u^{i}l_{i} ∈ F (L).

Consider the category of filtered Fontaine-Laffaille modules \underline{MF}_{p−1} from [13]. The objects of this category are finite dimensional k-vector spaces M with decreasing filtration of length p by subspaces M = M^{0} ⊃ M^{1} ⊃ ⋯ ⊃ M^{p−1} ⊃ M^{p} = 0 and σ-linear maps φ_{i} : M^{i} ⟶ M such that Ker φ_{i} ⊃ M^{i+1}, where 0 ⩽ i < p, and ∑_{i} Im φ_{i} = M. The morphisms in \underline{MF}_{p−1} are the morphisms of filtered vector spaces which commute with the corresponding morphisms φ_{i}, 0 ⩽ i < p. The category \underline{MF}_{p−1} is abelian. The object M of \underline{MF}_{p−1} is:

— etale (resp., multiplicative) if M^{1} = 0 (resp., M = M^{p−1});

— connected (resp., unipotent) if M has no etale (resp., multiplicative) subquotient. Introduce the full subcategories \underline{MF}_{p−1}^{et}, \underline{MF}_{p−1}^{m}, \underline{MF}_{p−1}^{c} and \underline{MF}_{p−1}^{u} of, resp., etale, multiplicative, connected and unippotent objects in \underline{MF}_{p−1}. These subcategories are closed under the operations of taking subobjects and quotient objects and, therefore, are also abelian. For any M ∈ \underline{MF}_{p−1}, there are standard exact sequences 0 ⟶ M^{et} ⟶ M ⟶ M^{c} ⟶ 0 and 0 ⟶ M^{u} ⟶ M ⟶ M^{m} ⟶ 0, where M^{et} (resp., M^{u}) is the maximal etale (resp., unipotent) subobject and M^{c} (resp., M^{m}) is the maximal connected (resp., multiplicative) quotient object. The categories \underline{L}_{cr}^{∗} and \underline{MF}_{p−1} do not differ very much. Indeed, introduce the functor Md : ˜ \underline{L}^{∗} ⟶ ˜ \underline{L}^{∗} induced on the level of filtered modules by (L, F (L)) ↦ (L/u^{p}L, F (L)/u^{p}L). Denote by Md(\underline{L}_{cr}^{∗}) the full subcategory of ˜ \underline{L}^{∗} consisting of the objects Md(L), where L ∈ \underline{L}_{cr}^{∗}. Define the functor F : \underline{MF}_{p−1} ⟶ ˜ \underline{L}^{∗} as follows. Let M ∈ \underline{MF}_{p−1} with the corresponding filtration M^{i} and σ-linear morphisms φ_{i}, 0 ⩽ i < p. Then on the level of objects, F(M) = (L, F (L), φ, N), where L = M ⊗_{k} W_{1}/u^{p}W_{1}, F (L) = ∑_{0⩽i<p} u^{p−1−i}W_{1}(M^{i} ⊗ 1) and for any m ∈ M_{i}, φ(u^{p−1−i}m_{i}) = φ_{i}(m_{i}). One can easily see that F is equivalence of the categories \underline{MF}_{p−1} and Md(\underline{L}_{cr}^{∗}). Now the difference between the categories \underline{L}_{cr}^{∗} and \underline{MF}_{p−1} is described by the following Proposition.

Proposition 1.13. For L_{1}, L_{2} ∈ \underline{L}_{cr}^{∗}, Md induces a surjection from Hom\underline{L}_{cr}^{∗}(L_{1}, L_{2}) to Hom˜\underline{L}^{∗}(Md(L_{1}), Md(L_{2})) and its kernel coincides with (i_{L_{2}∗} ∘ j_{L_{1}}^{∗})Hom\underline{L}^{∗}(L_{1}^{m}, L_{2}^{et}), where i_{L_{2}} : L_{2}^{et} ⟶ L_{2} (resp., j_{L_{1}} : L_{1} ⟶ L_{1}^{m}) is the maximal etale subobject in L_{2} (resp., multiplicative quotient object for L_{1}).

Proof. For L_{2} = (L_{2}, F (L_{2}), φ, N), let ϕ : L_{2} ⟶ L_{2} be such that ϕ(l) =

φ(u^{p−1}l) for any l ∈ L_{2}. Let L_{2}^{c} = \{l ∈ L_{2} | ϕ(l)^{n} ⟶_{n→∞} 0\} and let L′_{2} ∈ ˜ \underline{L}^{∗}

be the filtered module (L_{2}/u^{p}L^{c}, F (L_{2})/u^{p}L_{2}^{c}) with φ and N induced from L_{2}. Then there are natural strict epimorphisms

L_{2} ⟶^{α} L′_{2} ⟶^{β} Md(L_{2}),

where Ker α is associated with the filtered module (u^{p}L_{2}^{c}, u^{p}L_{2}^{c}) and Ker β — with (u^{p}L_{2}/u^{p}L_{2}^{c}, u^{p}L_{2}/u^{p}L_{2}^{c}). Clearly, φ|_{u^{p}L_{2}^{c}} is nilpotent and then by Lemma 1.5,

α_{∗} : Hom\underline{L}^{∗}(L_{1}, L_{2}) ⟶ Hom˜\underline{L}^{∗}(L_{1}, L′_{2})

is bijective. Note that the natural embedding L_{2}^{et} ⟶ L_{2} induces the identification u^{p}L_{2}/u^{p}L_{2}^{c} = u^{p}L_{2}^{et}/u^{p+1}L_{2}^{et}. Let L′′_{2} = (u^{p}L_{2}, u^{p}L_{2}) ∈ \underline{L}^{∗} with induced φ and N. Then L′′_{2} is multiplicative and there is a natural projection γ : L′′_{2} ⟶ Ker β such that Ker γ is associated with (u^{p+1}L_{2}^{et}, u^{p+1}L_{2}^{et}). Note that φ is nilpotent on u^{p+1}L_{2}^{et}. Applying Lemma 1.5 we obtain that

β_{∗} : Hom˜\underline{L}^{∗}(L_{1}, L′_{2}) ⟶ Hom˜\underline{L}^{∗}(L_{1}, Md(L_{2}))

is surjective and

Ker β_{∗} = Hom˜\underline{L}^{∗}(L_{1}, Kerβ) = Hom˜\underline{L}^{∗}(L_{1}, L′′) ≃ Hom(L_{1}^{m}, L′′_{2}).

It remains to note that Hom˜\underline{L}^{∗}(L_{1}^{m}, L′′_{2}) = Hom\underline{L}^{∗}(L_{1}^{m}, L_{2}^{et}) via the natural embedding of L′′ into L_{2}^{et}. □

Corollary 1.14. The functor Md ∘ F^{−1} induces equivalence of the categories \underline{L}_{cr}^{∗c} (resp., \underline{L}_{cr}^{∗u}) and \underline{MF}_{p−1}^{c} (resp., \underline{MF}_{p−1}^{u}).

1.4. Simple objects in \underline{L}^{∗}.

Definition. An object L of \underline{L}^{∗} is simple if any strict monomorphism i : L_{1} ⟶ L in \underline{L}^{∗} is either isomorphism or the zero morphism. Equivalently, L is simple iff any strict epimorphism j : L ⟶ L_{2} is either isomorphism or the zero morphism.

All simple objects in \underline{L}^{∗} can be described as follows. Let [0, 1]_{p} = {r ∈ ℚ | 0 ⩽ r ⩽ 1, v_{p}(r) = 0}, where v_{p} is a p-adic valuation. Then any r ∈ [0, 1]_{p} can be uniquely written as r = ∑_{i⩾1} a_{i}p^{−i}, where the digits 0 ⩽ a_{i} = a_{i}(r) < p form a periodic sequence. The minimal positive period of this sequence will be denoted by s(r). Let r̃ = 1 − r. Then r̃ ∈ [0, 1]_{p} and r̃ = ∑_{i⩾1} ã_{i}p^{−i}, where for all i ⩾ 1, the digits ã_{i} = a_{i}(r̃) are such that a_{i} + ã_{i} = p − 1.

Definition. For r ∈ [0, 1]_{p}, let L(r) = (L(r), F (L(r)), φ, N) be the following object of the category \underline{L}_{cr}^{∗}:

• L(r) = ⊕_{i∈ℤ/s(r)}W_{1}l_{i};

• F (L(r)) = ∑_{i∈ℤ/s(r)} W_{1}uã_{i}l_{i};

• for i ∈ ℤ/s(r), φ(uã_{i}l_{i}) = l_{i+1}.

• N is uniquely recovered from the condition N|_{φ(F (L))} = 0 mod u^{p}, cf. Proposition 1.2.

Remark. If r = 0 or r = 1 we obtain the objects L(0) and L(1) introduced in Subsection 1.2. Note also that L(r) is connected iff r ≠ 0 and unipotent iff r ≠ 1.

For n ∈ ℕ and r ∈ [0, 1]_{p}, set r(n) = ∑_{i⩾1} a_{i+n}(r)p^{−i}. Extend this definition to any n ∈ ℤ by setting r(n) := r(n + Ns(r)) for a sufficiently large N ∈ ℕ.

Proposition 1.15. a) If r ∈ [0, 1]_{p} then L(r) is simple;

b) if r_{1}, r_{2} ∈ [0, 1]_{p} then L(r_{1}) ≃ L(r_{2}) if and only if there is an n ∈ ℤ such that r_{1} = r_{2}(n);

c) if L is a simple object of the category \underline{L}^{∗} then there is an r ∈ [0, 1]_{p} such that L ≃ L(r).

Proof. Lemma 1.16 below implies that the simple objects in the categories \underline{L}_{cr}^{∗} and \underline{L}^{∗} are the same. By Corollary 1.14, the functor Md ∘ F^{−1} transforms simple objects of \underline{L}^{∗} to simple objects in \underline{MF}_{p−1}. It remains to note that an analogue of our Proposition for the category \underline{MF}_{p−1} is proved in [13]. □

Lemma 1.16. For any L ∈ \underline{L}^{∗}, there is an L^{cr} ∈ \underline{L}_{cr}^{∗} and a strict monomorphism ι^{cr} ∈ Hom\underline{L}^{∗}(L^{cr}, L) such that if ι′ ∈ Hom\underline{L}^{∗}(L′, L) is a strict monomorphism and L′ ∈ \underline{L}_{cr}^{∗} then there is a strict monomorphism α : L′ ⟶ L^{cr} such that ι′ = ι^{cr} ∘ α.

Proof of Lemma. Suppose L = (L, F (L), φ, N). Consider the k-linear space M := φ(F (L))/u^{p}φ(F (L)). Let L̃ = M ⊗_{k} (W_{1}/u^{p}W_{1}) = L/u^{p}L, F̃ = F (L)/u^{p}L and φ̃ : F̃ ⟶ M be the map induced by φ. Proceed by induction to define for all i ⩾ 1, the subspaces M_{i} ⊂ M and the W_{1}-submodules F̃_{i} ⊂ L̃ as follows. From the definition of N : L ⟶ L/u^{2p}L it follows easily that N induces a k-linear map Ñ_{1} : M ⟶ M and Ñ_{1}^{p} = 0. Therefore, M_{1} := Ker Ñ_{1} is a non-trivial subspace in M. Suppose i ⩾ 1 and M_{i} has been already defined. Let F̃_{i} be the submodule of the elements of the form u^{a}l in M ⊗_{k} (W_{1}/u^{p}W_{1}), where a ⩾ 0, l ∈ M_{i} and u^{a}l ∈ F̃. Then set M_{i+1} = φ̃(F̃_{i}). Verify that for all indices i, M_{i+1} ⊂ M_{i}. If i = 1 we must prove that Ñ_{1}(M_{2}) = 0. Indeed, M_{2} is spanned by φ(u^{a}l), where l ∈ M_{1} and u^{a}l ∈ F̃_{1}. But N(φ(u^{a}l)) = φ(uN(u^{a}l)) = φ(−u^{a+1}l + u^{a+1}N(l)) ∈ u^{p}L. If i > 1 then we can assume by induction that M_{i−1} ⊂ M_{i}. This implies that F̃_{i−1} ⊂ F̃_{i} and M_{i} ⊂ M_{i+1}.

We obtained a decreasing sequence of non-trivial finite dimensional k-linear spaces {M_{i} | i ⩾ 1}. For i ≫ 1, these spaces become a non-trivial constant space M^{cr} ⊂ M such that if F̃^{cr} = {u^{a}l ∈ F̃ | a ⩾ 0, l ∈ M^{cr}} then φ̃(F̃^{cr}) = M^{cr}. This subspace M^{cr} has the maximality property: if M′ ⊂ M is such that for F̃′ = {u^{a}l ∈ F̃ | a ⩾ 0, l ∈ M′}, φ̃(F̃′) = M′ then M′ ⊂ M^{cr}. Indeed, show as earlier that M′ ⊂ M_{1} and then proceed by induction proving that M′ ⊂ M_{i} for all i ⩾ 1. Now in notation from Subsection 1.3, there is an L^{cr} ∈ \underline{L}_{cr}^{∗} such that Md(L^{cr}) = (M^{cr} ⊗_{k} (W_{1}/u^{p}W_{1}), F̃^{cr}, φ̃, Ñ), where Ñ|_{M^{cr}} = 0. Then from Proposition 1.13 it follows the existence of a strict monomorphism ι^{cr} : L^{cr} ⟶ L. If ι′ : L′ = (L′, F (L′), φ, N) ⟶ L is strict monomorphism and L′ ∈ \underline{L}_{cr}^{∗} then Md(L′) is associated with the filtered module (M′ ⊗_{k} (W_{1}/u^{p}W_{1}), F̃′) and by the above maximality property of M^{cr}, M′ is a subspace in M^{cr} and Md(L′) is a strict subobject of Md(L^{cr}). This gives the required strict embedding α. The Lemma is proved. □

1.5. Extensions in \underline{L}^{∗}. Suppose r_{1}, r_{2} ∈ [0, 1]_{p}. Choose an s ∈ ℕ which is divisible by s(r_{1}) and s(r_{2}) and introduce the objects L_{1} = (L_{1}, F (L_{1}), φ, N) and L_{2} = (L_{2}, F (L_{2}), φ, N) of the category \underline{L}_{cr}^{∗} as follows: L_{1} = ⊕_{i∈ℤ/s}W_{1}l_{i}^{(1)}, F (L_{1}) = ∑_{i∈ℤ/s} W_{1}uã_{i}l_{i}^{(1)}, where r_{1} = ∑_{i⩾1} a_{i}p^{−i} with

the digits 0 ⩽ a_{i} < p, ã_{i} = (p − 1) − a_{i} and for all i ∈ ℤ/s, φ(uã_{i}l_{i}^{(1)}) = l_{i+1}^{(1)}; L_{2} = ⊕_{j∈ℤ/s}W_{1}l_{j}^{(2)}, F (L_{2}) = ∑_{j∈ℤ/s} W_{1}ub̃_{j}l_{j}^{(2)}, where r_{2} = ∑_{j⩾1} b_{j}p^{−j} with

the digits 0 ⩽ b_{j} < p, b̃_{j} = (p − 1) − b_{j}, and for all j ∈ ℤ/s, φ(ub̃_{j}l_{j}^{(2)}) = l_{j+1}^{(2)}.

Lemma 1.17. For κ = 1, 2, L_{κ} is isomorphic to the product of s/s(r_{κ}) copies of the (simple) object L(r_{κ}).

Proof. Take κ = 1. For γ ∈ 𝔽_{p^{s}} and ī ∈ ℤ/s(r_{1}), let mī(γ) = ∑_{i mod s(r_{1})=ī} σ^{i}(γ)l_{i}^{(1)} and M(γ) = ∑_{ī∈ℤ/s(r_{1})} W_{1}mī(γ) ⊂ L_{1}. Then all M(γ) = (M(γ), M(γ) ∩ F (L_{1}), φ, N) with induced φ and N are subobjects of L_{1} isomorphic to L(r_{1}). If γ_{1}, …, γ_{d} is an 𝔽_{p^{s(r_{1})}}-basis of 𝔽_{p^{s}} then M(γ_{1}) × ⋯ × M(γ_{d}) is isomorphic to L_{1}. (Use that d = s/s(r_{1}) and det(σ^{i}(γ_{j})) ≠ 0, where for a given ī, i is such that i mod s(r_{1}) = ī and 1 ⩽ j ⩽ d.) □

If L = (L, F (L), φ, N) ∈ \underline{L}^{∗} then we shall use the same notation L for the image (L, F (L), φ) of L under the forgetful functor from \underline{L}^{∗} to \underline{L}_{0}^{∗}. Clearly, this forgetful functor induces a group homomorphism Ext\underline{L}^{∗}(L_{2}, L_{1}) ⟶ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}). Suppose L = (L, F (L), φ) ∈ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}). Consider a σ(W_{1})-linear section S : l_{j}^{(2)} ↦ l_{j}, j ∈ ℤ/s, of the corresponding epimorphic map φ(F (L)) ⟶ φ(F (L_{2})). Then

a) L = L_{1} ⊕ (⊕_{j∈ℤ/s}W_{1}l_{j});

b) for all indices j ∈ ℤ/s, there are unique elements v_{j} ∈ L_{1}, such that F (L) = F (L_{1}) + ∑_{j∈ℤ/s} W_{1}(ub̃_{j}l_{j} + v_{j}) and φ(ub̃_{j}l_{j} + v_{j}) = l_{j+1}; c) F (L) ⊃ u^{p−1}L if and only if for all j ∈ ℤ/s, u^{b_{j}}v_{j} ∈ F (L_{1});

d) if S′ : l_{j}^{(2)} ↦ l′_{j} = l_{j} + φ(w_{j−1}), where j ∈ ℤ/s and w_{j−1} ∈ F (L_{1}), is another section of the epimorphism φ(F (L)) ⟶ φ(F (L_{2})) then for the corresponding elements v′_{j} ∈ L_{1}, v′_{j} − v_{j} = w_{j} − ub̃_{j}φ(w_{j−1}).

The constructions from above items a)-d) can be summarized as follows.

Lemma 1.18. Let Z(L_{2}, L_{1}) = {(v_{j})_{j∈ℤ/s} ∈ L_{1}^{s} | u^{b_{j}}v_{j} ∈ F (L_{1})} be a subgroup in L_{1}^{s} and let

B(L_{2}, L_{1}) = {(w_{j} − ub̃_{j}φ(w_{j−1}))_{j∈ℤ/s} | w_{j} ∈ F (L_{1})}

be a subgroup of Z(L_{2}, L_{1}). Then there is a natural isomorphism of abelian groups Z(L_{2}, L_{1})/B(L_{2}, L_{1}) ≃ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}).

Proposition 1.19. Any L ∈ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}) appears from a system of factors (v_{j})_{j∈ℤ/s} ∈ Z(L_{2}, L_{1}) satisfying the following normalization condition (C1) if v_{j} = ∑_{i,t} γ_{ijt}u^{t}l_{i}^{(1)} with γ_{ijt} ∈ k, then γ_{ijb̃_{j}} = 0.

Proof. Choose a section S of the projection φ(F (L)) ⟶ φ(F (L_{2})) with the minimal set γ(S) = {(i, j, b̃_{j}) | γ_{ijb̃_{j}} ≠ 0}. Suppose γ(S) ≠ ∅ (otherwise, the proposition is proved) and let (v_{j})_{j∈ℤ/s} be the corresponding system of factors. Suppose (i_{0}, j_{0}, b̃_{j_{0}}) ∈ γ(S) and γ = γ_{i_{0}j_{0}b̃_{j_{0}}} . Replace (v_{j})_{j∈ℤ/s} by an equivalent system (v′_{j})_{j∈ℤ/s} via the elements w_{j} ∈ F (L_{1}) such that w_{j} = 0 if j ≠ j_{0} − 1 and w_{j_{0}−1} = σ^{−1}(γ)uã_{i_{0}−1}l_{i_{0}−1}^{(1)}. If v′_{j} = ∑_{i,t} γ′_{ijt}u^{t}l_{i}^{(1)} then

— γ′_{i_{0}j_{0}b̃_{j_{0}}} = 0;

— γ′_{i_{0}−1,j_{0}−1,ã_{i_{0}−1}} = σ^{−1}(γ) + γ_{i_{0}−1,j_{0}−1,ã_{i_{0}−1}};

— for all remaining indices γ′_{ijt} = γ_{ijt}.

Then γ(S′) ⊂ γ(S) \ {(i_{0}, j_{0}, b̃_{j_{0}})} ∪ {(i_{0} − 1, j_{0} − 1, ã_{i_{0}−1})} and the minimality condition for S implies (i_{0} − 1, j_{0} − 1, ã_{i_{0}−1}) ∈ γ(S′) \ γ(S). Therefore, ã_{i_{0}−1} = b̃_{j_{0}−1}, γ_{i_{0}−1,j_{0}−1,ã_{i_{0}−1}} = 0, γ′_{i_{0}−1,j_{0}−1,b̃_{j_{0}−1}} = σ^{−1}(γ), and the new section S′ again satisfies the minimality condition. Repeating the above procedure we obtain for all n ∈ ℤ/s, that ã_{i_{0}−n} = b̃_{j_{0}−n}, that is r̃_{1}(i_{0}) = r̃_{2}(j_{0}). Choose β ∈ k such that σ^{s}(β) − β = γ and consider w_{j} ∈ F (L_{1}) such that for all 0 ⩽ n < s, w_{j_{0}+n} = σ^{n}(β)ub̃_{j_{0}+n}l_{i_{0}+n}^{(1)}. Then for the corresponding new system of factors (v′_{j})_{j∈ℤ/s}, where

v′_{j} = v_{j} + w_{j} − ub̃_{j}φ(w_{j−1}) = ∑_{i,t} γ′_{ijt}u^{t}l_{i}^{(1)},

one has γ′_{i_{0},j_{0},b̃_{j_{0}}} = 0, and γ_{ijt} = γ′_{ijt} if (i, j, t) ≠ (i_{0}, j_{0}, b̃_{j_{0}}). This contradicts to the minimality condition for S. □

Proposition 1.20. Any L ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) can be described via a system of factors (v_{j})_{j∈ℤ/s}, satisfying the above condition (C1) and the normalization condition

(C2) the coefficients γ_{ijt} = 0 if t > ã_{i}.

Proof. Suppose v^{(0)} = (v_{j})_{j∈ℤ/s} is such that v_{j_{0}} = γu^{t_{0}}l_{i_{0}}^{(1)} with γ ∈ k, t_{0} > ã_{i_{0}} and for j ≠ j_{0}, v_{j} = 0. It will be sufficient to prove that any such system of factors is trivial. Take the elements w_{j}^{(0)}, j ∈ ℤ/s, such that w_{j_{0}}^{(0)} = −γu^{t_{0}}l_{i_{0}}^{(1)} and w_{j}^{(0)} = 0 if j ≠ j_{0}. Then the corresponding equivalent system (v_{j}^{(1)})_{j∈ℤ/s} is such that v_{j}^{(1)} = 0 if j ≠ j_{0} + 1, and v_{j_{0}+1}^{(1)} = γ^{p}u^{t_{1}}l_{i_{0}+1}^{(1)}, where t_{1} = b̃_{j_{0}+1} + (t_{0} − ã_{i_{0}})p. This implies that t_{1} ⩾ p > ã_{i_{0}+1}, t_{1} − ã_{i_{0}+1} ⩾ t_{0} − ã_{i_{0}}, and t_{1} − ã_{i_{0}+1} > t_{0} − ã_{i_{0}} unless b̃_{j_{0}+1} = 0, t_{1} = p and ã_{i_{0}+1} = p − 1. Repeat this procedure by using for all n ⩾ 0, the appropriate elements w_{j}^{(n)}, j ∈ ℤ/s, to obtain the equivalent systems of factors (v_{j}^{(n)})_{j∈ℤ/s} such that v_{j}^{(n)} = 0 if j ≠ j_{0} + n, and v_{j_{0}+n}^{(n)} = γ^{p^{n}}u^{t_{n}}l_{i_{0}+n}^{(1)}. If (r̃_{2}, r̃_{1}, t_{0}) ≠ (0, 1, p) then t_{n} → ∞ and we can use the elements w_{j} = ∑_{n⩾0} w_{j}^{(n)}, j ∈ ℤ/s, to trivialize the original system of factors v^{(0)}. If (r̃_{2}, r̃_{1}, t_{0}) = (0, 1, p), we can trivialize v^{(0)} via the elements w_{j}, j ∈ ℤ/s, where for 0 ⩽ n < s, w_{j_{0}+n} = κ^{p^{n}}u^{p}l_{i_{0}+n}^{(1)} and κ ∈ k is such that σ^{s}(κ) − κ = γ. □

Proposition 1.21. Suppose L = (L, F (L), φ) ∈ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}) is given via a system of factors (v_{j})_{j∈ℤ/s} satisfying the normalization condition (C1). Then L comes from \underline{L}_{cr}^{∗} if and only if all v_{j} ∈ F (L_{1}).

Proof. Let N_{1} : L ⟶ L/u^{p}L be a W_{1}-differentiation such that for all j ∈ ℤ/s, N_{1}(l_{j}) = 0 (and, of course, N_{1}(l_{j}^{(1)}) = 0). If all v_{j} ∈ F (L_{1}), F (L) is generated by the elements ub̃_{j}l_{j} and uã_{j}l_{j}^{(1)}, j ∈ ℤ/s. If m is any of these elements then the basic identity N_{1}(φ(m)) = φ(uN_{1}(m)) is, clearly, satisfied. By Proposition 1.2, N_{1} can be extended to a unique W_{1}-differentiation N : L ⟶ L/u^{2p} and L = (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗}. Suppose now that L = (L, F (L), φ, N) ∈ \underline{L}_{cr}^{∗} and for all j ∈ ℤ/s, v_{j} = ∑_{i,t} γ_{ijt}u^{t}l_{i}^{(1)} with γ_{ijb̃_{j}} = 0. Consider the following congruence (use that

−ub̃_{j}l_{j} ≡ v_{j} mod F (L))

(1.5)  N(ub̃_{j}l_{j} + v_{j}) ≡ ∑_{i,t} γ_{ijt}(b̃_{j} − t)u^{t}l_{i}^{(1)} + ub̃_{j}N(l_{j})modF (L).

The condition L ∈ \underline{L}_{cr}^{∗} implies that N(ub̃_{j}l_{j} +v_{j}) ∈ F (L) mod u^{2p}L and N(l_{j}) ∈ u^{p}L mod u^{2p}L ⊂ F (L) mod u^{2p}L. This means that all (b̃_{j} −t)γ_{ijt}u^{t}l_{i}^{(1)} ∈ F (L_{1}). Therefore, for t ≠ b̃_{j}, γ_{ijt}u^{t}l_{i}^{(1)} ∈ F (L_{1}), and v_{j} ∈ F (L_{1}). The proposition is proved. □

Definition. A pair (i_{0}, j_{0}) ∈ (ℤ/s)^{2} is (r_{1}, r_{2})_{cr}-admissible if ã_{i_{0}} ≠ b̃_{j_{0}} and there is an m_{0} = m_{cr}(i_{0}, j_{0}) ∈ ℕ such that for 1 ⩽ m < m_{0}, ã_{i_{0}+m} = b̃_{j_{0}+m} but ã_{i_{0}+m_{0}} > b̃_{j_{0}+m_{0}}.

Remark. For any (r_{1}, r_{2})_{cr}-admissible pair of indices (i_{0}, j_{0}), one has r̃_{1}(i_{0}) > r̃_{2}(j_{0}) (or, equivalently, r_{1}(i_{0}) < r_{2}(j_{0})).

Definition. For (i_{0}, j_{0}) ∈ (ℤ/s)^{2} and γ ∈ k, denote by E_{cr}(i_{0}, j_{0}, γ) the extension L ∈ Ext\underline{L}_{cr}^{∗}(L_{2}, L_{1}) given by the system of factors (v_{j})_{j∈ℤ/s} such that v_{j_{0}} = γuã_{i_{0}} l_{i_{0}}^{(1)} and v_{j} = 0 if j ≠ j_{0}.

Proposition 1.22. Any element L ∈ Ext\underline{L}_{cr}^{∗}(L_{2}, L_{1}) can be obtained as a sum of E_{cr}(i, j, γ_{ij}), where (i, j) ∈ (ℤ/s)^{2} runs over the set of (r_{1}, r_{2})_{cr}-admissible pairs and all coefficients γ_{ij} ∈ k.

Proof. Propositions 1.19-1.21 imply that any L = (L, F (L), φ, N) from the group Ext\underline{L}_{cr}^{∗}(L_{2}, L_{1}) can be presented as a sum of extensions E_{cr}(i, j, γ_{ij}), where i, j ∈ ℤ/s are such that ã_{i} ≠ b̃_{j}, and γ_{ij} ∈ k. If m_{0} ∈ ℕ is such that for 1 ⩽ m < m_{0}, one has ã_{i+m} = b̃_{j+m} but ã_{i+m_{0}} < b̃_{j+m_{0}}, then the extension E_{cr}(i, j, γ_{ij}) is trivial, cf. the proof of Proposition 1.20. The proposition is proved. □

The above proposition describes the subgroup Ext\underline{L}_{cr}^{∗}(L_{2}, L_{1}) of Ext\underline{L}^{∗}(L_{2}, L_{1}). In particular, working modulo this subgroup we can describe the extensions in the whole category \underline{L}^{∗} via the systems of factors (v_{j})_{j∈ℤ/s} ∈ Z(L_{2}, L_{1}) such that all v_{j} = ∑_{i,t} γ_{ijt}u^{t}l_{i}^{(1)} satisfy the normalization conditions (C1) and

(C3) if t ⩾ ã_{i} then γ_{ijt} = 0.

Proposition 1.23. Suppose the system of factors (v_{j})_{j∈ℤ/s} satisfies the conditions (C1) and (C3). If it determines L = (L, F (L), φ) ∈ Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}) from the image of Ext\underline{L}^{∗}(L_{2}, L_{1}) then:

a) γ_{ijt} = 0 if t < ã_{i} − 1;

b) if t = ã_{i} − 1 and there is an m_{0} ∈ ℕ such that for all 1 ⩽ m < m_{0}, ã_{i+m} − 1 = b̃_{j+m} but ã_{i+m_{0}} − 1 > b̃_{j+m_{0}}, then γ_{ijt} = 0;

c) if t = ã_{i} − 1 and for all m ∈ ℤ/s, ã_{i+m} − 1 = b̃_{j+m} then γ_{ijt} = 0.

Proof. Suppose L = (L, F (L), φ, N) ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) and (v_{j})_{j∈ℤ/s} describes the image of L in Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}). By the definition of N, uN(ub̃_{j}l_{j} +v_{j}) ∈ F (L), and this implies that γ_{ijt} = 0 if t < ã_{i} − 1, t ≠ b̃_{j} (use congruence (1.5)). This proves a). Now we can set for all indices i and j, γ_{ij} := γ_{i,j,ã_{i}−1}. Let κ_{ij} ∈ k be such that N(l_{j}) ≡ ∑_{i} κ_{ij}l_{i}^{(1)}modu^{p}L and suppose γ_{ij} ≠ 0 (this implies that b̃_{j} ≠ ã_{i} − 1). For m ⩾ 0, consider the relations

(1.6)  N(l_{j+m+1}) = φ(uN(ub̃_{j+m}l_{j+m} + v_{j+m})).

If m = 0 then (1.6) implies κ_{i+1,j+1} = γ_{ij}^{p}(b̃_{j} −ã_{i} +1). Suppose that there is an m_{0} ⩾ 0 such that for all 1 ⩽ m < m_{0}, ã_{i+m} − 1 = b̃_{j+m} but ã_{i+m_{0}} − 1 ≠ b̃_{j+m_{0}}. Then (1.6) together with (1.5) (where j is replaced by j + m) imply that for 1 ⩽ m < m_{0},

κ_{i+m+1,j+m+1} = κ_{i+m,j+m}^{p} = γ_{ij}^{p^{m+1}}(b̃_{j} − ã_{i} + 1).

In particular, N(l_{j+m_{0}})mod u^{p}L contains l_{i+m_{0}}^{(1)} with the coefficient γ_{ij}^{p^{m_{0}}} (b̃_{j} − ã_{i} + 1). Therefore, uN(ub̃_{j+m_{0}} l_{j+m_{0}} + v_{j+m_{0}})mod u^{p}L contains l_{j+m_{0}}^{(1)} with the coefficient u^{b̃_{j+m_{0}} +1}γ_{ij}^{p^{m_{0}}} (b̃_{j} −ã_{i} +1). But this monomial must belong to F (L_{1}). This proves that if γ_{ij} ≠ 0 then b̃_{j+m_{0}} + 1 > ã_{i+m_{0}}. Finally, suppose that for all m ⩾ 1, ã_{i+m} −1 = b̃_{j+m}. Then ã_{i} −1 = ã_{i+s} −1 = b̃_{j+s} = b̃_{j} and γ_{ij} = γ_{i,j,ã_{i}−1} = γ_{i,j,b̃_{j}} = 0. □

Remark. With notation from the proof of above proposition the elements v_{j} = ∑_{i} γ_{ij}u^{ã_{i}−1}l_{i}^{(1)} determine a system of factors from Z(L_{2}, L_{1}) iff γ_{ij} = 0 when either ã_{i} = 0 or b̃_{j} = p − 1 (in this case v_{j} should belong to F (L_{1})).

Definition. A pair (i_{0}, j_{0}) ∈ (ℤ/s)^{2} is (r_{1}, r_{2})_{st}-admissible if:

• b̃_{j_{0}} ≠ p − 1 and ã_{i_{0}} ≠ 0, cf. above remark;

• ã_{i_{0}} − 1 ≠ b̃_{j_{0}};

• there is an m_{0} = m_{st}(i_{0}, j_{0}) ∈ ℕ such that for 1 ⩽ m < m_{0}, ã_{i_{0}+m}−1 = b̃_{j_{0}+m} but ã_{i_{0}+m_{0}} − 1 < b̃_{j_{0}+m_{0}}.

Definition. A pair (i_{0}, j_{0}) ∈ (ℤ/s)^{2} is (r_{1}, r_{2})_{sp}-admissible if i_{0} = 0 and for all m ∈ ℤ/s, ã_{m} − 1 = b̃_{j_{0}+m}.

Proposition 1.24. a) If (i_{0}, j_{0}) is an (r_{1}, r_{2})_{st}-admissible pair then r_{1}(i_{0}) + 1/(p − 1) > r_{2}(j_{0});

b) if (0, j_{0}) is an (r_{1}, r_{2})_{sp}-admissible pair then r_{1} + 1/(p − 1) = r_{2}(j_{0}).

Proof. a) Here for 1 ⩽ m < m_{0}, a_{i_{0}+m} + 1 = b_{j_{0}+m} and a_{i_{0}+m_{0}} ⩾ b_{j_{0}+m_{0}}. Therefore,

r_{1}(i_{0}) + 1/(p − 1) > ∑_{1⩽m⩽m_{0}}(a_{i_{0}+m} + 1)p^{−m} >

∑_{1⩽m⩽m_{0}} b_{j_{0}+m}p^{−m} + ∑_{m>m_{0}}(p − 1)p^{−m} ⩾ r_{2}(j_{0}).

The part b) can be obtained similarly. □

Using the calculations from the proof of Proposition 1.23 we obtain the following two statements.

Proposition 1.25. Suppose (i_{0}, j_{0}) ∈ (ℤ/s)^{2} is (r_{1}, r_{2})_{st}-admissible and γ ∈ k. Then there is a unique E_{st}(i_{0}, j_{0}, γ) ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) given by the system of factors (v_{j})_{j∈ℤ/s} such that v_{j_{0}} = γu^{ã_{i_{0}} −1}l_{i_{0}}^{(1)} and v_{j} = 0 if j ≠ j_{0}, and the map N, which is uniquely determined by the condition:

• if j = j_{0} + m with 1 ⩽ m ⩽ m_{st}(i_{0}, j_{0}) then

N(l_{j_{0}+m}) ≡ γ^{p^{m}}(b̃_{j_{0}} − ã_{i_{0}} + 1)l_{i_{0}+m}^{(1)}mod u^{p}L

and, otherwise, N(l_{j}) ≡ 0 mod u^{p}L.

Proposition 1.26. Suppose (0, j_{0}) ∈ (ℤ/s)^{2} is (r_{1}, r_{2})_{sp}-admissible and γ ∈ 𝔽_{q}, q = p^{s}. Then there is a unique E_{sp}(j_{0}, γ) ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) given by the zero system of factors and the map N, which is uniquely determined by the condition:

• N(l_{j_{0}+m}) ≡ γ^{p^{m}}l_{m}^{(1)}mod(u^{p}L), m ∈ ℤ/s.

The following proposition gives the uniqueness property of the decomposition of elements of Ext\underline{L}^{∗}(L_{2}, L_{1}) into a sum of standard extensions.

Proposition 1.27. Any element L ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) appears as a unique sum of the extensions E_{cr}(i, j, γ_{ij}^{cr}), E_{st}(i, j, γ_{ij}^{st}) and E_{sp}(j, γ_{0j}^{sp}), where all γ_{ij}^{cr}, γ_{ij}^{st} ∈ k but γ_{0j}^{sp} ∈ 𝔽_{q}, and γ_{ij}^{cr} = 0, resp. γ_{ij}^{st} = 0, γ_{0j}^{sp} = 0, if the corresponding pair of lower indices is not (r_{1}, r_{2})_{cr}-admissible, resp. (r_{1}, r_{2})_{st}-admissible, (r_{1}, r_{2})_{sp}- admissible.

Proof. By Proposition 1.23, any L ∈ Ext\underline{L}^{∗}(L_{2}, L_{1}) can be decomposed as a sum of the above special extensions. To prove the uniqueness of such decomposition, assume that L represents a trivial element of Ext\underline{L}^{∗}(L_{2}, L_{1}) and prove that all involved coefficients γ_{ij}^{cr}, γ_{ij}^{st} and γ_{0j}^{sp} are equal to 0. The image of L in Ext\underline{L}_{0}^{∗}(L_{2}, L_{1}) is given by the system of factors (v_{j}^{cr}+v_{j}^{st})_{j∈ℤ/s} such that

— v_{j}^{cr} = ∑_{i} γ_{ij}^{cr}uã_{i}l_{i}^{(1)};

— v_{j}^{st} = ∑_{i} γ_{ij}^{st}u^{ã_{i}−1}l_{i}^{(1)}.

Let w_{j} ∈ F (L_{1}) be such that for all j, v_{j} = w_{j} − ub̃_{j}φ(w_{j−1}). If w_{j} ≡ ∑_{i} κ_{ij}uã_{i}l_{i}^{(1)}mod uF (L) with κ_{ij} ∈ k, then for all i and j,

(1.7)  γ_{ij}^{cr}uã_{i} + γ_{ij}^{st}u^{ã_{i}−1} ≡ κ_{ij}uã_{i} − κ_{i−1,j−1}^{p}ub̃_{j}mod u^{ã_{i}+1}.

Suppose (i_{0}, j_{0}) is (r_{1}, r_{2})_{st}-admissible. Then ã_{i_{0}} − 1 ≠ b̃_{j_{0}} and comparing the coefficients for u^{ã_{i_{0}} −1} in (1.7) we deduce that γ_{i_{0}j_{0}}^{st} = 0. Therefore, all γ_{ij}^{st} = 0. Suppose (i_{0}, j_{0}) is (r_{1}, r_{2})_{cr}-admissible. Then for m_{0} = m_{cr}(i_{0}, j_{0}), ã_{i_{0}} ≠ b̃_{j_{0}}, ã_{i_{0}+m} = b̃_{j_{0}+m} if 1 ⩽ m < m_{0}, and ã_{i_{0}+m_{0}} > b̃_{j_{0}+m_{0}}. Then (1.7) implies that γ_{i_{0}j_{0}}^{cr} = κ_{i_{0}j_{0}}, κ_{i_{0}+m,j_{0}+m} = κ_{i_{0}+m−1,j_{0}+m−1}^{p} for 1 ⩽ m < m_{0}, and κ_{i_{0}+m_{0}−1,j_{0}+m_{0}−1} = 0. Therefore, γ_{i_{0}j_{0}}^{cr} = 0. Finally, L is the trivial element of the group Ext\underline{L}^{∗}(L_{2}, L_{1}) and, therefore, for all j, N(l_{j}) ∈ u^{p}L. Then from the description of standard extensions E_{sp}(j, γ_{0j}^{sp}) in Proposition 1.26 it follows that all γ_{0j}^{sp} = 0. □

2. The functor CV^{∗} : \underline{L}^{∗} ⟶ \underline{CMΓ}_{F}

2.1. The object R_{st}^{0} ∈ ˜ \underline{L}^{∗}. Let R = \underleftarrow{lim}_{n}(Ō/p)_{n} be Fontaine’s ring; it

has a natural structure of k-algebra via the map k ⟶ R given by α ↦ \underleftarrow{lim}([σ^{−n}α]mod p), where for any γ ∈ k, [γ] ∈ W (k) ⊂ Ō is the Teichmüller representative of γ. Let m_{R} be the maximal ideal of R. Choose x_{0} = (x_{0}^{(n)} mod p)_{n⩾0} ∈ R and ε = (ε^{(n)}mod p)_{n⩾0} such that for all n ⩾ 0, x_{0}^{(n+1)p} = x_{0}^{(n)} and ε^{(n+1)p} = ε^{(n)} with x_{0}^{(0)} = −p, ε^{(0)} = 1 but ε^{(1)} ≠ 1. We shall denote by v_{R} the valution on R such that v_{R}(x_{0}) = 1. Let Y be an indeterminate. Consider the divided power envelope R⟨Y ⟩ of R[Y] with respect to the ideal (Y). If for j ⩾ 0, γ_{j}(Y) is the j-th divided power of Y then R⟨Y ⟩ = ⊕_{j⩾0}Rγ_{j}(Y). Denote by R_{st} the completion ∏_{j⩾0} Rγ_{j}(Y) of R⟨Y ⟩ and set, Fil^{p}R_{st} = ∏_{j⩾p} Rγ_{j}(Y). Define the σ-linear morphism of the R-algebra R_{st} by the correspondence Y ↦ x_{0}^{p}Y; it will be denoted below by the same symbol σ. Introduce a W_{1}-module structure on R_{st} by the k-algebra morphism W_{1} ⟶ R_{st} such that u ↦ ι(u) := x_{0} exp(−Y) = x_{0} ∑_{j⩾0}(−1)^{j}γ_{j}(Y). Set F (R_{st}) = ∑_{0⩽i<p} x_{0}^{p−1−i}Rγ_{i}(Y) + Fil^{p}R_{st}. Define the continuous σ-linear morphism of R-modules φ : F (R_{st}) ⟶ R_{st} by setting for 0 ⩽ i < p, φ(x_{0}^{p−1−i}γ_{i}(Y)) = γ_{i}(Y)(1 − (i/2)x_{0}^{p}Y), and for i ⩾ p, φ(γ_{i}(Y)) = 0. Let N be a unique Rdifferentiation of R_{st} such that N(Y) = 1.

Proposition 2.1. a) If a ∈ R_{st} and b ∈ F (R_{st}) then

φ(ab) = σ(a)φ(b) mod x_{0}^{2p}R_{st};

b) φ mod x_{0}^{2p}R_{st} is a σ-linear morphism of W_{1}-modules;

c) for any b ∈ R_{st} and w ∈ W_{1}, N(wb) = N(w)b + wN(b);

d) for any l ∈ F (R_{st}), uN(l) ∈ F (R_{st}) and

N(φ(l)) = φ(uN(l)) mod x_{0}^{2p}R_{st}.

Proof. a) It is sufficient to verify it for a = Y and b = x_{0}^{p−1−i}γ_{i}(Y), 0 ⩽ i < p.

b) Use that the multiplication by σ(u) = u^{p} comes as the multiplication by ι(u)^{p} = x_{0}^{p} ≡ x_{0}^{p} exp(−x_{0}^{p}Y) = σ(ι(u)) mod x_{0}^{2p}R_{st}.

c) Use that N(ι(u)) = −ι(u).

d) It will be enough to check the identity for l = x_{0}^{p−1−i}γ_{i}(Y) with 1 ⩽ i < p. Then N(φ(l)) = γ_{i−1}(Y)(1 − (1/2)(i + 1)x_{0}^{p}Y). On the other hand, uN(l) = x_{0}^{p−1−(i−1)}γ_{i−1}(Y) exp(−Y) and φ(uN(l)) is equal to

γ_{i−1}(Y) (1 − \frac{i − 1}{2}x_{0}^{p}Y ) exp(−x_{0}^{p}Y) ≡ γ_{i−1}(Y) (1 − \frac{i + 1}{2}x_{0}^{p}Y ) mod x_{0}^{2p}

□

Introduce a Γ_{F} -action on R_{st} mod x_{0}^{p^{2}/(p−1)}R_{st} as follows.

For any τ ∈ Γ_{F}, let k(τ) ∈ ℤ be such that τ(x_{0}) = ε^{k(τ)}x_{0} and let l̃og(1+X) = X − X^{2}/2 + ⋯ − X^{p−1}/(p − 1) be the truncated logarithm. For any τ ∈ Γ_{F}, define a linear map τ : R_{st} ⟶ R_{st} by extending the natural action of τ on R and setting for τ ∈ Γ_{F} and j ⩾ 0,

τ(γ_{j}(Y)) := ∑_{0⩽i⩽min{j,p−1}} γ_{j−i}(Y)γ_{i}(l̃ogε).

Note that the cocycle relation ε^{k(τ_{1})}(τ_{1}ε)^{k(τ)} = ε^{k(τ_{1}τ)}, where τ_{1}, τ ∈ Γ_{F}, implies the cocycle relation

k(τ_{1})l̃ogε + k(τ)l̃og(τ_{1}(ε)) ≡ k(τ_{1}τ)l̃ogε mod x_{0}^{p^{2}/(p−1)}.

(Use that l̃og(1 + X)^{k} ≡ kl̃og(1 + X) mod(X^{p}) and ε ≡ 1 mod x_{0}^{p/(p−1)}.) In addition, for any k ∈ ℤ, the obvious congruence

(1 + X)^{k} = exp(k log(1 + X)) ≡ ẽxp(kl̃og(1 + X)) mod(X^{p})

implies that for any τ ∈ Γ_{F}, τ(x_{0} exp(−Y)) ≡ x_{0} exp(−Y) mod x_{0}^{p^{2}/(p−1)}. Therefore, the correspondences γ_{j}(Y) ↦ τ(γ_{j}(Y)) induce a Γ_{F} -action on W_{1}- algebra R_{st} mod x_{0}^{p^{2}/(p−1)}R_{st}, which extends the natural Γ_{F} -action on R.

Proposition 2.2. For any τ ∈ Γ_{F},

a) τ(F (R_{st})) = F (R_{st});

b) for any a ∈ F (R_{st}), τ(φ(a)) ≡ φ(τ(a) mod x_{0}^{p+1/(p−1)}R_{st};

c) for any b ∈ R_{st}, τ(N(b)) = N(τ(b)).

Proof. The proof is straightforward in cases a) and c). Part b) follows by direct calculation from the following Lemma. □

Lemma 2.3. σ(l̃ogε)/x_{0}^{p} ≡ l̃ogε mod x_{0}^{p+1/(p−1)}R.

Proof. Consider Fontaine’s element

t^{+} = log[ε] = ∑_{n⩾1}(−1)^{n−1} \frac{([ε] − 1)^{n}}{n} = ∑_{m∈ℤ} p^{m}[η_{m}] ∈ A_{cr}

where all η_{m} ∈ R. Then t^{+} ∈ Fil^{1}A_{cr} and σt^{+} = pt^{+}. This implies for all m ∈ ℤ, that η_{m} = σ^{−m}η_{0}. Consider H ⊂ A_{cr} consisting of the elements of the form ∑_{m∈ℤ} p^{m}[r_{m}] such that for m ⩽ 0, v_{R}(r_{m}) ⩾ p^{2}/(p − 1) (this is automatic for m ⩽ −2), v_{R}(r_{1}) ⩾ p^{2}/(p − 1) − 1 and v_{R}(r_{2}) ⩾ p^{2}/(p − 1) − 2. Then H is an additive subgroup in A_{cr}. Verify that

• for all n ⩾ p, ([ε] − 1)^{n}/n ∈ H.

Indeed, the congruence [ε] ≡ 1 + [a_{0}] mod pW (R) (where a_{0} = ε − 1) implies that [ε] = lim_{m→∞}(1 + [σ^{−m}a_{0}])^{p^{m}}. Therefore,

[ε] − 1 = ∑_{m⩾0}[a_{m}]p^{m} = [a_{0}] (1 + ∑_{m⩾1} p^{m}[b_{m}]) ,

where v_{R}(a_{m}) = p^{1−m}/(p − 1), v_{R}(b_{1}) = −1 and v_{R}(b_{2}) = −1 − 1/p. If n ≢ 0 mod p then ([ε] − 1)^{n} ≡ [a_{0n}] + p[a_{1n}] + p^{2}[a_{2n}] mod p^{3}W (R) with v_{R}(a_{0n}) = v_{R}(a_{1n}) + 1 = v_{R}(a_{2n}) + 2 = pn/(p − 1). This proves that ([ε] − 1)^{n}/n ∈ H for all n ≢ 0 mod p, n > p. As for all remaining n ⩾ p, just note that for all M ⩾ 1,

([ε] − 1)^{p^{M}} ≡ [a_{0}]^{p^{M}} (1 + p^{M+1}[b_{1}] + p^{M+2}[b_{2M})]) mod p^{M+3}W (R),

where v_{R}(b_{2M}) = −2.

The above calculations mean that t^{+} ≡ l̃og[ε] mod H. Therefore, if

l̃og[ε] = [ω_{0}] + p[ω_{1}] + p^{2}[ω_{2}] mod p^{3}W (R)

then ω_{0} = l̃ogε ≡ η_{0} mod x_{0}^{p^{2}/(p−1)}R,

ω_{1} ≡ η_{1} ≡ σ^{−1}η_{0} ≡ σ^{−1}l̃ogε mod x_{0}^{p^{2}/(p−1)−1}R

and ω_{2} ≡ η_{2} mod x_{0}^{p^{2}/(p−1)−2}R. Now note that l̃og[ε] ∈ Fil^{1}A_{cr} ⋂ W (R), that is l̃og[ε] is divisible by [x_{0}] + p in W (R). The division algorithm gives (ω_{1} − ω_{0}/x_{0})/x_{0} ≡ ω_{2} mod x_{0}^{1/(p−1)}R. Therefore, σ(ω_{1}) ≡ σ(ω_{0})/x_{0}^{p} mod x_{0}^{p}σ(ω_{2})R. The lemma is proved. □

By above results we can introduce R_{st}^{0} = (R_{st}^{0}, F (R_{st}^{0}), φ, N) ∈ ˜ \underline{L}^{∗}, where R_{st}^{0} = R_{st} mod x_{0}^{p}m_{R} and F (R_{st}^{0}) = F (R_{st}) mod x_{0}^{p}m_{R} with induced σ-linear map φ and W_{1}-differentiation N. The above defined Γ_{F} -action on R^{0} mod x_{0}^{p}m_{R} respects the structure of R_{st}^{0} as an object of the category ˜ \underline{L}^{∗}. In our setting the filtered Galois module R_{st}^{0} plays a role of Fontaine’s ring Â_{st}.

2.2. The functor V^{∗}. If L = (L, F (L), φ, N) ∈ ˜ \underline{L}^{∗} then the triple (L, F (L), φ) is an object of ˜ \underline{L}_{0}^{∗} which will be denoted below by the same symbol L.

Definition. Let R^{0} = (R^{0}, F (R^{0}), φ) ∈ ˜ \underline{L}_{0}^{∗}, where R^{0} = R/x_{0}^{p}m_{R}, F (R^{0}) = x_{0}^{p−1}R^{0}, the W_{1}-module structure on R^{0} is given via u ↦ x_{0} and ϕ is induced by the map r ↦ r/x_{0}^{p(p−1)}, r ∈ R.

For any L = (L, F (L), φ, N) ∈ \underline{L}^{∗}, consider the Γ_{F} -module V^{∗}(L) = Hom˜\underline{L}^{∗}(L, R_{st}^{0}). If f ∈ V^{∗}(L) and i ⩾ 0, introduce the k-linear morphisms f_{i} : L ⟶ R^{0} such that for any l ∈ L, f(l) = ∑_{i⩾0} f_{i}(l)γ_{i}(Y). The correspondence f ↦ f_{0} gives the homomorphism of abelian groups pr_{0} : V^{∗}(L) ⟶ V_{0}^{∗}(L) := Hom˜\underline{L}_{0}^{∗}(L, R^{0}).

Proposition 2.4. pr_{0} is isomorphism of abelian groups.

Proof. Clearly, pr_{0} is additive. Suppose f ∈ Ker pr_{0}. Then for all i ⩾ 0 and l ∈ L, f_{i}(l) = f_{0}(N^{i}(l))) = 0, i.e. f = 0. Suppose g ∈ Hom˜\underline{L}_{0}^{∗}(L, R^{0}). This means that g : L ⟶ R^{0} is a σ-linear morphism of W_{1}-modules, g(F (L)) ⊂ F (R^{0}) and for any l ∈ F (L), g(φ(l)) = (g(l)/x_{0}^{p−1})^{p}. Set for any l ∈ L, f(l) = g(l) + g(Nl)γ_{1}(Y) + ⋯ + g(N^{i}l)γ_{i}(Y) + . . .. Then for any l ∈ L, f(N(l)) = N(f(l)) and our Proposition is implied by the following Lemma. □

Lemma 2.5. a) For any l ∈ L, f(ul) = x_{0} exp(−Y)f(l);

b) for any l ∈ F (L)), φ(f(l)) = f(φ(l)).

Proof of Lemma. a) For any l ∈ L, f(ul) = ∑_{i⩾0} g(N^{i}(ul))γ_{i}(Y) =

x_{0} ∑_{i⩾0} g((N − id)^{i}l)γ_{i}(Y) = x_{0} ∑_{i,s}(−1)^{i−s}\binom{i}{s}g(N^{s}l)γ_{s}(Y)γ_{j}(Y)

= x_{0} ∑_{j,s}(−1)^{j}g(N^{s}l)γ_{s}(Y)γ_{j}(Y) = x_{0} exp(−Y)f(l).

b) Let l ∈ L. Prove by induction on i ⩾ 1 that

N^{i}(φ(l)) = φ((uN)^{i}(l)) = −\frac{i(i − 1)}{2}u^{p}φ(u^{i−1}N^{i−1}(l)) + φ(u^{i}N^{i}(l)).

Then

g(N^{i}(φ(l))) = −\frac{i(i − 1)}{2}x_{0}^{p} (\frac{g(u^{i−1}N^{i−1}(l))}{x_{0}^{p−1}})^{p} + (\frac{g(u^{i}N^{i}(l))}{x_{0}^{p−1}})^{p}

and f(φ(l)) is equal to ∑_{i⩾0} g(N^{i}(φ(l))γ_{i}(Y) =

∑_{i⩾0} (\frac{g(N^{i}l)}{x_{0}^{p−1−i}} )^{p} (γ_{i}(Y) − \frac{i(i + 1)}{2}x_{0}^{p}γ_{i+1}(Y)) = φ(f(l)).

□

Corollary 2.6. a) If rk_{W_{1}}L = s then |V^{∗}(L)| = p^{s};

b) the correspondence L ↦ V^{∗}(L) induces an exact functor V^{∗} from \underline{L}^{∗} to the category of 𝔽_{p}[Γ_{F}]-modules.

Proof. a) Proceed as in [1, 3]. Suppose the structure of the filtered φ-module L is given by a choice of a W_{1}-basis m_{1}, …, m_{s} of F (L) and a non-degenerate matrix A ∈ M_{s}(W_{1}) such that (m_{1}, …, m_{s}) = (φ(m_{1}), …, φ(m_{s}))A. Let X̄ = (X_{1}, …, X_{s}) be a vector with s independent variables and let R_{0} = Frac R. Consider the quotient A_{L} of the polynomial ring R_{0}[X̄] by the ideal generated by the coordinates of the vector (X̄A)^{(p)} − x_{0}^{p(p−1)}X̄. (For a matrix C the matrix C^{(p)} is obtained by raising all elements of C to p-th power.) Then A_{L} is etale R_{0}-algebra of rank p^{s} (use that (u^{p−1}I_{s})A^{−1} ∈ M_{s}(W_{1})) and all its R̄_{0}-points give rise to elements of the group Hom˜\underline{L}_{0}^{∗}(L, R), where

R = (R, x_{0}^{p−1}R, φ) ∈ ˜ \underline{L}_{0}^{∗} is such that for any r ∈ R, φ(x_{0}^{p−1}r) = r^{p}. It remains to note that φ|_{x_{0}^{p}m_{R}} is nilpotent, by Lemma 1.5, the natural projection R ⟶ R^{0} induces bijection from Hom˜\underline{L}_{0}^{∗}(L, R) to Hom˜\underline{L}_{0}^{∗}(L, R^{0}) = V_{0}^{∗}(L) and by Proposition 2.4, |V_{0}^{∗}(L)| = |V^{∗}(L)|.

b) This follows from a) because the functor L ↦ V_{0}^{∗}(L) is left exact. □

Introduce the ideal J̃ = ∑_{0⩽i<p} x_{0}^{p−i}m_{R}γ_{i}(Y) + Fil^{p}R_{st}^{0} of R_{st}^{0}. Then

F (R_{st}^{0}) ⊃ J̃ and φ|J̃ is nilpotent. For R̃_{st}^{0} = (R_{st}^{0}/J̃, F (R_{st}^{0})/J̃, φ mod J̃) ∈ ˜ \underline{L}_{0}^{∗}, there is a natural projection R_{st}^{0} ⟶ R̃_{st}^{0} in ˜ \underline{L}_{0}^{∗} and for any L ∈ \underline{L}_{0}^{∗}, Hom˜\underline{L}_{0}^{∗}(L, R_{st}^{0}) = Hom˜\underline{L}_{0}^{∗}(L, R̃_{st}^{0}). This implies the following description of the Γ_{F} -modules V^{∗}(L) where L ∈ \underline{L}^{∗} (use the identification pr_{0} of Proposition 2.4).

Corollary 2.7.

V^{∗}(L) = \{ ∑_{0⩽i<p} N^{∗i}(f_{0})γ_{i}(Y) mod J̃ | f_{0} ∈ Hom˜\underline{L}_{0}^{∗}(L, R^{0})\}

Remark. a) In the above description of V^{∗}(L), for any l ∈ L, N^{∗}(f_{0})(l) = f_{0}(N(l)). In addition, all N^{∗i}(f_{0})γ_{i}(Y) depend just on N_{1} = N mod u^{p}L.

b) If L ∈ \underline{L}^{∗u} then in the above Corollary we can replace R^{0} and J̃ by, respectively, R^{u} = (R/x_{0}^{p}R, x_{0}^{p−1}R/x_{0}^{p}R, φ) ∈ ˜ \underline{L}_{0}^{∗} and the ideal J̃^{u} = ∑_{0⩽i<p} Rx_{0}^{p−i}γ_{i}(Y) + Fil^{p}R_{st}^{0}. In particular, for unipotent modules the whole theory can be developed in the context of k[u]/u^{p}-modules.

2.3. The category \underline{CMΓ}_{F} and the functor CV^{∗}.

Definition. The objects of the category \underline{CMΓ}_{F} are the triples H = (H, H^{0}, j), where H, H^{0} are finite ℤ_{p}[Γ_{F}]-modules, Γ_{F} acts trivially on H^{0} and j : H ⟶ H^{0} is an epimorphic map of ℤ_{p}[Γ_{F}]-modules. If H_{1} = (H_{1}, H_{1}^{0}, j_{1}) ∈ \underline{CMΓ}_{F} then Hom_{\underline{CMΓ}_{F}} (H_{1}, H) consists of the couples (f, f^{0}), where f : H_{1} ⟶ H and f^{0} : H_{1}^{0} ⟶ H^{0} are morphisms of Γ_{F} -modules such that jf = f^{0}j_{1}.

The category \underline{CMΓ}_{F} is pre-abelian, cf. Appendix A, and its objects have a natural group structure. In particular, with above notation, Ker(f, f^{0}) = (Kerf, j_{1}(Kerf)) together with the natural embedding to H_{1}. Similarly, Coker(f, f^{0}) = (H/f(H_{1}), H^{0}/j(f(H_{1}))). For example, the map (id, 0) : (H, H) ⟶ (H, 0) has the trivial kernel and cokernel. In addition, the monomorphism (f_{1}, f_{1}^{0}) : H_{1} ⟶ H is strict if and only if f_{1}(Ker j_{1}) = f_{1}(H_{1}) ∩ Ker j. Suppose H_{2} = (H_{2}, H_{2}^{0}, j_{2}) and (f_{2}, f_{2}^{0}) : H ⟶ H_{2} is an epimorphism. Then it is strict if and only if f_{2}^{0} induces epimorphic map from Kerj to Ker j_{2}. In \underline{CMΓ}_{F} we can use formalism of short exact sequenes and the corresponding 6-terms Hom_{\underline{CMΓ}_{F}} −Ext_{\underline{CMΓ}_{F}} exact sequences, cf. Appendix A.

Definition. Suppose L ∈ \underline{L}^{∗} and i^{et} : L^{et} ⟶ L is the maximal etale subobject. Then CV^{∗} : \underline{L}^{∗} ⟶ \underline{CMΓ}_{F} is the functor such that CV^{∗}(L) = (V^{∗}(L), V^{∗}(L^{et}), V^{∗}(i^{et})).

The simple objects in \underline{CMΓ}_{F} are of the form either (H, 0, 0), where H is a simple ℤ_{p}[Γ_{F}]-module, or (𝔽_{p}, 𝔽_{p}, id), where 𝔽_{p} is provided with the trivial Γ_{F} -action. In this context it will be very convenient to use the following formalism. For s ∈ ℕ, consider Serre’s fundamental characters χ_{s} : Γ_{F} ⟶ k^{∗}. Here for τ ∈ Γ_{F}, χ_{s}(τ) = (τx_{s})/x_{s} mod x_{0}^{p}, where x_{s} ∈ R is such that x_{s}^{p^{s}−1} = x_{0}. If χ is any continuous (1-dimensional) character of Γ_{F} then there are s, m ∈ ℕ such that 0 < m ⩽ p^{s} − 1 and χ = χ_{s}^{m}. Set r(χ) = m/(p^{s} − 1). Then r(χ) depends only on χ and the correspondence χ ↦ r(χ) gives a bijection from the set of all continuous (1-dimensional) characters of Γ_{F} with values in k^{∗} to the set [0, 1]_{p} \ {0}. For r ∈ [0, 1]_{p}, r ≠ 0, introduce the Γ_{F} -module 𝔽(r) such that 𝔽(r) = 𝔽_{p^{s(r)}}, where s(r) is the period of the p-digit expansion of r, cf. Subsection 1.2, with the Γ_{F} -action given by the character χ such that r(χ) = r. We have:

— all 𝔽(r) are simple ℤ_{p}[Γ_{F}]-modules;

— Γ_{F} -modules 𝔽(r_{1}) and 𝔽(r_{2}) are isomorphic if and only if there is an n ∈ ℤ such that r_{1} = r_{2}(n);

— any simple ℤ_{p}[Γ_{F}]-module is isomorphic to some 𝔽(r). It will be natural to set F(r) := (𝔽(r), 0, 0) for all r ∈ (0, 1]_{p}, and to set separately F(0) := (𝔽_{p}, 𝔽_{p}, id). With above notation we have the following property, where the objects L(r) were introduced in Subsection 1.3.

Proposition 2.8. For any r ∈ [0, 1]_{p}, CV^{∗}(L(r)) = F(r).

Proof. The proof goes along the lines of Subsection 4.2 of [1], cf. also the beginning of Subsection 2.4 below. □

2.4. A criterion. Suppose L_{1}, L_{2} are given in notation of Subsection 1.4 and q = p^{s}. Then for i = 1, 2, V^{∗}(L_{i}) = V_{i} are 1-dimensional vector spaces over 𝔽_{q} with Γ_{F} -action given by the character χ_{i} : Γ_{F} ⟶ k^{∗} such that r(χ_{i}) = r_{i}. (Note that (q−1)r_{i} ∈ ℤ and, therefore, χ_{i}(Γ_{F}) ⊂ 𝔽_{q}^{∗}.) Choose π_{s} ∈ F̄ such that π_{s}^{q−1} = −p. Then F_{s} = F (π_{s}) is a tamely ramified extension of F of degree q−1 and all points of V_{i} are defined over F_{s}. We can identify V_{i} with the 𝔽_{p}[Γ_{F}]- module 𝔽_{q}π̄_{s}^{(q−1)r_{i}} ⊂ Ō/pŌ, where π̄_{s} = π_{s} mod p. These identifications allow us to fix the points h_{i}^{0} := π̄_{s}^{(q−1)r_{i}} ∈ V_{i} and to identify V_{i} with the Γ_{F} -module {αh_{i}^{0} | α ∈ 𝔽_{q}}. Suppose h_{1} ∈ V_{1}. Define the homomorphism

F_{h_{1}} : Ext_{𝔽_{p}[Γ_{F}]}(V_{1}, V_{2}) ⟶ Z^{1}(Γ_{F_{s}}, 𝔽_{q}) = Hom(Γ_{F_{s}}, 𝔽_{q}),

where Γ_{F_{s}} = Gal(F̄/F_{s}), as follows. If V ∈ Ext_{𝔽_{q}[Γ_{F}]}(V_{1}, V_{2}) and h ∈ V is a lift of h_{1} then for any τ ∈ Γ_{F}, F_{h_{1}}(V)(τ) = a_{τ} ∈ 𝔽_{q}, where τh − h = a_{τ} h_{2}^{0}.

Clearly, F_{h_{1}}(V) does not depend on a choice of h and it is the zero function if and only if the projection V ⟶ V_{1} admits a Γ_{F} -equivariant section. In other words, we have the following criterion.

Proposition 2.9. V is the trivial extension if and only if for all h_{1} ∈ V_{1}, one has F_{h_{1}}(V) = 0.

2.5. Galois modules V^{∗}(E_{cr}(i_{0}, j_{0}, γ)). Suppose we have an object L = (L, F (L), φ, N) of the category \underline{L}_{cr}^{∗}. Then there is a special σ(W_{1})-basis l_{1}, …, l_{s} of φ(F (L)) such that for some integers 0 ⩽ c_{1}, …, c_{s} < p and a matrix A ∈ GL_{s}(k), the elements u^{c_{1}}l_{1}, …, u^{c_{s}}l_{s} form a W_{1}-basis of F (L) and (φ(u^{c_{1}}l_{1}), …, φ(u^{c_{s}}l_{s})) = (l_{1}, …, l_{s})A. For 1 ⩽ i ⩽ s, set c̃_{i} = (p − 1) − c_{i}. The following Proposition is a special case of Corollary 2.7 (remind that R^{0} = R/x_{0}^{p}m_{R}).

Proposition 2.10. With above notation, V^{∗}(L) is the 𝔽_{p}[Γ_{F}]-module of all (θ_{1}, …, θ_{s}) mod x_{0}^{p}m_{R} ∈ (R^{0})^{s} such that

(θ_{1}^{p}/x_{0}^{pc̃_{1}}, …, θ_{s}^{p}/x_{0}^{pc̃_{s}}) = (θ_{1}, …, θ_{s})A.

Remark. In [1, 2] it was proved (in the context of the Fontaine-Laffaille theory) that the family of 𝔽_{p}[Γ_{F}]-modules V^{∗}(L), where L ∈ \underline{L}_{cr}^{∗}, coincides with the family of all killed by p subquotients of crystalline representations of Γ_{F} with weights from [0, p). This result can be also extracted from Subsection 4, where we establish that the family of 𝔽_{p}[Γ_{F}]-modules V^{∗}(L), where L ∈ \underline{L}^{∗}, coincides with the family of all killed by p subquotients of semi-stable representations of Γ_{F} with weights from [0, p).

For an (r_{1}, r_{2})_{cr}-admissible pair (i_{0}, j_{0}) ∈ (ℤ/s)^{2} and γ ∈ k, use the description of E_{cr}(i_{0}, j_{0}, γ) from Subsection 1.4. Then by Corollary 2.7, V = V^{∗}(E_{cr}(i_{0}, j_{0}, γ)) is identified with the additive group of all taken modulo x_{0}^{p}m_{R} solutions in R of the following system of equations

X_{i}^{(1)p}/x_{0}^{pa_{i}} =  X_{i+1}^{(1)},  for all i ∈ ℤ/s;
X_{j}^{p}/x_{0}^{pb_{j}}  = X_{j+1} − δ_{jj_{0}}γ^{p}X_{i_{0}+1}^{(1)}, for all j ∈ ℤ/s

Note that the first group of equations describes V_{1} = V^{∗}(L_{1}) and the correspondences X_{i}^{(1)} ↦ 0 and X_{j} ↦ X_{j}^{(2)} with i, j ∈ ℤ/s, define the map V ⟶ V_{2}, where V_{2} = V^{∗}(L_{2}) is associated with all taken modulo x_{0}^{p}m_{R} solutions in R of the equations X_{j}^{(2)p}/x_{0}^{pb_{j}} = X_{j+1}^{(2)}, j ∈ ℤ/s. As it was noted in Subsection 2.2, the corresponding Γ_{F} -action on V, V_{1} and V_{2} comes from the natural Γ_{F} -action on R^{0}. Take x_{s} ∈ R such that x_{s}^{q−1} = x_{0} and x_{s} ↦ π_{s}mod p under the natural identification R/x_{0}^{p}R ≃ Ō/pŌ. (This identification is given by the correspondence

r = \underleftarrow{lim}_{n}(r_{n} mod p) ↦ r^{(1)} := lim_{n→∞}r_{n+1}^{p^{n}}.) For i, j ∈ ℤ/s, set x_{0}^{r_{1}(i)} := x_{s}^{(q−1)r_{1}(i)}

and x_{0}^{r_{2}(j)} := x_{s}^{(q−1)r_{2}(j)}, and introduce the variables Z_{i}^{(1)} = x_{0}^{−pr_{1}(i)}X_{i}^{(1)}, Z_{j} = x_{0}^{−pr_{2}(j)}X_{j}, Z_{j}^{(2)} = x_{0}^{−pr_{2}(j)}X_{j}^{(2)}. Then the elements of V appear as the taken modulo m_{R} solutions in R_{0} := Frac(R) of the following system of equations

Z_{i}^{(1)p}  =  Z_{i+1}^{(1)},  for all i ∈ ℤ/s;
Z_{j}^{p}  =  Z_{j+1},  for all j ≠ j_{0} + 1;
Z_{j_{0}+1} − Z_{j_{0}+1}^{q} = γ^{p}Z_{i_{0}+1}^{(1)}x_{0}^{p(r_{1}(i_{0})−r_{2}(j_{0}))}

Note that for the points h_{1}^{0} ∈ V_{1} and h_{2}^{0} ∈ V_{2} chosen in Subsection 2.4, one has Z_{i}^{(1)}(h_{1}^{0}) = Z_{i}^{(2)}(h_{2}^{0}) = 1, where i ∈ ℤ/s. Suppose α ∈ 𝔽_{q} and h_{1} = αh_{1}^{0} ∈ V_{1}. Let F_{s} = k((x_{s})) ⊂ R_{0} = Frac R. The field-of-norms functor gives a natural embedding of the absolute Galois group Γ_{F_{s}} of F_{s} into Γ_{F_{s}}, where F_{s} = F (π_{s}). Then the restriction F_{h_{1}}(V)|_{Γ_{Fs}} of the cocycle

{F_{h_{1}}(V)(τ) = A_{τ,α}(i_{0}, j_{0}, γ) ∈ 𝔽_{q} | τ ∈ Γ_{F_{s}}}

from Subsection 2.4 can be described as follows. Let U ∈ R_{0} be such that U − U^{q} = γx_{0}^{r_{1}(i_{0})−r_{2}(j_{0})}. Then for any τ ∈ Γ_{F_{s}}, σ^{j_{0}}(A_{τ,α}(i_{0}, j_{0}, γ)) = σ^{i_{0}}(α)(τ(U) − U) and therefore

A_{τ,α}(i_{0}, j_{0}, γ) = σ^{i_{0}−j_{0}}(α)σ^{−j_{0}}(τU − U).

The following Lemma is an immediate consequence of the definition of (r_{1}, r_{2})_{cr}-admissible pairs.

Lemma 2.11. With above notation let C = −(q − 1)(r_{1}(i_{0}) − r_{2}(j_{0})). Then C is a prime to p integer and 1 ⩽ C ⩽ q − 1.

2.6. Galois modules V^{∗}(E_{st}(i_{0}, j_{0}, γ)). For an (r_{1}, r_{2})_{st}-admissible pair (i_{0}, j_{0}) ∈ (ℤ/s)^{2} and γ ∈ k, use the description of E_{st}(i_{0}, j_{0}, γ) from Subsection 1.5. By Subsection 2.2, V = V^{∗}(E_{st}(i_{0}, j_{0}, γ)) is identified (as an abelian group) with the solutions ({X_{i}^{(1)} | i ∈ ℤ/s}, {X_{j} | j ∈ ℤ/s}) ∈ R^{2s} of the following system of equations

(2.1) X_{i}^{(1)p}/x_{0}^{pa_{i}}  = X_{i+1}^{(1)}, for all i ∈ ℤ/s; X_{j}^{p}/x_{0}^{pb_{j}} + δ_{jj_{0}}γ^{p}X_{i_{0}}^{(1)p}/x_{0}^{pa_{i_{0}} +p} = X_{j+1}, for all j ∈ ℤ/s

The structure of V as an element of Ext_{𝔽_{p}[Γ_{F}]}(V_{1}, V_{2}) can be described along the lines of Subsection 2.5. The action of Γ_{F} on V comes from the natural Γ_{F} -action on R̃_{st}^{0}, and the embedding of V into (R_{st}^{0})^{2s} given by the following correspondences:

– if i ∈ ℤ/s then X_{i}^{(1)} ↦ X_{i}^{(1)} mod x_{0}^{p}m_{R};

– if j ∉ {j_{0} + 1, …, j_{0} + m_{0}} then X_{j} ↦ X_{j} mod x_{0}^{p}m_{R};

– for 1 ⩽ m ⩽ m_{0}, X_{j_{0}+m} ↦ X_{j_{0}+m} + γ^{p^{m}}(b̃_{j_{0}} − ã_{i_{0}} + 1)X_{i_{0}+m}^{(1)}Y mod x_{0}^{p}m_{R}.

Similarly to Subsection 2.5, introduce new variables by the relations Z_{i}^{(1)} = x_{0}^{−pr_{1}(i)}X_{i}^{(1)}, Z_{i} = x_{0}^{−pr_{2}(i)}X_{i} and Z_{i}^{(2)} = x_{0}^{−pr_{2}(i)}X_{i}^{(2)}, i ∈ ℤ/s, and rewrite system of equations (2.1) in the following form:

Z_{i}^{(1)p}  =  Z_{i+1},  for all i ∈ ℤ/s;
Z_{j}^{p}  =  Z_{j+1},  for all j ≠ j_{0} + 1;
Z_{j_{0}+1} − Z_{j_{0}+1}^{q} = γ^{p}Z_{i_{0}+1}^{(1)}x_{0}^{p(r_{1}(i_{0})−r_{2}(j_{0})−1)}

If α ∈ 𝔽_{q} and h_{1} = αh_{1}^{0} ∈ V_{1}, then the restriction to Γ_{F_{s}} of the cocycle {F_{h_{1}}(V)(τ) = A_{τ,α}(i_{0}, j_{0}, γ) | τ ∈ Γ_{F_{s}}} can be described as follows. Let U ∈ R_{0} be such that

U − U^{q} = γx_{0}^{r_{1}(i_{0})−r_{2}(j_{0})−1}.

Then for any τ ∈ Γ_{F_{s}}, σ^{j_{0}}(A_{τ,α}(i_{0}, j_{0}, γ)) = σ^{i_{0}}(α)(τU − U). Thus

A_{τ,α}(i_{0}, j_{0}, γ) = σ^{i_{0}−j_{0}}(α)σ^{−j_{0}}(τU − U).

The following Lemma is a direct consequence of the definition of (r_{1}, r_{2})_{st}- admissible pairs, cf. also Proposition 1.24

Lemma 2.12. Let C = −(q − 1)(r_{1}(i_{0}) − r_{2}(j_{0}) − 1). Then C is a prime to p integer such that 1 ⩽ C < (q − 1)(1 + 1/(p − 1)).

2.7. Galois modules E_{sp}(j_{0}, γ). In this subsection (0, j_{0}) is some (r_{1}, r_{2})_{sp}- admissible pair (i.e. r_{1} + 1/(p − 1) = r_{0}(j_{0})) and γ ∈ 𝔽_{q}. Then V = V^{∗}(E_{sp}(j_{0}, γ)) is identified as an abelian group with the solutions

({X_{i}^{(1)} | i ∈ ℤ/s}, {X_{j}^{(2)} | j ∈ ℤ/s}) ∈ R^{2s}

of the following system of equations

X_{i}^{(1)p}/x_{0}^{pa_{i}} = X_{i+1}^{(1)}, for all i ∈ ℤ/s,
X_{j}^{(2)p}/x_{0}^{pb_{j}} = X_{j+1}^{(2)}, for all j ∈ ℤ/s.

The corresponding Γ_{F} -action comes from the natural Γ_{F} -action on R_{st}^{0} and the embedding of V into (R_{st}^{0})^{2s} given by the following correspondences:

– if i ∈ ℤ/s then X_{i}^{(1)} ↦ X_{i}^{(1)} mod x_{0}^{p}m_{R};

– if m ∈ ℤ/s then X_{j_{0}+m}^{(2)} ↦ X_{j_{0}+m}^{(2)} + γ^{p^{m}}X_{m}^{(1)}Y mod x_{0}^{p}m_{R}.

If α ∈ 𝔽_{q} and h_{1} = αh_{1}^{0} ∈ V_{1} then the cocycle

{F_{h_{1}}(V)(τ) = A_{τ,α}^{sp}(j_{0}, γ) | τ ∈ Γ_{F_{s}}}

can be described as follows. Note that the point h_{1} corresponds to the collection ({σ^{i}(α)x_{0}^{pr_{1}(i)}|i ∈ ℤ/s}, {σ^{i−j_{0}}(αγ)x_{0}^{pr_{1}(i−j_{0})}Y |i ∈ ℤ/s}). Then for τ ∈ Γ_{F_{s}}, τ(h_{1}) corresponds to the collection

({σ^{i}(α)x_{0}^{pr_{1}(i)} | i ∈ ℤ/s}, {σ^{i−j_{0}}(αγ)x_{0}^{pr_{1}(i−j_{0})}(Y + k(τ)l̃ogε) | i ∈ ℤ/s}).

Therefore, τ(h_{1}) − h_{1} corresponds to the collection

({ 0 | i ∈ ℤ/s}, {σ^{i−j_{0}}(αγ)x_{0}^{pr_{2}(i)}k(τ) | i ∈ ℤ/s}),

which corresponds to σ^{−j_{0}}(αγ)h_{2}^{0}. Therefore, A_{τ,α}^{sp}(j_{0}, γ) = σ^{−j_{0}}(αγ)k(τ). Notice that for any τ ∈ Γ_{F_{s}} ⊂ Γ_{F_{s}}, A_{τ,α}^{sp}(j_{0}, γ) = 0.

2.8. Fully faithfulness of CV^{∗}. In this subsection we prove the following important property.

Proposition 2.13. The functor CV^{∗} is fully faithful.

Proof. We must prove that for all L_{1}, L_{2} ∈ \underline{L}^{∗}, the functor CV^{∗} induces a bijective map

Π(L_{1}, L_{2}) : Hom\underline{L}^{∗}(L_{2}, L_{1}) ⟶ Hom_{\underline{CMΓ}_{F}} (CV^{∗}(L_{1}), CV^{∗}(L_{2})).

By induction on lengths of composition series for L_{1} and L_{2} it will be sufficient to verify that for any two simple objects L_{1} and L_{2}:

• Π(L_{1}, L_{2}) is bijective;

• the functor CV^{∗} induces injective map

EΠ (L_{1}, L_{2}) : Ext\underline{L}^{∗}(L_{2}, L_{1}) ⟶ Ext_{\underline{CMΓ}_{F}} (CV^{∗}(L_{1}), CV^{∗}(L_{2})).

The first fact has been already checked in Subsection 2.3. In order to verify the second property, notice that for any two objects L_{1}, L_{2} ∈ \underline{L}^{∗}, the natural map

Ext_{\underline{CMΓ}_{F}} (CV^{∗}(L_{1}), CV^{∗}(L_{2})) ⟶ Ext_{\underline{MΓ}_{F}} (V^{∗}(L_{1})), V^{∗}(L_{2}))

is injective. Therefore, we can prove injectivity of EΠ(L_{1}, L_{2}) on the level of functor V^{∗}. In addition, for n_{1}, n_{2} ∈ ℕ, Ext\underline{L}^{∗}(L_{2}^{n_{1}}, L_{1}^{n_{2}}) = Ext\underline{L}^{∗}(L_{2}, L_{1})^{n_{1}n_{2}} (the formation of Ext is compatible with direct sums). So, by Lemma 1.17, we can replace L_{1} and L_{2} by the objects introduced in Subsection 1.5 (where they are denoted also by L_{1} and L_{2}). By Proposition 1.27, any element of Ext\underline{L}^{∗}(L_{2}, L_{1}) appears as a sum of standard extensions of the form E_{cr}(i, j, γ_{ij}), E_{st}(i, j, γ_{ij}) and E_{sp}(j, γ_{j}^{sp}). Here: a) (i, j) ∈ (ℤ/s)^{2} is either (r_{1}, r_{2})_{cr}-admissible or (r_{1}, r_{2})_{st}-admissible and all γ_{ij} ∈ k; b) j ∈ ℤ/s is such that (0, j) is (r_{1}, r_{2})_{sp}-admissible and γ_{j}^{sp} ∈ 𝔽_{q}.

Remark. A couple (i, j) can’t be both (r_{1}, r_{2})_{cr}-admissible and (r_{1}, r_{2})_{st}- admissible, but it can be (r_{1}, r_{2})_{cr}-admissible and (r_{1}, r_{2})_{sp}-admissible at the same time.

By Subsections 2.5 -2.7, we can attach to these standard extensions the 1cocycles A_{τ,α}(i, j, γ_{ij}) and A_{τ,α}^{sp}(j, γ_{j}^{sp}), where τ ∈ Γ_{F_{s}}. It remains to prove that the sum of these cocycles is trivial only if all corresponding coefficients γ_{ij} and γ_{j}^{sp} are equal to 0. First, we need the following lemma.

Lemma 2.14. Suppose for all (i, j) ∈ (ℤ/s)^{2}, the elements U_{ij} ∈ R_{0} = Frac R are such that U_{ij} − U_{ij}^{q} = γ_{ij}x_{s}^{−C_{ij}}, where all γ_{ij} ∈ k and all C_{ij} are prime to p natural numbers. For τ ∈ Γ_{F_{s}}, let B_{τ} (i, j, γ_{ij}) = τ(U_{ij}) − U_{ij} ∈ 𝔽_{q}. If for all α ∈ 𝔽_{q} and all τ ∈ Γ_{F_{s}},

(2.2)  ∑_{i,j,∈ℤ/s} σ^{i−j}(α)σ^{−j}B_{τ} (i, j, γ_{ij}) = 0

then all γ_{ij} = 0.

Proof of Lemma. For different prime to p natural numbers C_{ij} the extensions F_{s}(U_{ij}) behave independently. Therefore, we can assume that all C_{ij} = C are the same. Let j_{0} = j_{0}(j) be such that 0 ⩽ j_{0} < s and j_{0} ≡ −j mod s. Then (2.2) means that for any α ∈ 𝔽_{q},

B_{α} := ∑_{i,j∈ℤ/s} σ^{i−j}(α)σ^{j_{0}}(U_{ij}) ∈ F_{s}.

Then

B_{α} − B_{α}^{q} = ∑_{j∈ℤ/s} ( ∑_{i∈ℤ/s} σ^{i−j}(α)γ_{ij}^{p^{−j}} ) x_{s}^{−p^{j_{0}} C}.

Looking at the Laurent series of B_{α} ∈ F_{s} we conclude that all B_{α} ∈ 𝔽_{q}. This means that for all j ∈ ℤ/s and α ∈ 𝔽_{q}, ∑_{i∈ℤ/s} σ^{i}(α)γ_{ij} = 0 and, therefore, all γ_{ij} = 0. The lemma is proved □

Now suppose that for all α ∈ 𝔽_{q} and τ ∈ Γ_{F_{s}}, the sum of cocycles A_{τ,α}(i, j, γ_{ij}) and A_{τ,α}^{sp}(j, γ_{j}^{sp}) is zero. Restrict this sum to the subgroup Γ_{F_{s}}. Then all spterms will disappear and by above Lemma 2.14 all γ_{ij} = 0. So, for all τ ∈ Γ_{F_{s}} and α ∈ 𝔽_{q}, ∑_{j∈ℤ/s} σ^{−j}(αγ_{j}^{sp}) = 0, and this implies that all γ_{j}^{sp} = 0. □

Corollary 2.15. The functor V^{∗} is fully faithful on the subcategories of unipotent objects \underline{L}^{∗u} and of connected objects \underline{L}^{∗c}.

Proof. Indeed, on both categories the map Π(L_{1}, L_{2}) is already bijective on the level of functor V^{∗}. □

2.9. Ramification estimates. Suppose L ∈ \underline{L}^{∗} and H = V^{∗}(L). For any rational number v ⩾ 0, denote by Γ_{F}^{(v)} the ramification subgroup of Γ_{F} in upper numbering, [22].

Proposition 2.16. If v > 2 − \frac{1}{p} then Γ_{F}^{(v)} acts trivially on H.

A proof can be obtained along the lines of the paper [17] (which adjusts Fontaine’s approach from [14]). Alternatively, one can apply author’s method from [3]: if τ ∈ Γ^{(v)} with v > 2 − 1/p then there is an automorphism ψ of R such that ψ(x_{0}) = τ(x_{0}) and ψ induces the trivial action on H; therefore we can assume that τ comes from the absolute Galois group of k((x_{0})) and the characteristic p approach from [3] gives the ramification estimate which coincides with the required by the theory of field-of-norms.

Corollary 2.17. If F̃ is the common field-of-definition of points of 𝔽_{p}[Γ_{F}]- modules V(L) for all L ∈ \underline{L}^{∗}, then v_{p}(D(F̃/F)) < 3 − \frac{1}{p}, where D(F̃/F) is the different of the field extension F̃/F.

3. Semistable representations with weights from [0, p) and filtered W-modules

3.1. The ring S. Let v = u + p ∈ W and let S be the p-adic closure of the divided power envelope of W with respect to the ideal generated by v. Use the same symbols σ and N for natural continuous extensions of σ and N from W to S. For i ⩾ 0, denote by Fil^{i}S the i-th divided power of the ideal (v) in S. Then for 0 ⩽ i < p, there are σ-linear morphisms ϕ_{i} = σ/p^{i} : Fil^{i}S ⟶ S. Note that ϕ_{0} = σ and agree to use the notation φ for ϕ_{p−1}. One can see also that S is the p-adic closure of W (k)[v_{0}, v_{1}, …, v_{n}, …], where v_{0} = v and for all n ⩾ 0, v_{n+1}^{p}/p = v_{n}. Consider the ideals m_{S} = (p, v, v_{1}, …, v_{n}, …), I = (p, v_{1}, v_{2}, …) and J = (p, v_{1}v, v_{2}, …, v_{n}, …) of S. Then

— m_{S} is the maximal ideal in S;

— I = Fil^{p}S + pS ⊃ J;

— φ(I) ⊂ S and φ(J) ⊂ pS;

— φ(v^{p−1}) ≡ 1 − v_{1}(mod J) and φ(v_{1}) ≡ 1(mod J).

3.2. The ring of semi-stable periods Â_{st}. Let R be Fontaine’s ring and let x_{0}, ε ∈ R be the elements chosen in Subsection 2.1. Denote by A_{cr} the Fontaine crystalline ring. It is the p-adic closure of the divided power envelope of W (R) with respect to the ideal ([x_{0}] + p) of W (R), where [x_{0}] ∈ W (R) is the Teichmüller representative of x_{0}. Then for i ⩾ 0, Fil^{i}A_{cr} is the i-th divided power of the ideal ([x_{0}] + p) in A_{cr}. Denote by σ : A_{cr} ⟶ A_{cr} the natural morphism induced by the p-th power on R. Then for 0 ⩽ i < p, there are σ-linear maps ϕ_{i} = σ/p^{i} : Fil^{i}A_{cr} ⟶ A_{cr}. We shall often use the simpler notation φ = ϕ_{p−1} and F (A_{cr}) = Fil^{p−1}A_{cr}. Notice that A_{cr} is provided with the natural continuous Γ_{F} -action. Let X be an indeterminate. Then Â_{st} is the p-adic closure of the ring A_{cr}[γ_{i}(X) | i ⩾ 0] ⊂ A_{cr}[X] ⊗_{ℤ_{p}} ℚ_{p}, where for all i ⩾ 0, γ_{i}(X) = X^{i}/i!. The ring Â_{st} has the following additional structures:

• the S-module structure given by the natural W (k)-algebra structure and the correspondence u ↦ [x_{0}]/(1 + X);

• the ring endomorphism σ, which is the extension of the above defined endomorphism σ of A_{cr} via the condition σ(X) = (1 + X)^{p} − 1;

• the continuous A_{cr}-derivation N : Â_{st} ⟶ Â_{st} such that N(X) = X + 1;

• for any i ⩾ 0, the ideal Fil^{i}Â_{st}, which is the closure of the ideal ∑_{i_{1}+i_{2}⩾i} (Fil^{i_{1}}A_{cr}) γ_{i_{2}}(X);

• the action of Γ_{F}, which is the extension of the Γ_{F} -action on A_{cr} such that for all τ ∈ Γ_{F}, τ(X) = [ε]^{k(τ)}(X + 1) − 1. Here all k(τ) ∈ ℤ_{p} are such that τ(x_{0}) = ε^{k(τ)}x_{0}.

Note that for 0 ⩽ m < p, σ(Fil^{m}Â_{st}) ⊂ p^{m}Â_{st} and, as earlier, we can set ϕ_{m} = p^{−m}σ|_{Fil_{m}Â_{st}} and introduce the simpler notation φ = ϕ_{p−1} and F (Â_{st}) = Fil^{p−1}Â_{st}.

3.3. Construction of semi-stable representations of Γ_{F} with weights from [0, p). For 0 ⩽ m < p, consider the category S̃_{m} of quadruples M = (M, Fil^{m}M, ϕ_{m}, N), where Fil^{m}M ⊂ M are S-modules, ϕ_{m} : Fil^{m}M ⟶ M is a σ-linear map and N : M ⟶ M is a W (k)-linear endomorphism such that for any s ∈ S and m ∈ M, N(sx) = N(s)x + sN(x) The morphisms of the category S̃_{m} are S-linear morphisms of filtered modules commuting with the corresponding morphisms ϕ_{m} and N. Notice that for 0 ⩽ m < p, Â_{st} has a natural structure of the object of the category S̃_{m}. As earlier, we shall use the simpler notation φ = ϕ_{p−1} and F (M) = Fil^{p−1}M. For 0 ⩽ m < p, the Breuil category S_{m} of strongly divisible S-modules of weight ⩽ m is a full subcategory of S̃_{m} consisting of the objects M = (M, Fil^{m}M, ϕ_{m}, N) such that

(1) M is a free S-module of finite rank;

(2) (Fil^{m}S)M ⊂ Fil^{m}M;

(3) (Fil^{m}M) ∩ pM = pFil^{m}M;

(4) ϕ_{m}(Fil^{m}M) spans M over S;

(5) Nϕ_{m} = pϕ_{m}N;

(6) (Fil^{1}S)N(Fil^{m}M) ⊂ Fil^{m}M.

For M ∈ S_{m}, let T_{st}^{∗}(M) be the Γ_{F} -module of all S-linear and commuting with ϕ_{m} and N, maps f : M ⟶ Â_{st} such that f(Fil^{m}M) ⊂ Fil^{m}Â_{st}. Then one has the following two basic facts:

• T_{st}^{∗}(M) is a continuous ℤ_{p}[Γ_{F}]-module without p-torsion, its ℤ_{p}-rank equals rk_{S}M, and V_{st}^{∗}(M) = T_{st}^{∗}(M) ⊗_{ℤ_{p}} ℚ_{p} is semi-stable Γ_{F} -module with Hodge-Tate weights from [0, m];

• any semi-stable representation of Γ_{F} with Hodge-Tate weights from [0, m], 0 ⩽ m < p, appears in the form V_{st}^{∗}(M) for a suitable M ∈ S_{m}.

By Theorem 1.3 [6] these facts follow from the existence of strongly divisible lattices in S ⊗_{W} F -modules associated with weakly admissible (ϕ_{0}, N)-modules with filtration of length m. Breuil proved this for all m ⩽ p − 2 but his method can be easily extended to cover the case m = p − 1 as well, cf. also [7].

3.4. The category \underline{L}^{f}. In this section we introduce W-analogues of Breuil’s S-modules from the category S_{p−1} and prove that they can be also used to construct semi-stable representations of Γ_{F} with Hodge-Tate weights from [0, p).

Definition. Let ˜ \underline{L} be the category of L = (L, F (L), φ, N_{S}), where L ⊃ F (L) are W-modules, φ : F (L) ⟶ L is a σ-linear morphism of W-modules and N_{S} : L ⟶ L_{S} := L ⊗_{W} S is such that for all w ∈ W and l ∈ L, N_{S}(wl) = N(w)l + (w ⊗ 1)N_{S}(l). For L_{1} = (L_{1}, F (L_{1}), φ, N_{S}) ∈ ˜ \underline{L}, the morphisms Hom˜\underline{L}(L, L_{1}) are W-linear f : L ⟶ L_{1} such that f(F (L)) ⊂ F (L_{1}), fφ = φf and fN_{S} = N_{S}(f ⊗ 1).

Let A_{st} = (Â_{st}, F (Â_{st}), φ, N_{S}), where N_{S} = N ⊗ 1. Then A_{st} is an object of the category ˜ \underline{L}.

Suppose L = (L, F (L), φ, N_{S}) ∈ ˜ \underline{L}. Set L_{S} := L ⊗_{W} S, F (L_{S}) = (F (L) ⊗ 1)S + (L ⊗ 1)Fil^{p}S, and φ_{S} : F (L_{S}) ⟶ F (L_{S}) is a unique σ-linear map such that φ_{S}|_{F (L)⊗1} = φ ⊗ 1 and for any s ∈ Fil^{p}S and l ∈ L, φ_{S}(l ⊗ s) = (φ(v^{p−1}l) ⊗ 1)φ(s)/φ(v^{p−1}).

Definition. Denote by \underline{L}^{f} the full subcategory in ˜ \underline{L} consisting of the quadruples L = (L, F (L), φ, N_{S}) such that

• L is a free W-module of finite rank;

• v^{p−1}L ⊂ F (L), F (L) ∩ pL = pF (L) and L = φ(F (L)) ⊗_{σW} W;

• for any l ∈ F (L), vN_{S}(l) ∈ F (L_{S}) and φ_{S}(vN(l)) = cN_{S}(φ(l)), where c = 1 + u^{p}/p.

It can be easily seen that for L = (L, F (L), φ, N_{S}) ∈ \underline{L}^{f} and the map N = N_{S} ⊗ 1 : L_{S} ⟶ L_{S}, the quadruple L_{S} = (L_{S}, F (L_{S}), φ_{S}, N) is the object of the category S_{p−1} The main result of this Subsection is the following statement.

Proposition 3.1. For any M = (M, F (M), φ, N) ∈ S_{p−1}, there is an L = (L, F (L), φ, N_{S}) ∈ \underline{L}^{f} such that M = L_{S}.

Corollary 3.2. a) If L ∈ \underline{L}^{f} and T_{st}^{∗}(L) = Hom˜\underline{L}(L, Â_{st}) with the induced structure of ℤ_{p}[Γ_{F}]-module then V_{st}^{∗}(L) = T_{st}^{∗}(L) ⊗_{ℤ_{p}} ℚ_{p} is a semi-stable ℚ_{p}[Γ_{F}]-module with Hodge-Tate weights from [0, p) and dim_{ℚ_{p}} V_{st}^{∗}(L) = rk_{W}L.

b) For any semi-stable ℚ_{p}[Γ_{F}]-module V_{st}^{∗} with Hodge-Tate weights from [0, p), there is an L ∈ \underline{L}^{f} such that V_{st}^{∗} ≃ V_{st}^{∗}(L).

Proof of Proposition 3.1. Let d be a rank of M over S. If L ⊂ M is a free W-submodule of rank d and M is generated by the elements of L over S we say that L is W-structural (with respect to M). Let F (L) = F (M) ∩ L.

Lemma 3.3. If L is W-structural for M then

a) F (L) ⊃ v^{p−1}L;

b) F (L) ∩ pL = pF (L);

c) F (L) is a free W-module of rank d.

Proof. a) v^{p−1}L ⊂ (Fil^{p−1}S)M ∩ L ⊂ F (M) ∩ L = F (L).

b) F (L)∩pL = L∩F (M)∩pL = F (M)∩pL = F (M)∩pM ∩pL = pF (M)∩pL = pF (L).

c) F (L) has no p-torsion. Therefore, it will be sufficient to prove that F (L)/pF (L) is a free k[[u]]-module of rank d. Consider the following natural embeddings of k[[v]]-modules

L/pL ⊃ F (L)/pF (L) ⊃ v^{p−1}L/pv^{p−1}L ≃ L/pL (Use b) and that pL ∩ v^{p−1}L = pv^{p−1}L.) It remains to note that L/pL is free of rank d over k[[v]]. The Lemma is proved. □

Suppose L is W-structural for M.

Lemma 3.4. If L is W-structural then φ(F (L)) spans M over S.

Proof. The equality S = W + Fil^{p}S implies that M = L + (Fil^{p}S)L = L + (Fil^{p}S)M. Therefore,

F (M) = F (M) ∩ L + (Fil^{p}S)M = F (L) + (Fil^{p}S)L

(use that F (M) ⊃ (Fil^{p}S)M) and in notation of Subsection 3.1 one has

F (M) = F (L) + v_{1}L + JM.

This implies that φ(F (L)), φ(v_{1}L) and φ(JM) span M over S. But for any l ∈ L, φ(v_{1}l) = φ(v_{1})φ(v^{p−1}l)/φ(v^{p−1}) = (1 − v_{1})^{−1}φ(v^{p−1}l) ≡ φ(v^{p−1}l)mod m_{S}M. For similar reasons, φ(JM) ⊂ pM ⊂ m_{S}M. This means that φ(F (L)) spans M modulo m_{S}M. The lemma is proved. □

By above lemma it remains to prove the existence of a W-structural L for M such that φ(F (L)) ⊂ L. Let ϕ_{0} be a σ-linear endomorphism of the S-module M ∈ S_{p−1} such that for all m ∈ M, ϕ_{0}(m) = φ(v^{p−1}m)/φ(v^{p−1}). Clearly, ϕ_{0}(m_{S}M) ⊂ m_{S}M and, therefore, it induces a σ-linear endomorphism σ_{0} of the k-vector space M_{k} = M/m_{S}M.

Lemma 3.5. Suppose n ∈ ℤ_{⩾0}, L is W-structural and φ(F (L)) ⊂ L + p^{n}M. Then there is a W-structural L′ for M such that φ(F (L′)) ⊂ L′ + p^{n}JM.

Proof. Denote by F (L)_{k} the image of F (L) in the k-vector space M/m_{S}M = L/(m_{S} ∩ W)L = L_{k}. Let s = dim_{k} F (L)_{k}, then s ⩽ d = dim_{k} L_{k}. Choose a W-basis e^{(1)}, …, e^{(d)} of L and a W-basis f^{(1)}, …, f^{(d)} of F (L) such that

• for 1 ⩽ i ⩽ s, f^{(i)} = e^{(i)} and for s < i ⩽ d, f^{(i)} ∈ vL.

It will be convenient to use the following vector notation: ē = (ē_{1}, ē_{2}), where ē_{1} = (e^{(1)}, …, e^{(s)}) and ē_{2} = (e^{(s+1)}, …, e^{(d)}), and f̄ = (f̄_{1}, f̄_{2}), where f̄_{1} = ē_{1} and f̄_{2} = (f^{(s+1)}, …, f^{(d)}). Then in obvious notation one has (φ(f̄_{1}), φ(f̄_{2})) = (ē_{1}, ē_{2})C, where C ∈ GL_{d}(S). Clearly, C ≡ C_{0} + p^{n}v_{1}C_{1} mod p^{n}J with C_{0} ∈ GL_{d}(W) and C_{1} ∈ M_{d}(W). Clearly, φ(F (L)) ⊂ L + p^{n}JM iff C_{1} ≡ 0 mod m_{S}. Choose ḡ = (ḡ_{1}, ḡ_{2}) ∈ L^{d} and set

ē′_{1} = (e^{′(1)}, …, e^{′(s)}) = ē_{1} + p^{n}(v_{1} − v^{p−1})ḡ_{1}
ē′_{2} = (e^{′(s+1)}, …, e^{′(d)}) = ē_{2} + p^{n}(v_{1} − v^{p−1})ḡ_{2}

Clearly, the coordinates of ē′ = (ē′_{1}, ē′_{2}) give an S-basis of M and we can introduce the structural W-module L′ = ∑_{i} We^{′(i)} for M.

Prove that the elements e^{′(i)}, 1 ⩽ i ⩽ s, and f^{(i)}, s < i ⩽ d, generate F (L′) mod p^{n}JM. Indeed, we have

(3.1)  L + p^{n}IM = L′ + p^{n}IM

and this implies that the image F (L)_{k} of F (L) in L_{k} coincides with its analogue F (L′)_{k}. In addition, for 1 ⩽ i ⩽ s,

e^{′(i)} ∈ L′ ∩ (F (L) + p^{n}IM) ⊂ L′ ∩ F (M) = F (L′).

Therefore, it would be sufficient to prove that (vL′) ∩ F (L′) mod p^{n}JM is generated by the images of ve^{′(i)}, 1 ⩽ i ⩽ s, and f^{(s+1)}, …, f^{(d)}. But relation (3.1) implies that vL + p^{n}JM = vL′ + p^{n}JM and

(vL′) ∩ F (L′) mod p^{n}JM = (vL) ∩ F (L) mod p^{n}JM.

It remains to note that for 1 ⩽ i ⩽ s, ve^{′(i)} ≡ ve^{(i)} mod p^{n}JM. Therefore, we can define special bases for L′ and F (L′) by the relations f̄′_{1} = ē′_{1} and f̄′_{2} = f̄_{2} and obtain that

(φ(f̄′_{1}), φ(f̄′_{2})) = (φ(f̄_{1}), φ(f̄_{2})) + p^{n}v_{1}(σ_{0}ḡ_{1}, ¯0) mod p^{n}JM

and

(φ(f̄′_{1}), φ(f̄′_{2})) ≡ (ē′_{1}, ē′_{2})C_{0} + p^{n}v^{p−1}(ḡ_{1}, ḡ_{2})C_{0}+
+p^{n}v_{1}((ē_{1}, ē_{2})C_{1} − (ḡ_{1}, ḡ_{2})C_{0} + (σḡ_{1}, ¯0)) mod p^{n}JM

So, φ(F (L′)) ⊂ L′ + p^{n}JM if and only if there is an ḡ = (ḡ_{1}, ḡ_{2}) ∈ L^{d} such that (σ_{0}ḡ_{1}, ¯0) ≡ (ḡ_{1}, ḡ_{2})C_{0} + h̄ mod (m_{S} ∩ W)L, where h̄ = (ē_{1}, ē_{2})C_{1} ∈ L and C_{0} mod m_{S} ∈ GL_{d}(k). The existence of such vector ḡ is implied by Lemma 3.6 below. □

Lemma 3.6. Suppose V is a d-dimensional vector space over k with a σ-linear endomorphism σ_{0} : V ⟶ V and ā = (ā_{1}, ā_{2}) ∈ V^{d}, where ā_{1} ∈ V^{s} and ā_{2} ∈ V^{d−s}. Then for any C ∈ GL_{d}(k) there is an ḡ = (ḡ_{1}, ḡ_{2}) ∈ V^{d} with ḡ_{1} ∈ V^{s} and ḡ_{2} ∈ V^{d−s} such that

(3.2)  (σ_{0}ḡ_{1}, ¯0) = ḡC + ā.

Proof. Let C^{−1} = (\begin{matrix}D_{11} & D_{12} \\ D_{21} & D_{22} \end{matrix}) with the block matrices of sizes s×s, (d−s)×s,

s × (d − s) and (d − s) × (d − s). Then the equality (3.2) can be rewritten as

\begin{aligned}(σ_{0}ḡ_{1})D_{11} & = & ḡ_{1} + ā′_{1} \\ (σ_{0}ḡ_{1})D_{21} & = & ḡ_{2} + ā′_{2} \end{aligned}

where (ā′_{1}, ā′_{2}) = āC^{−1}. Clearly, it will be sufficient to solve the first equation in ḡ_{1}, but this is a special case of Lemma 1.1. □

Lemma 3.7. Suppose n ⩾ 0 and L is W-structural for M such that φ(F (L)) ⊂ L + p^{n}JM. Then there is a W-structural L′ for M such that φ(F (L′)) ⊂ L′ + p^{n+1}M.

Proof. Suppose the coordinates of ē ∈ M^{d} form a W-basis of L and D ∈ M_{d}(W) is such that the coordinates of f̄ = ēD form a W-basis of F (L). Then φ(f̄) = ē + p^{n}h̄, where h̄ ≡ ¯0 mod JM. Let ē′ = ē + p^{n}h̄ and let L′ be a Wsubmodule in M spanned by the coordinates of ē′. Clearly, L′ is W-structural. Prove that F (L′) is spanned by the coordinates of ē′D. Indeed, suppose ē and ē′ have the coordinates e^{(i)} and, resp., e^{′(i)}, 1 ⩽ i ⩽ s. Then for all i, e^{′(i)} = e^{(i)} +p^{n}h^{(i)}, where h^{(i)} ∈ JM ⊂ (Fil^{p}S)M. This means that a W-linear combination of e^{(i)} belongs to F (M) if and only if the same linear combination of e^{′(i)} belongs to F (M). This implies that ē′D spans F (L′) over W because ēD spans F (L) over W. Then φ(F (L′)) ⊂ L′ + p^{n+1}M because φ(h̄) ∈ pM (use that φ(J) ⊂ pS) and

φ(ē′D) = φ(ēD + p^{n}h̄D) = ē + p^{n}h̄ + p^{n}φ(h̄)σ(D) ≡ ē′modp^{n+1}M
□

It remains to notice that applying above Lemmas 3.6 and 3.7 one after another we shall obtain a sequence of W-structural modules L_{n} such that for all n ⩾ 0,

L_{n} +p^{n+1}M = L_{n+1} +p^{n+1}M, where L_{0} ⊗_{W} S = M. Therefore, L = \underleftarrow{lim}_{n}L_{n}/p^{n}

is W-structural and φ(L) ⊂ L. The proposition is completely proved. □

3.5. The categories \underline{L}^{t} and \underline{L}^{ft}.

Definition. W-module L is p-strict if it is isomorphic to ⊕_{1⩽i⩽s}W/p^{n_{i}}, where n_{1}, …, n_{s} ∈ ℕ.

In particular, if L is p-strict and pL = 0 then L is a free W_{1}-module. The p-strict modules can be efficiently studied via devissage due to the following property.

Lemma 3.8. L is p-strict if and only if pL and L/pL are p-strict.

Proof. Specify Breuil’s proof of a similar statement but for more complicated ring S = W^{DP} from [6]. □

Definition. Denote by \underline{L}^{t} the full subcategory in ˜ \underline{L} consisting of the quadruples L = (L, F (L), φ, N_{S}) such that

• L is p-strict;

• v^{p−1}L ⊂ F (L), F (L) ∩ pL = pF (L) and L = φ(F (L)) ⊗_{σW} W;

• for any l ∈ F (L), vN_{S}(l) ∈ F (L_{S}) and φ_{S}(vN_{S}(l)) = cN_{S}(φ(l)), where c = 1 + u^{p}/p.

Definition. Denote by \underline{L}^{t}[1] the full subcategory in \underline{L}^{t}, which consists of objects killed by p.

The category \underline{L}^{t}[1] is not very far from the category \underline{L}^{∗} introduced in Section 1. Indeed, suppose L = (L, F (L), φ, N_{S}) ∈ \underline{L}^{t}[1]. Note that N_{S}(L) ⊂ L_{S_{1}} := L ⊗_{W_{1}} S_{1} = L/u^{p}L ⊕ (L ⊗ 1)Fil^{p}S_{1}. (Remind that S_{1} = S/pS = W_{1}/u^{p}W_{1} ⊕ Fil^{p}S_{1}.) With this notation we have the following property.

Proposition 3.9. There is a unique N : L ⟶ L/u^{2p} such that

a) for any l ∈ L, N(l) ⊗ 1 = cN_{S}(l) in L_{S_{1}}, where c = 1 + u^{p}/p ∈ S^{∗};

b) (L, F (L), φ, N) ∈ \underline{L}^{∗}.

Proof. Let N_{1} := cN_{S} : L ⟶ L_{S_{1}}. Then for any w ∈ W_{1} and l ∈ L, one has N_{1}(wl) = N(w)l+wN_{1}(l) (use that N(c) = 0 in S_{1}) and there is a commutative diagram (use that σ(c) = 1 in S_{1})

\xymatrix{F (L) \ar[r]^{φ} \ar[d]_{uN_{1}} & L \ar[d]^{N_{1}} \\ F (L)_{S} \ar[r]^{φ_{S}} & L_{S_{1}}}

Prove that N_{1}(φ(F (L)) ⊂ L/u^{p}L and, therefore, N_{1}(L) ⊂ L/u^{p}L. Indeed, (uN_{1})(F (L)) ⊂ uN_{1}(L) ∩ F (L)_{S} ⊂ (uL/u^{p}L ⊕ (uL)Fil^{p}S_{1}) ∩ (F (L)/u^{p}L ⊕ LFil^{p}S_{1}) ⊂ F (L)/u^{p}L ⊕ (uL)Fil^{p}S_{1}. This implies that N_{1}(φ(F (L)) ⊂ φ_{S}(uN_{1}(F (L))) ⊂ L/u^{p}L because φ_{S}(uFil^{p}S_{1}) = 0. So, by Proposition 1.3 there is a unique N : L ⟶ L/u^{2p} such that Nmod u^{p} = N_{1} and (L, F (L), φ, N) ∈ \underline{L}^{∗}. □

Corollary 3.10. With above notation the correspondence

(L, F (L), φ, N_{S}) ↦ (L, F (L), φ, N)

induces the equivalence of categories Π : \underline{L}^{t}[1] ⟶ \underline{L}^{∗}.

Proof. We must verify that our correspondence is surjective on objects and bijective on morphisms. The first holds because N_{S} = c^{−1}N mod u^{p} and the second — because a W_{1}-linear map f commutes with N iff it commutes with N mod u^{p} (use Proposition 1.2) iff f ⊗_{W_{1}} S_{1} commutes with N_{S}. □

Corollary 3.11. The category \underline{L}^{t} is preabelian.

Proof. Corollary 3.10 and Proposition 1.3 imply that \underline{L}^{t}[1] is pre-abelian. This can be extended then to the whole category \underline{L}^{t} by Breuil’s method from [6] via above Lemma 3.8. □

Note that if L = (L, F (L), φ, N_{S}) and M = (M, F (M), φ, N_{S}) are objects of \underline{L}^{t} and f ∈ Hom\underline{L}(L, M) then:

• Kerf = (K, F (K), φ, N_{S}), where K = Ker(f : L ⟶ M) and F (K) = F (L) ∩ K with induced φ and N_{S};

• Cokerf = (C, F (C), φ, N_{S}), where C = M/M′, M′ is equal to (f(L) ⊗_{W} W[u^{−1}]) ∩ M and F (C) = F (M)/(M′ ∩ F (M)) with induced φ and N_{S};

• f is strict monomorphic means that f : L ⟶ M is monomorphism of Wmodules, (f(L) ⊗_{W} W[u^{−1}]) ∩ M = f(L) (or, equivalently, M/f(L) is p-strict) and f(F (L)) = L ∩ F (M);

• f is strict epimorphic means that f is epimorphism of p-strict modules and f(F (L)) = F (M).

According to Appendix A, we can use the concept of p-divisible group \{L^{(n)}, i_{n}\}_{n⩾0} in \underline{L}^{t}. In this case L^{(n)} = (L_{n}, F (L_{n}), φ, N_{S}), where all L_{n} are free W/p^{n}-modules of the same rank equal to the height of this p-divisible group. We have obvious equivalence of the category \underline{L}^{f} and the category of p-divisible groups of finite height in \underline{L}^{t}.

Definition. Denote by \underline{L}^{ft} the full subcategory in \underline{L}^{t}, which consists of strict subobjects of p-divisible groups in \underline{L}^{t}. By \underline{L}^{ft}[1] we denote the full subcategory in \underline{L}^{ft} consisting of all objects killed by p.

It is easy to see that \underline{L}^{ft} contains all strict subquotients of the corresponding p-divisible groups. Contrary to the case of filtered modules coming from crystalline representations, the categories \underline{L}^{ft} and \underline{L}^{t} do not coincide but they have the same simple objects. Note that the functor Π from Corollary 3.10 identifies simple objects of the categories \underline{L}^{t} and \underline{L}^{∗} and for any two objects L_{1}, L_{2} ∈ \underline{L}^{t}[1], we have a natural isomorphism Ext_{\underline{L}^{t}[1]}(L_{1}, L_{2}) = Ext\underline{L}^{∗}(Π(L_{1}), Π(L_{2})). One can use the methods of Subsection 1.2 to extend the concepts of etale, connected, unipotent and multiplicative objects to the whole category \underline{L}^{t}. The starting point for this extension is the case of W (k)-modules, which is well-known from the classical Dieudonne theory [10]. Then we obtain the following standard properties:

• for any L ∈ \underline{L}^{t}, there are a unique maximal etale subobject (L^{et}, i^{et}) and a unique maximal connected quotient object (L^{c}, j^{c}) in \underline{L}^{t} such that the sequence

0 ⟶ L^{et} ⟶^{i^{et}} L ⟶^{j^{c}} L^{c} ⟶ 0 is exact and the correspondences L ↦ L^{et} and L ↦ L^{c} are functorial; if L ∈ \underline{L}^{ft} then L^{et} and L^{c} are also objects of \underline{L}^{ft};

• for any L ∈ \underline{L}^{t}, there are a unique maximal unipotent subobject (L^{u}, i^{u}) and a unique maximal multiplicative quotient object (L^{m}, j^{m}) in \underline{L}^{t} such that

the sequence 0 ⟶ L^{u} ⟶^{i^{u}} L ⟶^{j^{m}} L^{m} ⟶ 0 is exact and the correspondences L ↦ L^{u} and L ↦ L^{m} are functorial; if L ∈ \underline{L}^{ft} then L^{u} and L^{u} are also objects of \underline{L}^{ft}.

Denote by \underline{L}^{et,t}, \underline{L}^{c,t}, \underline{L}^{u,t} and \underline{L}^{m,t} the full subcategories in \underline{L}^{t} consisting of, resp., etale, connected, unipotent and multiplicative objects. We have also the corresponding full subcategories \underline{L}^{et,ft}, \underline{L}^{c,ft}, \underline{L}^{u,ft} and \underline{L}^{m,ft} in \underline{L}^{ft}. The results of Subsection 1.5 and Appendix A imply that in the category \underline{L}^{ft}:

• there is a unique etale p-divisible group L^{∞}(0) := {L^{(n)}(0), i_{n}}_{n⩾0} of height 1 such that L^{(1)}(0) = L(0);

• there is a unique multiplicative p-divisible group of height 1, L^{∞}(1) := {L^{(n)}(1), i_{n}}_{n⩾0} such that L^{(1)}(1) = L(1);

• for any p-divisible group L^{∞} there are functorial exact sequences of p-divisible groups

0 ⟶ L^{∞,et} ⟶ L^{∞} ⟶ L^{∞,c} ⟶ 0

0 ⟶ L^{∞,u} ⟶ L^{∞} ⟶ L^{∞,m} ⟶ 0

Here L^{∞,et} and L^{∞,m} are products of several copies of L^{∞}(0) and, resp., L^{∞}(1), and L^{∞,c} and L^{∞,u} are p-divisible groups in the categories \underline{L}^{c,ft} and, resp., \underline{L}^{u,ft}.

4. Semistable modular representations with weights [0, p)

In this section we prove that all killed by p subquotients of Galois invariant lattices of semistable ℚ_{p}[Γ_{F}]-modules with Hodge-Tate weights [0, p) can be obtained via the functor V^{∗} from Section 2.

4.1. The functor V^{t} : \underline{L}^{t} ⟶ \underline{MΓ}_{F}. For n ⩾ 1, introduce the objects A_{st,n} = (Â_{st,n}, F (Â_{st.n}), φ, N_{S}) of the category ˜ \underline{L}, with Â_{st,n} = Â_{st}/p^{n}Â_{st}, F (Â_{st,n}) = F (Â_{st})/p^{n}F (Â_{st}) and induced φ and N_{S}. Let A_{st,∞} = (A_{st,∞}, F (A_{st,∞}), φ, N_{S}) be the inductive limit of all A_{st,n}. For L ∈ \underline{L}^{t}, set V^{t}(L) = Hom˜\underline{L}(L, A_{st,∞}) with the induced structure of Γ_{F} - module. This gives the functor V^{t} : \underline{L}^{t} ⟶ \underline{MΓ}_{F}. We shall use the same notation for its restriction to the category \underline{L}^{ft}.

Proposition 4.1. Suppose L = (L, F (L), φ, N_{S}) ∈ \underline{L}^{t}. Then N_{S}|_{φ(F (L))} is nilpotent.

By devissage and Corollary 3.10 this is implied by the following statement for the objects of the category \underline{L}^{∗}.

Lemma 4.2. If L = (L, F (L), φ, N) ∈ \underline{L}^{∗} then N^{p}(φ(F (L)) ⊂ u^{p}L.

Proof. For any l ∈ F (L), N(φ(l)) = φ(uN(l)). Use induction to prove that for 1 ⩽ m ⩽ p, N^{m}(φ(l)) ≡ φ(u^{m}N^{m}(l))mod u^{p}L and use then that φ(u^{p}N^{p}(l)) ∈ φ(uF (L)) ⊂ u^{p}L. □

Proposition 4.3. For n ⩾ 1, ⊕_{j⩾0}A_{cr,n}γ_{j}(log(1 + X)) is the maximal W (k)- submodule of Â_{st,n} where N is nilpotent.

Proof. For any j ⩾ 1, one has N(γ_{j}(log(1 + X)) = γ_{j−1}(log(1 + X)) and N is nilpotent on ⊕_{j⩾0}A_{cr,n}γ_{j}(log(1 + X)). Therefore, it will be sufficient to prove that

Ker (N^{p}|Â_{st,1}) = ⊕_{0⩽j<p}A_{cr,1}γ_{j}(log(1 + X)).

Let C = 𝔽_{p}⟨X⟩ be the divided power envelope of 𝔽_{p}[X] with respect to the ideal (X). Then C = 𝔽_{p}[X_{0}, X_{1}, …, X_{n}, …]_{<p} is the ring of polynomials in X_{i} := γ_{p^{i}}(X), where for all i ⩾ 0, X_{i}^{p} = 0. Let m_{C} be the maximal ideal of C and Y = log(1 + X) ∈ C. Then Y ≡ X_{0} − X_{1} mod m_{C}^{2} and for all j ⩾ 0, γ_{p^{j}}(Y) ≡ X_{j} − X_{j+1} mod m_{C}^{2}. This implies that with Y_{j} = γ_{j}(Y) for all j ⩾ 0,

C = 𝔽_{p}[X_{0}, Y_{0}, …, Y_{n}, …]_{<p} = 𝔽_{p}⟨Y ⟩[X]_{<p} = ⊕_{0⩽i<p}𝔽_{p}⟨Y ⟩γ_{i}(X).

So, Â_{st,1} = ⊕_{\begin{matrix}j⩾0 \\ 0⩽i<p \end{matrix}}A_{cr,1}γ_{i}(X)γ_{j}(Y). Remind N(X) = X + 1 and for j ⩾ 1,

N(γ_{j}(Y)) = γ_{j−1}(Y). Using that N^{p} is an A_{cr,1}-derivation, N^{p}(X) = X + 1 and N^{p}(γ_{j+p}(Y)) = γ_{j}(Y), we obtain that for any P = ∑_{i,j} α_{ij}X^{i}γ_{j}(Y) ∈ 𝔽_{p}⟨Y ⟩[X]_{<p} with α_{ij} ∈ 𝔽_{p},

N^{p}(P) = ∑_{i,j} α_{ij}iX^{i}γ_{j}(Y) + ∑_{i,j}(i + 1)α_{i+1,j}γ_{j}(Y)X^{i} + ∑_{i,j} α_{i,j+p}X^{i}γ_{j}(Y).

If P ∈ KerN^{p} then for all involved indices i, j,

iα_{ij} + (i + 1)α_{i+1,j} + α_{i,j+p} = 0.

This implies that α_{ij} = 0 if either i ≠ 0 or j ⩾ p. Indeed, take i = p − 1. Then −α_{p−1,j} + α_{p−1,j+p} = 0. Because for j ≫ 0, α_{p−1,j} = 0 it implies that all α_{p−1,j} = 0. Then proceed similarly with i = p − 2 and so on. This proves that all α_{ij} = 0 if i ≠ 0. It remains to note that for i = 0, our relations give α_{0,j+p} = 0 for all j ⩾ 0. □

As earlier, consider the category ˜ \underline{L}_{0}. Remind that its objects are the triples (L, F (L), φ), where L ⊃ F (L) are W-modules and φ : F (L) ⟶ L is a σ-linear morphism. For any object L = (L, F (L), φ, N_{S}) ∈ ˜ \underline{L}, agree to use the same notation L for the corresponding object (L, F (L), φ) ∈ ˜ \underline{L}_{0}. For all n ⩾ 0, set A_{cr,n} = (A_{cr,n}, F (A_{cr,n}), φ) ∈ ˜ \underline{L}_{0} with A_{cr,n} = A_{cr}/p^{n}A_{cr}, F (A_{cr,n}) = F (A_{cr})/p^{n}F (A_{cr}) and induced φ. Here the W-module structure on A_{cr,n} is defined by the morphism of W (k)-algebras W ⟶ A_{cr,n} such that u ↦ [x_{0}]. Denote by A_{cr,∞} the inductive limit of all A_{cr,n}. Suppose L ∈ \underline{L}^{t} and f ∈ Hom˜\underline{L}(L, A_{st,n}). Then by Propositions 4.1 and 4.3,

f(φ(F (L))) ⊂ ⊕_{j⩾0}A_{cr,n}γ_{j}(log(1 + X)).

Consider the formal embedding of the algebra A_{st,n} into the completion ∏_{j⩾0} A_{cr,n}γ_{j}(log(1 + X)) of ⊕_{j⩾0}A_{cr,n}γ_{j}(log(1 + X)) such that X ↦ ∑_{j⩾1} γ_{j}(log(1+X)). Then any element of A_{st,n} can be uniquely written in the form ∑_{j⩾0} a_{j}γ_{j}(log(1 + X)), where all a_{j} ∈ A_{cr,n}. Note that the W-module structure on A_{st,n} is given via the map

u ↦ [x_{0}]/(1 + X) = [x_{0}] ∑_{j⩾0}(−1)^{j}γ_{j}(log(1 + X)).

For j ⩾ 0, introduce the W (k)-linear maps f_{j} ∈ Hom(L, A_{cr,n}) such that for any l ∈ L, one has f(l) = ∑_{j⩾0} f_{j}(l)γ_{j}(log(1 + X)). Then using methods from [6] obtain the following property.

Proposition 4.4. a) The correspondence f ↦ f_{0} induces isomorphism of abelian groups V^{t}(L) = Hom_{˜\underline{L}_{0}}(L, A_{cr,n});

b) for any j ⩾ 0 and l ∈ L, f_{j}(l) = f_{0}(N^{j}(l)).

Corollary 4.5. The functor V^{t} is exact.

Proof. Let \underline{L}_{0}^{t} be the full subcategory of ˜ \underline{L}_{0} consisting of the triples (L, F (L), φ) coming from all L = (L, F (L), φ, N) ∈ \underline{L}^{t}. By Proposition 4.4 it will be sufficient to prove that the functor V_{0}^{t} : \underline{L}_{0}^{t} ⟶ (Ab), such that V_{0}^{t}(L) = Hom_{˜\underline{L}_{0}}(L, A_{cr,∞}), is exact. The verification can be done by devissage along the lines of paper [13]. □

Remark. One can simplify the verification of above corollary by replacing A_{cr,1} by the corresponding object Ã_{cr,1} related to the module Ã_{cr,1} = (R/x_{0}^{p})T_{1} ⊕ (R/x_{0}^{p}) introduced in Subsection 4.2 below.

Corollary 4.6. For L ∈ \underline{L}^{f}, let \{L^{(n)}, i_{n}\}_{n⩾0} be the corresponding p-divisible group in the category \underline{L}^{ft}. Then in notation of Corollary 3.2, T_{st}^{∗}(L) =

\underleftarrow{lim}_{n}V^{t}(L^{(n)}).

4.2. The functor V[1]^{∗}. Note the following case of Proposition 4.4.

Proposition 4.7. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{t}[1]. Then there is an isomorphism of abelian groups V^{t}(L) ≃ Hom_{˜\underline{L}_{0}}(L, A_{cr,1}). In addition, Γ_{F} acts on V^{t}(L) via its natural action on A_{st,1} and the identification ι_{L} : Hom_{˜\underline{L}_{0}}(L, A_{cr,1}) ⟶ Hom˜\underline{L}(L, A_{st,1}) such that if f_{0} ∈ Hom_{˜\underline{L}_{0}}(L, A_{cr,1}) then for any l ∈ L,

ι_{L}(f_{0})(l) = ∑_{j⩾0} f_{0}(N^{j}(l))γ_{j}(log(1 + X))

Introduce the functor V[1]^{∗} := V^{t}|_{\underline{L}^{t}[1]}∘Π^{−1} : \underline{L}^{∗} ⟶ \underline{MΓ}_{F}, where Π : \underline{L}^{t}[1] ⟶ \underline{L}^{∗} is the equivalence of categories from Corollary 3.10.

Proposition 4.8. On the subcategory of unipotent objects \underline{L}^{∗u} of \underline{L}^{∗} the functors V[1]^{∗} and V^{∗} coincide.

Proof. The definition of A_{cr} implies that A_{cr,1} = (R/x_{0}^{p})[T_{1}, T_{2}, …]_{<p}, where for all indices i ⩾ 1, T_{i} comes from γ_{p^{i}}([x_{0}] + p) and T_{i}^{p} = 0. Set F (A_{cr,1}) = Fil^{p−1}A_{cr,1} = (x_{0}^{p−1}R/x_{0}^{p}R) ⊕ (R/x_{0}^{p})I_{1}, where the ideal I_{1} is generated by all T_{i}. Then the corresponding map φ : F (A_{cr,1}) ⟶ A_{cr,1} is uniquely determined by the conditions φ(x_{0}^{p−1}) = 1 − T_{1}, φ(T_{1}) = 1 and φ(T_{i}) = 0 if i ⩾ 2. In particular, φ(A_{cr,1}) ⊂ (R/x_{0}^{p})T_{1} ⊕ (R/x_{0}^{p}). Let Ã_{cr,1} = A_{cr,1}/J_{1} with the induced structure of filtered φ-module Ã_{cr,1}, where the ideal J_{1} of A_{cr,1} is generated by the elements T_{1}x_{0}^{p} and T_{i} with i ⩾ 2. Then the projection A_{cr,1} ⟶ Ã_{cr,1} induces for any object L = (L, F (L), φ, N) of the category \underline{L}^{∗}, the identification (use that φ|_{J} = 0)

Hom_{˜\underline{L}_{0}}(L, A_{cr,1}) = Hom_{˜\underline{L}_{0}}(L, Ã_{cr,1}).

Introduce a_{0}, a_{−1} ∈ Hom(L, R/x_{0}^{p}) such that for any m ∈ L, f_{0}(m) = a_{−1}(m)T_{1} + a_{0}(m). Note that a_{0} and a_{−1} are W_{1}-linear, where the multiplication by u on L correspondes to the multiplication by x_{0} in R/x_{0}^{p}. Then for any m ∈ F (L), the requirement f_{0}(φ(m)) = φ(f_{0}(m)) is equivalent to the conditions

(4.1)  \begin{aligned}a_{0}(φ(m)) & = & a_{−1}(m)^{p} + \frac{a_{0}(m)^{p}}{x_{0}^{p(p−1)}} \\ a_{−1}(φ(m)) & = & −\frac{a_{0}(m)^{p}}{x_{0}^{p(p−1)}} \end{aligned}

Note that these conditions depend only on m̄ = m mod u^{p}L.

Consider the operator V : L ⟶ L from Subsection 1.5. Clearly, V (u^{p}L) ⊂ uF (L) and for L̄ := L/u^{p}L, we obtain the induced operator V̄ : L̄ ⟶ L̄ (use that F (L)/uF (L) ⊂ L/u^{p}L). For any m ∈ L̄, relations (4.1) can be rewritten as follows:

\begin{aligned}a_{0}(m) & = & \frac{a_{0}(V̄ m)^{p}}{x_{0}^{p(p−1)}} + a_{−1}(V̄ m)^{p} \\ a_{−1}(m) & = & −\frac{a_{0}(V̄ m)^{p}}{x_{0}^{p(p−1)}} \end{aligned}

Therefore, if L is unipotent then for any m ∈ L̄,

a_{−1}(m) = −a_{0}(m) + a_{−1}(V̄ m)^{p} = −a_{0}(m) + a_{−1}(V̄^{2}m)^{p^{2}} = ⋯ = −a_{0}(m).

This implies that for any m ∈ F (L̄), a_{0}(φ(m)) = a_{0}(m)^{p}/x_{0}^{p(p−1)}. In other words, we have a natural identification

Hom_{˜\underline{L}_{0}}(L, R̃^{u}) = Hom_{˜\underline{L}_{0}}(L, Ã_{cr,1})

coming from the map of filtered φ-modules R̃^{u} ⟶ Ã_{cr,1} given by the Rlinear map R/x_{0}^{p} ⟶ Ã_{cr,1} = (R/x_{0}^{p})T_{1} ⊕ (R/x_{0}^{p}) such that for any r ∈ R/x_{0}^{p}, r ↦ (−rT_{1}, r). (For the definition of R̃ ∈ \underline{L}_{0}^{∗} cf. Subsection 2.2.) This implies that for all unipotent L ∈ \underline{L}^{∗u}, there is a natural identification of Γ_{F} -modules V[1]^{∗}(L) = V^{∗}(L). Indeed, the above embedding R/x_{0}^{p} ⟶ Ã_{cr,1} can be extended to the embedding of R_{st}/x_{0}^{p}R_{st} to

Ã_{st,1} = ∏_{j⩾0} Ã_{cr,1}γ_{j}(log(1 + X)),

which induces the above identification. □

4.3. Splittings Θ and Θ̃. Suppose L = (L, F (L), φ, N) ∈ \underline{L}^{∗}. Then there is a standard short exact sequence

(4.2)  0 ⟶ L^{u} ⟶^{i} L ⟶^{j} L^{m} ⟶ 0,

where (L^{u}, i) is the maximal unipotent subobject and (L^{m}, j) is the maximal multiplicative quotient of L. If L^{m} = (L^{m}, F (L^{m}), φ, N) then F (L^{m}) = L^{m} = L_{0} ⊗_{𝔽_{p}} W_{1}, where L_{0} = {l ∈ L^{m} | φ(l) = l}. Suppose S : L^{m} ⟶ F (L) ⊂ L is a W_{1}-linear section. Then for any l_{0} ∈ L_{0}, S(l_{0}) = φ(S(l_{0})) + g(l_{0}), where g ∈ Hom(L_{0}, L^{u}). If S′ : L^{m} ⟶ F (L) is another W_{1}-linear section then for any l_{0} ∈ L_{0}, S′(l_{0}) = φ(S′(l_{0})) + g′(l_{0}). Here g′ ∈ Hom(L_{0}, L^{u}) is such that for some h ∈ Hom(L_{0}, L^{u}), one has

(g′ − g)(l_{0}) = h(l_{0}) − φ(h(l_{0})).

Proposition 4.9. a) There is a section S such that g(L_{0}) ⊂ uL^{u}.

b) If g(L_{0}), g′(L_{0}) ⊂ uL^{u} then h(L_{0}) ⊂ uF (L^{u}).

Proof. a) It will be sufficient to prove that for any l ∈ L^{u}, there is an h ∈ F (L^{u}) such that l ≡ h − φ(h) mod uL^{u}. Suppose n_{0} ⩾ 1 is such that V^{n_{0}}(L^{u}) ⊂ uF (L^{u}). Then for all n ⩾ n_{0}, V^{n}(L^{u}) ⊂ uF (L^{u}). Let h = −(V l+V^{2}l+⋯+V^{n_{0}+1}l). By the definition of the operator V for all 1 ⩽ i ⩽ n_{0} + 1, V^{i}l ∈ F (L^{u}) and φ(V^{i}l) ≡ V^{i−1}l mod uL^{u}. Therefore, h ∈ F (L^{u}) and φ(h) ≡ −(l + V l + ⋯ + V^{n_{0}}l) ≡ −l + h mod uL^{u}.

b) We must prove that if h ∈ F (L^{u}) and h − φ(h) ∈ uL^{u} then h ∈ uF (L^{u}). Indeed, we have V (h) − h ∈ V (uL^{u}) ⊂ uF (L^{u}) and for all n ⩾ 1, V^{n}(h) ≡ hmod uF (L^{u}) implies that h ∈ uL^{u}. Therefore, φ(h) ∈ uL^{u} and h ∈ uF (L^{u}).

□

Proposition 4.10. With above notation the short exact sequence

0 ⟶ V[1]^{∗}(L^{m}) ⟶ V[1]^{∗}(L) ⟶ V[1]^{∗}(L^{u}) ⟶ 0

obtained from (4.2) by applying V[1]^{∗}, has a canonical functorial splittings Θ : V[1]^{∗}(L^{u}) ⟶ V[1]^{∗}(L) and Θ̃ : V[1]^{∗}(L) ⟶ V[1)]^{∗}(L^{m}) in the category \underline{MΓ}_{F}.

Proof. It will be sufficient to prove the existence of a functorial splitting

Θ : Hom_{˜\underline{L}_{0}}(L^{u}, Ã_{cr,1}) ⟶ Hom_{˜\underline{L}_{0}}(L, Ã_{cr,1})

of the epimorphism Hom_{˜\underline{L}_{0}}(L, Ã_{cr,1}) → Hom_{˜\underline{L}_{0}}(L^{u}, Ã_{cr,1}), obtained from exact sequence (4.2). Suppose f_{0} = (a_{−1}, a_{0}) : L^{u} ⟶ (R/x_{0}^{p})T_{1} ⊕ (R/x_{0}^{p}) belongs to Hom_{˜\underline{L}_{0}}(L^{u}, Ã_{cr,1}). Here a_{−1}, a_{0} ∈ Hom_{W_{1}}(L^{u}, R/x_{0}^{p}) and for any l ∈ L^{u}, a_{−1}(l) = −a_{0}(l), cf. Subsection 4.2. Let S : L^{m} ⟶ L be a W_{1}-linear section such that for any l ∈ L_{0}, S(l_{0}) = φ(S(l_{0})) + g(l_{0}), where g ∈ Hom(L_{0}, uL^{u}). Extend f_{0} to Θ(f_{0}) = (a_{−1}, a_{0}) : L ⟶ (R/x_{0}^{p})T_{1} ⊕ (R/x_{0}^{p}) by setting a_{0}(S(l_{0})) = −a_{−1}(S(l_{0})) = X, where X is a unique element of R/x_{0}^{p} such that X − X^{p}/x_{0}^{p(p−1)} = a_{0}(g(l_{0})). One can prove that Θ(f_{0}) ∈ Hom_{˜\underline{L}_{0}}(L, Ã_{cr,1}) by verifying relations (4.1) with m = S(l_{0}). □

4.4. A modification of Breuil’s functor. Remind that Breuil’s functor V^{t} : \underline{L}^{t} ⟶ \underline{MΓ}_{F} attaches to any L ∈ \underline{L}^{t}, the Γ_{F} -module V(L) = Hom˜\underline{L}(L, A_{st,∞}).

Proposition 4.11. The functor V^{t} is fully faithful on the subcategory of unipotent objects \underline{L}^{t,u}.

Proof. Indeed, by Subsection 2.3, V[1]^{∗} is fully faithful. Then the exactitude of V^{t} together with Proposition 4.8 implies that V^{t}|\underline{L}^{u,t} is fully faithful. □

Proposition 4.10 implies that V^{t} is very far from to be fully faithful on the whole \underline{L}^{t}: if L ∈ \underline{L}^{t}[1] and 0 ⟶ L^{u} ⟶ L ⟶ L^{m} ⟶ 0 is the standard exact sequence then the corresponding sequence of Γ_{F} -modules admits a functorial splitting. Introduce a modification Ṽ^{ft} : \underline{L}^{ft} ⟶ \underline{MΓ}_{F} of Breuil’s functor. Suppose L ∈ \underline{L}^{ft}. From the definition of the category \underline{L}^{ft} in Subsection 3 it follows the existence of L′ ∈ \underline{L}^{ft} such that pL′ = L. More precisely, there are a strict monomorphism i_{L′} : L ⟶ L′ and a strict epimorphism j_{L′} : L′ ⟶ L such that p id_{L′} = i_{L′} ∘ j_{L′}. (Note that j_{L′} ∘ i_{L′} = p id_{L}.) Consider the following short exact sequences

(4.3)  0 ⟶ L ⟶^{i_{L′}} L′ ⟶^{C_{p}} _{p}L′ ⟶ 0
(4.4)  0 ⟶ L′_{p} ⟶^{K_{p}} L′ ⟶^{j_{L′}} L ⟶ 0

and consider the corresponding sequence of Γ_{F} -modules and their morphisms

V^{t}(_{p}L^{′u}) ⟶^{Θ} V^{t}(_{p}L′) ⟶^{V^{t}(C_{p})} V^{t}(L′) ⟶^{V^{t}(K_{p})} V^{t}(L′_{p}) \underrightarrow{Θ̃} V^{t}(L_{p}^{′m}).

As earlier, for any L ∈ \underline{L}^{ft}, L^{u} is the maximal unipotent subobject and L^{m} is the maximal multiplicative quotient object for L.

Lemma 4.12. Ker(Θ̃ ∘ V^{t}(K_{p})) ⊃ Im(V^{t}(C_{p}) ∘ Θ).

Proof. The section Θ depends functorially on objects of the category \underline{L}^{t}[1] ⊃ \underline{L}^{ft}[1]. Therefore, we have the following commutative diagram

\xymatrix{V^{t}(_{p}L^{′u}) \ar[r]^{V^{t}(C_{p}^{u}∘K_{p}^{u})} \ar[d]^{Θ} & V^{t}(L_{p}^{′u}) \ar[d]^{Θ} \\ V^{t}(_{p}L′) \ar[r]^{V^{t}(C_{p}∘K_{p})} & V^{t}(L′_{p})}

and Θ̃ ∘ V^{t}(K_{p}) ∘ V^{t}(C_{p}) ∘ Θ = (Θ̃ ∘ Θ) ∘ V^{t}(C_{p}^{u} ∘ K_{p}^{u}) = 0. □

Definition. Set V_{L′}^{t}(L) = Ker(Θ̃ ∘ V^{t}(K_{p}))/Im (V^{t}(C_{p}) ∘ Θ).

Proposition 4.13. With above notation one has:

a) V_{L′}^{t}(L) = Coker V^{t}(C_{p}) = V^{t}(L) if L ∈ \underline{L}^{u,ft};

b) V_{L′}^{t}(L) = Ker V^{t}(K_{p}) = V^{t}(L) if L ∈ \underline{L}^{m,ft};

c) for any L ∈ \underline{L}^{ft}, we have the induced exact sequence of Γ_{F} -modules 0 ⟶ V^{t}(L^{m}) ⟶ V_{L′}^{t}(L) ⟶ V^{t}(L^{u}) ⟶ 0. This sequence depends functorially on the pair (L, L′).

Proof. The parts a) and b) are obtained directly from definitions. In order to prove c), note that pL′ = L implies that pL^{′u} = L^{u} and pL^{′m} = L^{m}. This gives a functorial sequence

0 ⟶ V_{L^{′m}}^{t}(L^{m}) ⟶ V_{L′}^{t}(L) ⟶ V_{L^{′u}}^{t}(L^{u}) ⟶ 0.

Then standard diagram chasing proves that this sequence is exact. □

Proposition 4.14. Suppose for a given L ∈ \underline{L}^{ft}, the objects L′, L′′ ∈ \underline{L}^{ft} are such that pL′ = pL′′ = L. Then there is a natural isomorphism f(L′, L′′) of Γ_{F} -modules such that the following diagram is commutative

\xymatrix{0 \ar[r] & V^{t}(L^{m}) \ar[d]^{id} \ar[r] & V_{L′′}^{t}(L) \ar[d]^{f(L′,L′′)} \ar[r] & V^{t}(L^{u}) \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & V^{t}(L^{m}) \ar[r] & V_{L′}^{t}(L) \ar[r] & V^{t}(L^{u}) \ar[r] & 0}

(The lines of this diagram are given by Prop 4.13)

Proof. By replacing L′′ by L′∏_{L}L′′ with respect to strict epimorphisms j_{L′} and

j_{L′′}, we can assume that there is a map f : L′′ ⟶ L′ which induces the identity map pL′′ = L ⟶ pL′ = L. Then the existence of f(L′, L′′) follows from functoriality and diagram chasing implies that it induces the identity maps on V^{t}(L^{u}) and V^{t}(L^{m}). □

Definition. For L, L′ ∈ \underline{L}^{ft} such that pL′ = L, set Ṽ^{ft}(L) = V_{L′}^{t}(L).

The correspondence L ⟶ Ṽ^{ft}(L) induces the additive exact functor Ṽ^{ft} : \underline{L}^{ft} ⟶ \underline{MΓ}_{F}.

4.5. φ-filtered module Ã_{cr,2} ∈ ˜ \underline{L}_{0}. Let ξ = [x_{0}] + p ∈ W (R) ⊂ A_{cr}, and for n ∈ ℕ, γ_{n}(ξ) = ξ^{n}/n!

Lemma 4.15. If n ⩾ 2p then φ(γ_{n}(ξ)) ∈ p^{2}A_{cr}.

Proof. We have φ(γ_{n}(ξ)) = (p^{n−p+1}/n!)([x_{0}]^{p}/p + 1)^{n}. Therefore, it will be sufficient to verify that for n ⩾ 2p, v_{p}(n!) + p + 1 ⩽ n. Using the estimate v_{p}(n!) < n/(p − 1) we obtain that the required inequality holds for p ⩾ 5 if n ⩾ p + 3 and for p = 3 if n ⩾ 8. It remains to check that our inequality holds for p = 3 and n ∈ {6, 7}. □

Let J_{2} be the closed ideal in A_{cr} generated by [x_{0}]^{p}ξ^{p}/p and all ξ^{n}/n! with n ⩾ 2p. Then J_{2} ⊂ F (A_{cr}) and φ(J_{2}) ⊂ p^{2}A_{cr}. Introduce Ã_{cr,2} = A_{cr}/(J_{2} + p^{2}A_{cr}) and consider the corresponding induced filtered φ-module Ã_{cr,2} = (Ã_{cr,2}, F (Ã_{cr,2}), φ) ∈ ˜ \underline{L}_{0}. Clearly, for any L ∈ \underline{L}_{0}^{t}, the natural projection A_{cr,2} → Ã_{cr,2} induces the identification Hom_{˜\underline{L}_{0}}(L, A_{cr,2}) =

Hom_{˜\underline{L}_{0}}(L, Ã_{cr,2}).

Consider the structure of Ã_{cr,2} more closely.

Let T_{1} = ξ^{p}/p. With obvious notation the elements of Ã_{cr,2} can be written uniquely modulo the subgroup [x_{0}^{p}R]T_{1} + [x_{0}^{2p}R] + p[x_{0}^{p}R] + p^{2}W (R) in the form [r_{−1}]T_{1} + [r_{0}] + p[r_{1}], where r_{−1}, r_{0}, r_{1} ∈ R. Informally, we shall use that r_{−1}, r_{1} ∈ R/x_{0}^{p} and r_{0} ∈ R/x_{0}^{2p}. The W (R)-module structure on Ã_{cr,2} is induced by usual operations on Teichmuller’s representatives and the relation pT_{1} ≡ [x_{0}]^{p} mod p^{2}W (R). (Use that T_{1} ≡ [x_{0}]^{p}/p + p[x_{0}]^{p−1}mod p^{2}W (R).) The S-module structure on Ã_{cr,2} is induced by the W (k)-algebra morphism S ⟶ W (R) such that u ↦ [x_{0}]. Then F (Ã_{cr,2}) is generated over W (R) by the images of T_{1} and ξ^{p−1}. Note that ξ^{p−1} ≡ [x_{0}]^{p−1} − p[x_{0}]^{p−2} mod p^{2}W (R). The map φ : F (Ã_{cr,2}) ⟶ Ã_{cr,2} is uniquely determined by the knowledge of φ(T_{1}) and φ(ξ^{p−1}). Note that

φ(T_{1}) = (\frac{1 + [x_{0}]^{p}}{p} )^{p} ≡ 1 + [x_{0}]^{p} mod(J + p^{2}A_{cr,2} + p[m_{R}])

φ(ξ^{p−1}) = (1 + \frac{[x_{0}]^{p}}{p} )^{p−1} ≡ 1 − T_{1} mod (J + p^{2}A_{cr,2} + p[m_{R}])

Suppose L ∈ \underline{L}^{ft}[1] and L′ ∈ \underline{L}^{ft} is such that pL′ = L. Consider short exact sequences (4.3) and (4.4). Then the points f ∈ V^{t}(_{p}L′) and V^{t}(C_{p})(f) ∈ V^{t}(L′) are related via the commutative diagram

\xymatrix{L′ \ar[d]^{C_{p}} \ar[r]^{V^{t}(C_{p})(f)} & Ã_{cr,2} \ar[d] \\ _{p}L′ \ar[r]^{f} & Ã_{cr,1}}

where the right vertical arrow is induced by the correspondence

[r_{−1}]T_{1} + [r_{0}] + p[r_{1}] ↦ [r_{−1}]T_{1} + [r_{0} mod x_{0}^{p}].

Similarly, the points g ∈ V^{t}(L′) and V^{t}(K_{p})(g) ∈ V^{t}(L′_{p}) are related via the commutative diagram

\xymatrix{L′ \ar[r]^{g} & Ã_{cr,2} \\ L′_{p} \ar[u]^{K_{p}} \ar[r]^{V^{t}(K_{p})(g)} & Ã_{cr,1} \ar[u]}

where the right vertical arrow is induced by the correspondence

[r_{−1}]T_{1} + [r_{0}] ↦ [r_{−1}x_{0}^{p}] + p[r_{0}].

4.6. Filtered φ-modules A_{cr,1}^{0} and A_{cr,2}^{0}. Let A_{cr,2}^{0} be the W (R)- submodule of Ã_{cr,2} consisting of elements [r_{−1}]T_{1} + [r_{0}] + p[r_{1}] such that r_{−1} = −r_{0} mod x_{0}^{p}. Then F (A_{cr,2}^{0}) = F (Ã_{cr,2}) ∩ A_{cr,2}^{0} is generated over W (R) by [x_{0}^{p−1}]T_{1} + ξ^{p−1} and the congruence

φ([x_{0}^{p−1}]T_{1} + ξ^{p−1}) ≡ −T_{1} + 1 mod (J_{2} + p^{2}A_{cr,2} + p[m_{R}])

implies that φ(F (A_{cr,2}^{0})) ⊂ A_{cr,2}^{0} and A_{cr,2}^{0} = (A_{cr,2}^{0}, F (A_{cr,2}^{0}), φ) ∈ ˜ \underline{L}_{0}. Note that pA_{cr,2}^{0} = (pA_{cr,2}^{0}, pF (A_{cr,2}^{0}), φ) ∈ ˜ \underline{L}_{0}. Then in notation from Subsection 4.4, one has:

• Im Θ = Hom_{˜\underline{L}_{0}}(_{p}L′, pA_{cr,2}^{0});

• V^{t}(C_{p})(Im Θ) = Hom_{˜\underline{L}_{0}}(L′, pA_{cr,2}^{0});

• KerΘ̃ = Hom_{˜\underline{L}_{0}}(L′_{p}, pA_{cr,2}^{0});

• Ker(Θ̃ ∘ V^{t}(K_{p})) = Hom_{˜\underline{L}_{0}}(L′, A_{cr,2}^{0}).

Therefore, Ṽ^{ft}(L) = V_{L′}^{t}(L) = Hom_{˜\underline{L}_{0}}(L′, A_{cr,2}^{0}/pA_{cr,2}^{0}) = Hom_{˜\underline{L}_{0}}(L, A_{cr,2}^{0}/pA_{cr,2}^{0}).

4.7. The functor C̃V^{ft}. Let L ∈ \underline{L}^{ft} and let i^{et} : L^{et} ⟶ L be the maximal etale subobject of L.

Definition. C̃V^{ft} : \underline{L}^{ft} ⟶ \underline{CMΓ}_{F} is the functor induced by the correspon-

dence L ↦ C̃V^{ft}(L) = (Ṽ^{ft}(L), Ṽ^{ft}(L^{et}), Ṽ^{ft}(i^{et})).

The functor C̃V^{ft} is not very far from Breuil’s functor V^{t} but it satisfies the following important property.

Proposition 4.16. The functor C̃V^{ft} is fully faithful.

Proof. By standard devissage it will be sufficient to verify this property for the restriction C̃V^{ft}|_{\underline{L}^{ft}[1]}. Due to Proposition 2.13 it will be sufficient to verify that the functor Ṽ^{ft}|_{\underline{L}^{ft}[1]} ∘ Π^{−1} coincides with the functor V^{∗} from Subsection 2.2. This can be proved similarly to the proof of the corresponding fact for unipotent objects in Subsection 4.2 as follows. Let

A_{st,2}^{0} = ∏_{j⩾0}A_{cr,2}^{0}γ_{j}(log(1 + X)) ⊂ Ã_{st,2} = ∏_{j⩾0}Ã_{cr,2}γ_{j}(log(1 + X))

with induced structures of the objects A_{st,2}^{0} and Ã_{st,2} of the category ˜ \underline{L}. Then from Subsection 4.6 it follows that

V^{t}(L) = Hom˜\underline{L}(L, A_{st,2}^{0}/pA_{st,2}^{0}).

One can see easily that the correspondence

[r_{0} mod x_{0}^{p}]T_{1} + [r_{0}] + p[r_{1}] ↦ (r_{0} + x_{0}^{p}r_{1}) mod x_{0}^{p}m_{R}

induces the morphism A_{cr,2}^{0}/pA_{cr,2}^{0} ⟶ R^{0} in the category ˜ \underline{L}_{0}. This morphism induces a unique identification of the abelian groups V^{t}(L) and Hom(L, R^{0}) = V^{∗}(L). Now going to a suitable factor of the object A_{st,2}^{0}/pA_{st,2}^{0} we obtain that this identification is compatible with the Γ_{F} -actions on both abelian groups.

□

Now we can describe all Galois invariant lattices of semi-stable ℚ_{p}[Γ_{F}]-modules with weights from [0, p).

Corollary 4.17. Suppose V is a semi-stable representation of Γ_{F} with weights from [0, p), dim_{ℚ_{p}}V = s and T is a Γ_{F} -invariant lattice in V. Then there is

a p-divisible group \{L^{(n)}, i_{n}\}_{n⩾0} of height s in \underline{L}^{ft} such that \underleftarrow{lim}_{n} C̃V^{ft}(L^{(n)}) =

(T, T^{et}, i^{et}) ∈ \underline{CMΓ}_{F}.

5. Proof of Theorem 0.1.

As earlier, p is a fixed prime number, p ≠ 2. Starting Subsection 5.2 we assume p = 3.

5.1. For all prime numbers l, choose embeddings of algebraic closures ℚ̄ ⊂ ℚ̄_{l} and use them to identify the inertia groups I_{l} = Gal(ℚ̄_{l}/ℚ_{l,ur}), where ℚ_{l,ur} is the maximal unramified extension of ℚ_{l}, with the appropriate subgroups in Γ_{ℚ} = Gal(ℚ̄/ℚ). Introduce the category \underline{MΓ}_{ℚ}^{t}. Its objects are the pairs H_{ℚ} = (H, H̃_{st}), where H is a finite ℤ_{p}[Γ_{ℚ}]-module unramified outside p and H̃_{st} = (H_{st}, H_{st}^{0}, i) ∈ \underline{CMΓ}_{F}^{st}, where H|_{I_{p}} = H_{st}, F = W (𝔽̄_{p})[1/p] and \underline{CMΓ}_{F}^{st} is the image of the functor

C̃V^{ft} from Subsection 4.7. The morphisms in \underline{MΓ}_{ℚ}^{t} are compatible morphisms of Galois modules. Clearly, the category \underline{MΓ}_{ℚ}^{t} is special pre-abelian, cf. Appendix A. Let \underline{MΓ}_{ℚ}^{t}[1] be the full suibcategory of killed by p objects in \underline{MΓ}_{ℚ}^{t}. Denote by K(p) an algebraic extension of ℚ such that for any H_{ℚ} = (H, H̃_{st}) ∈ \underline{MΓ}_{ℚ}^{t}[1], Γ_{K(p)} = Gal(ℚ̄/K(p)) acts trivially on H. In other words, K(p) can be taken as a common field-of-definition of points of all such Γ_{ℚ}-modules H. Now assume that

(C) K(p) is totally ramified at p.

Under this assumption we have a natural identification Gal(K(p)/ℚ) = Gal(K(p)F/F), that is the Galois group of the global extension K(p)/ℚ comes as the Galois group of its completion over F. Therefore, we can identify \underline{MΓ}_{ℚ}^{t}[1] with the full subcategory of \underline{CMΓ}_{F}^{st}, consisting of (H_{st}, H_{st}^{0}, i) such that pH_{st} = 0 and all points of H_{st} are defined over K(p)F. In other words, the objects of \underline{MΓ}^{t}[1] can be described via our local results about killed by p subquotients of semistable representations of Γ_{F}.

Denote by \underline{MΓ}_{ℚ}^{ft}[1] a full subcategory in \underline{MΓ}_{ℚ}^{t}[1] which consists of killed by p subquotients of p-divisible groups in the category \underline{MΓ}_{ℚ}^{t}. Let F′ be the maximal tamely ramified extension of F in K(p)F. Then Gal(F′/F) is abelian group of order prime to p (use that the residue field of F is algebraically closed) and Gal(K(p)F/F′) is a p-group. This gives an abelian extension K′ of ℚ in K(p) of prime-to-p degree and such that K(p)/K′ is a p-extension. This extension is unramified outside p and, therefore, it coincides (use class field theory) with ℚ(ζ_{p}). In particular, all simple objects in \underline{MΓ}_{ℚ}^{t}[1] are of the form F(j) = (𝔽_{p}(j), 0, 0) if 1 ⩽ j < p and F(0) = (𝔽_{p}(0), 𝔽_{p}(0), id ) if j = 0. Let \underline{L}_{ℚ}^{ft}[1] and \underline{L}_{ℚ}^{t}[1] be the full subcategories of \underline{L}^{t}[1] mapped by the functor

C̃V^{ft} to the objects of \underline{MΓ}_{ℚ}^{ft}[1] and, resp., \underline{MΓ}_{ℚ}^{t}[1]. Clearly, \underline{L}_{ℚ}^{ft}[1] is a full subcategory in \underline{L}^{ft}[1] and the only simple objects in these categories are L(r), where r ∈ {j/(p − 1) | j = 0, 1, …, p − 1}. Suppose H^{∞} = \{H_{ℚ}^{(n)}, i_{n}\}_{n⩾0} is a p-divisible group in the category \underline{MΓ}_{ℚ}^{t}. Here all H_{ℚ}^{(n)} = (H^{(n)}, H̃_{st}^{(n)}) are objects of the category \underline{MΓ}_{ℚ}^{t}. Let L ∈ \underline{L}_{ℚ}^{ft}[1]

be such that C̃V^{ft}(L) = H̃_{st}^{(1)}. Note that the maximal etale subobject L^{et} of L is isomorphic to L(0)^{n_{et}}, where n_{et} = n_{et}(L) ∈ ℤ_{⩾0}, and L/L^{et} has no simple subquotients isomorphic to L(0). Similarly, the corresponding maximal multiplicative quotient L^{m} is isomorphic to L(1)^{n_{m}}, where n_{m} = n_{m}(L) ∈ ℤ_{⩾0}, and the kernel of the canonical projection L ⟶ L^{m} has no simple subquotients isomorphic to L(1). Therefore, for any M ∈ \underline{L}_{ℚ}^{ft}[1],

Ext_{\underline{L}_{ℚ}^{ft}[1]}(L(0), M) = Ext_{\underline{L}_{ℚ}^{ft}[1]}(M, L(1)) = 0.

This implies that for any H ∈ \underline{MΓ}_{ℚ}^{ft}[1],

Ext_{\underline{MΓ}_{ℚ}^{ft}[1]}(H, F(0)) = Ext_{\underline{MΓ}_{ℚ}^{ft}[1]}(F(1), H) = 0.

Therefore, by Theorem A.5 of Appendix A there is an embedding of p-divisible groups H^{∞,m} ⊂ H^{∞}, where H^{(1)m} = F(1)^{n_{m}}, and there is a projection of p-divisible groups H^{∞} ⟶ H^{∞,et}, where H^{(1)et} = F(0)^{n_{et}}. For similar reasons,

Ext_{\underline{MΓ}_{ℚ}^{ft}[1]}(F(0), F(0)) = Ext_{\underline{MΓ}_{ℚ}^{ft}[1]}(F(1), F(1)) = 0

and by Theorem A.4 of Appendix A, the corresponding p-divisible groups H_{ℚ}^{∞,m} and H_{ℚ}^{∞,et} are unique. Therefore they coincide with the products of trivial pdivisible groups (ℚ_{p}/ℤ_{p})(p − 1) and, resp., (ℚ_{p}/ℤ_{p})(0). We state this result in the following form.

Proposition 5.1. Under assumption (C), for any p-divisible group H^{∞} in the category \underline{MΓ}_{ℚ}^{t} there is a filtration of p-divisible groups

H^{∞} ⊃ H_{1}^{∞} ⊃ H_{0}^{∞}

such that H_{0}^{∞} = (ℚ_{p}/ℤ_{p})(p − 1)^{n_{m}}, H^{∞}/H_{1}^{∞} = (ℚ_{p}/ℤ_{p})(0)^{n_{et}} and all simple subquotients of H_{1}^{∞}/H_{0}^{∞} belong to {𝔽_{p}(j)|1 ⩽ j ⩽ p − 2}.

5.2. Assume that p = 3.

Lemma 5.2. K(3) = ℚ(\sqrt[3]{3}, ζ_{9}), where ζ_{9} is 9-th primitive root of 1.

This Lemma will be proved in Subsection 5.3 below. In particular, K(3) satisfies the assumption (C).

Proposition 5.3. If H^{∞} is a 3-divisible group in \underline{MΓ}_{ℚ}^{t} then in its filtration from Proposition 5.1 the 3-divisible group Ĥ^{∞} = H_{1}^{∞}/H_{0}^{∞} is a product of finitely many trivial 3-divisible groups (ℚ_{3}/ℤ_{3})(1).

Proof. Let ˆ \underline{L}_{ℚ} be the full subcategory of \underline{L}_{ℚ}^{ft}[1] consisting of objects L such that L^{m} = L^{et} = 0. This category has only one simple object L(1/2). Let ˆ \underline{MΓ}_{ℚ}

be the full subcategory in \underline{MΓ}_{ℚ}^{ft}[1] consisting of the objects C̃V^{ft}(L), where L ∈ ˆ \underline{L}_{ℚ}. Then ˆ \underline{L}_{ℚ} and ˆ \underline{MΓ}_{ℚ} are antiequivalent categories and Ĥ^{(1)} ∈ \underline{MΓ}_{ℚ}. By Theorems A.4 and A.5 our Proposition is implied by the following result. □

Proposition 5.4. Ext_{ˆ\underline{L}_{ℚ}}(L(1/2), L(1/2)) = 0.

Proof. Consider the equivalence of the categories Π : \underline{L}^{t} ⟶ \underline{L}^{∗} from Corollary 3.10. This equivalence transforms the functor C̃V^{ft} into the functor CV^{∗} from Section 2, cf. the proof of Proposition 4.16. Therefore, the objects L of the category Π(\underline{L}_{ℚ}^{t}) := \underline{L}_{ℚ}^{∗} are characterised by the condition that all points of V^{∗}(L) are defined over the field K(3)F. The objects L of the category Π(ˆ\underline{L}_{ℚ}) := ˆ\underline{L}_{ℚ}^{∗} are characterised by the additional properties: they are all obtained by subsequent extensions via L(1/2) and V^{∗}(L) appears as a subquotient of semistable representation of Γ_{F} with Hodge-Tate weights from [0,2]. Introduce the object L(1/2, 1/2) = (L, F (L), φ, N) of the category \underline{L}^{∗} as follows:

• L = W_{1}l ⊕ W_{1}l_{1};

• F (L) is spanned by ul_{1} and ul + l_{1};

• φ(ul_{1}) = l_{1}, φ(ul + l_{1}) = l;

• N(l_{1}) ≡ 0 mod u^{3}L, N(l) ≡ l_{1} mod u^{3}L. Clearly, L(1/2, 1/2) has a natural structure of an element of the group Ext\underline{L}^{∗}(L(1/2), L(1/2)).

Lemma 5.5. a) L(1/2, 1/2) ∈ \underline{L}_{ℚ}^{∗};

b) Ext\underline{L}_{ℚ}^{∗}(L(1/2), L(1/2)) ≃ ℤ/3 and is generated by the class of L(1/2, 1/2);

c) Ext\underline{L}_{ℚ}^{∗}(L(1/2), L(1/2, 1/2)) = Ext\underline{L}_{ℚ}^{∗}(L(1/2, 1/2), L(1/2)) = 0.

This Lemma will be proved in Subsection 5.4 below. Lemma 5.5 implies that Ext\underline{L}_{ℚ}^{∗}(L(1/2, 1/2), L(1/2, 1/2)) = 0 and, therefore, any object L of \underline{L}_{ℚ}^{∗} is the product of several copies of L(1/2) and L(1/2, 1/2).

Suppose L = L_{1} × L(1/2, 1/2) ∈ ˆ \underline{L}_{ℚ}^{∗}. Then there is a 3-divisible group H̃^{∞} in \underline{MΓ}_{ℚ}^{t} such that H̃^{(1)} = H′ × H(1/2, 1/2), where H′ and H(1/2, 1/2) = CV^{∗}(L(1/2, 1/2)) belong to ˆ \underline{MΓ}_{ℚ}. Clearly, we have Ext_{\underline{MΓ}_{ℚ}^{ft}[1]}(H′, H(1/2, 1/2)) = 0 and applying Theorem A.5 we obtain a 3-

divisible group H^{∞} in \underline{MΓ}_{ℚ}^{t} such that H^{(1)} = H(1/2, 1/2). This implies the existence of 2-dimensional semi-stable (and non-crystalline) representation of Γ_{F} with the only simple subquotient 𝔽_{3}(1), that is for any Galois invariant lattice T of such representation, the Γ_{F} -module T/3T has semi-simple envelope 𝔽_{3}(1) × 𝔽_{3}(1). This situation appears as a very special case of Breuil’s description of 2-dimensional semi-stable (and non-crystalline) representations. According to Theorem 6.1.1.2 of [5] the corresponding semi-simple envelope is either 𝔽_{3}(0) × 𝔽_{3}(1) or 𝔽_{3}(1) × 𝔽_{3}(2). The proposition is proved. □

Now our main Theorem appears as the following Corollary.

Corollary 5.6. If Y is a projective variety with semi-stable reduction modulo 3 and good reduction modulo all primes l ≠ 3 then h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ}).

Proof. Indeed, let V be the ℚ_{3}[Γ_{F}]-module of 2-dimensional etale cohomology of Y. Then it is a semi-stable representation of F and its Γ_{F} -invariant lattice determines a 3-divisible group in the category \underline{MΓ}_{ℚ}^{t}. By Proposition 5.3 this 3divisible group can be built from the Tate twists (ℚ_{3}/ℤ_{3})(i), i = 0, 1, 2. Equivalently, all Γ_{F} -equivariant subquotients of V are ℚ_{3}(i) with i = 0, 1, 2. Applying the Riemann Conjecture (proved by Deligne) to the reductions Y mod l with l ≠ 3, we obtain that ℚ(0) and ℚ(2) do not appear. Therefore, V is the product of finitely many ℚ_{3}(1) and h^{2}(Y_{ℂ}) = h^{1,1}(Y_{ℂ}). □

5.3. Proof of Lemma 5.2. Use the ramification estimate from Subsection 2.9 to deduce that the normalized discriminant of K(3) over ℚ satisfies the inequality |D(K(3)/ℚ)|^{[K(3):ℚ]^{−1}} < 3^{3−1/3} = 18.72075. Then Odlyzko estimates imply that [K(3) : ℚ] < 230 [11]. Let K_{0} = ℚ(ζ_{9}) and K_{1} = ℚ(\sqrt[3]{3}, ζ_{9}). Then K_{0} is the maximal abelian extension of ℚ in K(3) and K_{1} ⊂ K(3). We have also the inequality [K(3) : K_{1}] < 60 and, therefore, Gal(K(3)/ℚ) is soluble. Prove that K_{1} = K(3). Suppose the field K_{2} is the maximal abelian extension of K_{1} in K(3). One can apply the computer package SAGE to prove that the group of classes of K_{1} is trivial. Therefore, K_{2} is totally ramified at 3 and Gal(K_{2}/ℚ) coincides with the Galois group of the corresponding 3-completions. In particular, the maximal tamely ramified subextension of these completions comes from ℚ(ζ_{3}) and, therefore, K_{2}/K_{1} is 3-extension. Therefore, there is an η ∈ O_{K_{1}}^{∗} such that K_{1}(\sqrt[3]{η}) ⊂ K_{2}. Then a routine computation shows that the normalized discriminant for K_{1}(\sqrt[3]{η}) over ℚ is less than 3^{3−1/3} if and only if η ≡ 1mod O_{K_{1}}^{∗3}(1 + 3O_{K_{1}})^{×}. The Lemma will be proved if we show that such η ∈ O_{K_{1}}^{∗3}. (This is equivalent to the Leopoldt Conjecture for the field K_{1}.) This was proved via a SAGE computer program written by R.Henderson (Summer-2009 Project at Durham University supported by Nuffield Foundation). This program, cf. Appendix B, constructed a basis ε_{i}mod O_{K_{1}}^{∗3}, 1 ⩽ i ⩽ 9, of O_{K_{1}}^{∗}/O_{K_{1}}^{∗3} such that 18v_{3}(ε_{i}−1) takes values in the set {1, 2, 4, 5, 7, 8, 10, 13, 16}. In other words, v_{3}(η − 1) ⩾ 1 > 16/18 implies that η ∈ O_{K_{1}}^{∗3}. Lemma 5.2 is proved.

5.4. Proof of Lemma 5.5. a) Use the notation from the definition of the functor V^{t} in Subsection 4. If f_{0} ∈ V^{t}(L(1/2, 1/2)) then the correspondence f_{0} ↦ (f_{0}(l_{1}), f_{0}(l)) identifies V^{t}(L(1/2, 1/2)) with the 𝔽_{3}-module of couples (X_{10}, X_{0}) ∈ (R/x_{0}^{6})^{2} such that X_{10}^{3}/x_{0}^{3} = X_{10} and (X_{0}^{3} + X_{10})/x_{0}^{3} = X_{0}. Then the 𝔽_{3}[Γ_{F}]-module V^{t}(L(1/2, 1/2)) is identified with the module formed by the images of all (X_{10}, X_{0} + X_{10}Y) ∈ (R_{st}^{0})^{2} in R̃_{st}^{0} = R_{st}^{0}/(x_{0}^{3}m_{R} + x_{0}^{2}m_{R}Y + x_{0}m_{R}Y^{2}). In particular, the corresponding Γ_{F} -action on V^{t}(L(1/2, 1/2)) comes from the natural Γ_{F} -action on the residues of X_{10} and X_{0} modulo x_{0}^{3}m_{R}. Notice there is a natural Γ_{F} -equivariant identification

ι : m_{R}/(x_{0}^{3}m_{R}) ⟶ m̄/3m̄,

where m̄ is the maximal ideal of the valuation ring of ℚ̄_{3}. This isomorphism ι comes from the map r ↦ r^{(1)}, where for r = \underleftarrow{lim}_{n}(r_{n} mod p), r^{(1)} := lim_{n→∞}r_{n+1}^{p^{n}}.

Then Hensel’s Lemma implies the existence of unique Z_{10}, Z_{0} ∈ m̄ such that the following equalities hold ι(X_{10} mod x_{0}^{3}m_{R}) = Z_{10} mod 3m̄, ι(X_{0} mod x_{0}^{3}m_{R}) = Z_{0} mod 3m̄, Z_{10}^{3} + 3Z_{10} = 0 and Z_{0}^{3} + 3Z_{0} = −Z_{10}. Clearly, F (Z_{10}, Z_{0}) = F (ζ_{9}). Therefore, if τ ∈ Γ_{F} is such that τ(ζ_{9}) = ζ_{9} then τ(X_{10}) = X_{10} and τ(X_{0}) = X_{0}. Finally, it follows directly from definitions that if τ(\sqrt[3]{3}) = \sqrt[3]{3} then τ acts as identity on the image of Y in R̃_{st}^{0}. The part a) of the Lemma is proved.

b) Suppose L = (L, F (L), φ, N) ∈ Ext\underline{L}_{ℚ}^{∗}(L(1), L(1)). Then L = W_{1}l ⊕ W_{1}l_{1}, there is an w ∈ W_{1} such that F (L) is spanned by ul_{1} and ul+wl_{1} over W_{1}, and one has φ(ul_{1}) = l_{1}, φ(ul + wl_{1}) = l, N(l_{1}) ∈ u^{3}L and N(l_{1}) ≡ w^{3}l_{1} mod u^{3}L. Notice that L splits in \underline{L}^{∗} iff w ∈ uW_{1}. Therefore, we can assume that w = α ∈ k. Then the field-of-definition of all points of V^{t}(L) contains the field-of-definition of all solutions (X_{1}, X) mod x_{0}^{3}m_{R} ∈ (R/x_{0}^{3}m_{R})^{2} of the following congruences: X_{1}^{3}/x_{0}^{3} ≡ X_{1} mod x_{0}^{3}m_{R} and (X^{3} + α^{3}X_{1})/x_{0}^{3} ≡ X mod x_{0}^{3}m_{R}. Let x_{1} ∈ R be such that x_{1}^{2} = x_{0}. Then we can take X_{1} = x_{1}^{3} and for T = X/x_{1}^{3} one has the following Artin-Schreier-type congruence:

T^{3} − T ≡ −α^{3}/x^{6} mod m_{R}.

Using calculations from above part a) we can conclude that L ∈ \underline{L}_{ℚ}^{∗} if and only if the field-of-definition of T mod m_{R} over k((x_{1})) belongs to the field-of-definition of T_{0} mod m_{R} over k((x_{1})), where T_{0}^{3} − T_{0} ≡ −x_{1}^{−6} mod m_{R}. By Artin-Schreier theory this happens if and only if α ∈ 𝔽_{3} and,therefore, L ≃ L(1/2, 1/2). c) Suppose L = (L, F (L), φ, N) ∈ Ext\underline{L}_{ℚ}^{∗}(L(1/2), L(1/2, 1/2)). Then we can assume that:

— L = W_{1}l ⊕ W_{1}l_{1} ⊕ W_{1}m;

— F (L) is spanned over W_{1} by ul_{1}, ul+l_{1} and um+wl+w_{1}l_{1} with w, w_{1} ∈ W_{1};

— φ(ul_{1}) = l_{1}, φ(ul + l_{1}) = l and φ(um + wl + w_{1}l_{1}) = m. Then the condition u^{2}m ∈ F (L) implies that wl_{1} ∈ F (L), or w ∈ uW_{1} and we can assume that w = 0. Then the submodule W_{1}m + W_{1}l_{1} determines a subobject L′ of L, L′ ∈ \underline{L}_{ℚ}^{∗} and using calculations from b) we conclude that w_{1} ∈ 𝔽_{3} mod uW_{1}. Therefore, we can assume that w_{1} = α ∈ 𝔽_{3} and for m′ = m − αl we have m′ ∈ F (L) and φ(um′) = m′, i.e. L is a trivial extension. Now suppose L = (L, F (L), φ, N) ∈ Ext\underline{L}_{ℚ}^{∗}(L(1/2, 1/2), L(1/2)). Then we can assume that:

— L = W_{1}m ⊕ W_{1}m_{1} ⊕ W_{1}l;

— F (L) is spanned over W_{1} by ul, um_{1}+wl and um+m_{1}+w_{1}l with w, w_{1} ∈ W_{1};

— φ(ul) = l, φ(um_{1} + wl) = m_{1} and φ(um + m_{1} + w_{1}l) = m. Again, the condition u^{2}m ∈ F (L) implies that w ∈ uW_{1} and, therefore, we can assume that w = 0. Then the quotient module L/W_{1}m_{1} is the quotient of L in the category \underline{L}^{∗}. This quotient must belong to the subcategory \underline{L}_{ℚ}^{∗}. This implies that w_{1} ∈ 𝔽_{3} mod uW_{1}, and, as earlier, L becomes a trivial extension. The Lemma is completely proved.

Appendix A. p-divisible groups in pre-abelian categories

A.1. Short exact sequences in pre-abelian categories.

A.1.1. Pre-abelian categories. Introduce the concept of special pre-abelian category following mainly [28], cf. also [25, 26, 29]. Remind that a category S is pre-abelian if it is additive and for any morphism u ∈ Hom_{S}(A, B), there exist Ker u = (A_{1}, i) and Coker u = (B_{1}, j), where i ∈ Hom_{S}(A_{1}, A) and j ∈ Hom_{S}(B, B_{1}). For any objects A, B ∈ S, let A ∏ B and A ∐ B be their product and coproduct, respectively. There is a canonical isomorphism A ∏ B ≃ A ∐ B in S. More generally, for given morphisms:

• α ∈ Hom_{S}(C, A), β ∈ Hom_{S}(C, B), there is a fibered coproduct (A ∐_{C} B, i_{A}, i_{B}), with i_{A} ∈ Hom_{S}(A, A ∐_{C} B), i_{B} ∈ Hom_{S}(B, A ∐_{C} B) which

completes the diagram A ⟵^{α} C ⟶^{β} B to a cocartesian square;

• f ∈ Hom_{S}(A, C) and g ∈ Hom_{S}(B, C), there is a fibered product (A ∏_{C} B, p_{A}, p_{B}), with p_{A} ∈ Hom_{S}(A ∏_{C} B, A), p_{B} ∈ Hom_{S}(A ∏_{C} B, B),

which completes the diagram A ⟶^{f} C ⟵^{g} B to a cartesian square.

Suppose i ∈ Hom_{S}(A_{1}, A), f ∈ Hom_{S}(A_{1}, B) and (B ∐_{A_{1}} A, i_{A}, i_{B}) is their fibered coproduct. If (A_{2}, j) = Coker i then there is a morphism j_{B} :

B ∐_{A_{1}} A → A_{2} such that the following diagram

\xymatrix{A_{1} \ar[d]^{f} \ar[r]^{i} & A \ar[d]^{i_{A}} \ar[r]^{j} & A_{2} \ar[d]^{id} \\ B \ar[r]^{i_{B}} & B ∐_{A_{1}} A \ar[r]^{j_{B}} & A_{2}}

is commutative (use the zero morphism from B to A_{2}). A formal verification shows that (A_{2}, j_{B}) = Coker i_{B}. Suppose j ∈ Hom_{S}(A, A_{2}), g ∈ Hom_{S}(B, A_{2}) and (B ∏_{A_{2}} A, p_{B}, p_{A}) is their fibered product. If (A_{1}, i) = Ker j then there is an i_{B} : A_{1} → B ∏_{A_{2}} A (use the zero map from A_{1} to B) such that the following diagram

\xymatrix{A_{1} \ar[r]^{i} & A \ar[r]^{j} & A_{2} \\ A_{1} \ar[r]^{i_{B}} \ar[u]^{id} & B ∏_{A_{2}} A \ar[r]^{p_{B}} \ar[u]^{p_{A}} & B \ar[u]^{g}}

is commutative and (A_{1}, i_{B}) = Ker p_{B}.

A.1.2. Strict morphisms. A morphism u ∈ Hom_{S}(A, B) is strict if the canonical morphism Coim u = Coker(Ker u) → Im u = Ker(Coker u) is isomorphism. One can verify that always Ker u = (A_{1}, i) is a strict monomorphism and Coker u = (B_{1}, j) is a strict epimorphism. By definition, a sequence of objects and morphisms

(A.1)  0 ⟶ A_{1} ⟶^{i} A ⟶^{j} A_{2} ⟶ 0

in S is short exact if (A_{1}, i) = Ker j and (A_{2}, j) = Coker i. In particular, any strict monomorphism (resp. strict epimorphism) can be included in a short exact sequence.

Definition. A pre-abelian category is special if it satisfies the following two axioms: SP1. if α : C → A is strict monomorphism then i_{B} : B → A∐_{C}B is also strict

monomorphism; SP2. if f : A → C is strict epimorphism then p_{B} : A∏_{C}B → B is also strict

epimorphism.

Remark. A typical example of pre-abelian special category is the category of modules with filtration.

Consider short exact sequence (A.1) in S. If f ∈ Hom_{S}(A_{1}, B) then we have the following commutative diagram

(A.2)  \xymatrix{0 \ar[r] & A_{1} \ar[d]^{f} \ar[r]^{i} & A \ar[d]^{i_{A}} \ar[r]^{j} & A_{2} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & B \ar[r]^{i_{B}} & A ∐_{A_{1}} B \ar[r]^{j_{B}} & A_{2} \ar[r] & 0}

Then j_{B} = Coker i_{B} is strict epimorphism and by axiom SP1, i_{B} is strict monomorphism. Then Ker j_{B} = Ker(Coker i_{B}) = Imi_{B} = (B, i_{B}) and, therefore, the lower row of the above diagram is exact. Dually, for any g ∈ Hom_{S}(B, A_{2}) there is a commutative diagram

(A.3)  \xymatrix{0 \ar[r] & A_{1} \ar[r]^{i} & A \ar[r]^{j} & A_{2} \ar[r] & 0 \\ 0 \ar[r] & A_{1} \ar[r]^{i_{B}} \ar[u]^{id} & A ∏_{A_{2}} B \ar[r]^{p_{B}} \ar[u]^{p_{A}} & B \ar[u]^{g} \ar[r] & 0}

where i_{B} = Ker j_{B} is strict monomorphism, by Axiom SP2, p_{B} is strict epimorphism and the lower row of this diagram is exact. With relation to above diagram (A.2) we have the following ptoperties.

Lemma A.1. a) The natural map δ : Ker f ⟶ Ker i_{A} is isomorphism;

b) if f is strict epimorphism then Ker i_{A} is also strict epimorphism.

Proof. a) Suppose that Ker f = (K_{1}, α_{1}) and Ker i_{A} = (K, α). Then δ : Ker f ⟶ Ker i_{A} appears from the universal property of (K, α) because i_{A} ∘ i ∘ α_{1} = i_{B} ∘ f ∘ α_{1} = 0. The relation j ∘ α = j_{B} ∘ i_{A} ∘ α = 0 implies the existence of α̃ : K ⟶ A_{1} such that i ∘ α̃ = α. Then i_{B} ∘ f ∘ α̃ = i_{A} ∘ α = 0 and f ∘ α̃ = 0 (use that i_{B} is monomorphism). By the universal property of (K_{1}, α_{1}) this gives the map δ_{1} : K ⟶ K_{1} such that α_{1} ∘ δ_{1} = α̃ and this map is inverse to δ.

b) Suppose f is a strict epimorphism, then (B, f) = Coker α_{1}. Let (C̃, j̃) = Coker α. By functoriality, there is ε : B ⟶ C̃ such that ε ∘ f = j̃∘ i. Then j̃ and ε define a unique ω : A ∐_{A_{1}} B ⟶ C̃ such that ω ∘ i_{B} = ε and ω ∘ i_{A} = j̃. But i_{A} ∘ α = 0 implies by the universal property of (C̃, j̃) the map ω_{1} : C̃ ⟶ A ∐_{A_{1}} B and one can verify that it is inverse to ω. □

Remark. If f is strict monomorphism then i_{A} is also strict monomorphism by axiom SP2.

With relation to diagram (A.3) we have the following Lemma which is dual to above Lemma A.1.

Lemma A.2. a) The natural map Coker p_{A} ⟶ Coker g is isomorphism; b) if g is strict epimorphism then p_{A} is also strict epimorphism.

Proof. The proof is dual to the proof of Lemma A.1. □

Lemma A.3. A composition of two strict monomorphisms (resp., epimorphisms) is again strict monomorphism (resp., epimorphism).

Proof. It will be sufficient to consider only the case of monomorphisms. Suppose i ∈ Hom_{S}(A_{1}, A) and i_{1} ∈ Hom_{S}(A, B) are two strict monomorphisms. Construct the following commutative diagram:

(A.4)  \xymatrix{0 \ar[r] & A_{1} \ar[d]^{β} \ar[r]^{i} & A \ar[d]^{i_{1}} \ar[r]^{j} & A_{2} \ar[d]_{i_{A_{2}}} \ar[r] & 0 \\ 0 \ar[r] & K \ar[r]^{α} & B \ar[d]^{j_{1}} \ar[r]^{i_{B}} & A_{2} ∐_{A} B \ar[d]^{j} \ar[r] & 0 \\  &  & C_{1} \ar[r]^{γ} & C}

Here j = Coker i is strict epimorphism and the upper right square is cocartesian. Therefore, i_{B} is strict epimorphism and we obtain the second line which is short exact. The morphisms i_{1} and i_{A_{2}} are strict monomorphisms and we can complete our diagram by (C_{1}, j_{1}) = Coker i_{1} and (C, j) = Coker i_{A_{2}}. The maps β and γ are obtained by functoriality and we have proved that they are isomorphisms. Therefore, i_{1} ∘ i = β ∘ α is strict (use that α = Ker i_{B} is strict). □

A.1.3. Bifunctor Ext_{S}. If S is pre-abelian category then in the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & A_{1} \ar[r] \ar[d]^{id} & A \ar[r] \ar[d]^{f} & A_{2} \ar[r] \ar[d]^{id} & 0 \\ 0 \ar[r] & A_{1} \ar[r] & A′ \ar[r] & A_{2} \ar[r] & 0}

the morphism f is isomorphism. Therefore, we can introduce the set of equivalence classes of short exact sequences Ext_{S}(A_{2}, A_{1}). This set is functorial in both arguments due to axioms SP1 and SP2. Suppose the objects of S are provided with commutative group structure respected by morphisms of S. Then for any A, B ∈ S, Ext_{S}(A, B) has a natural group structure, where the class of split short exact sequences plays a role of neutral element. Remind that the sum ε_{1} + ε_{2} of two extensions

ε_{1} : 0 ⟶ A_{1} ⟶^{i′} A′ ⟶^{j′} A_{2} ⟶ 0 and ε_{2} : 0 ⟶ A_{1} ⟶^{i′′} A′′ ⟶^{j′′} A_{2} ⟶ 0

is the lower line of the following commutative diagram relating the rows l = ε_{1} ⊕ ε_{2}, ∇^{∗}(l) and (+)_{∗}∇^{∗}(l),

\xymatrix{l : 0 \ar[r] & A_{1} ∏ A_{1} \ar[r]^{i′ ∏ i′′} & A′ ∏ A′′ \ar[r]^{j′ ∏ j′′} & A_{2} ∏ A_{2} \ar[r] & 0 \\ ∇^{∗}(l) : 0 \ar[r] & A_{1} ∏ A_{1} \ar[d]^{+} \ar[u]^{id} \ar[r]^{i′ ∏ i′′} & A′ ∏_{A_{2}} A′′ \ar[d] \ar[u] \ar[r]^{j′ ∏ j′′} & A_{2} \ar[d]^{id} \ar[u]^{∇} \ar[r] & 0 \\ (+)_{∗}∇^{∗}(l) : 0 \ar[r] & A_{1} \ar[r] & A′′ \ar[r] & A_{2} \ar[r] & 0}

Here ∇ is the diagonal morphism, + is the morphism of the group structure on S. For any f ∈ Hom_{S}(A_{1}, B) and g ∈ Hom_{S}(B, A_{2}) the corresponding morphisms f_{∗} : Ext_{S}(A_{2}, A_{1}) → Ext_{S}(A_{2}, B) and g^{∗} : Ext_{S}(A_{2}, A_{1}) → Ext_{S}(B, A_{1}) are homomorphisms of abelian groups. The proof is completely formal and goes along the lines of [27]. Suppose ε ∈ Ext_{S}(A_{2}, A_{1}), then the extension ε+(−id)^{∗}ε splits. We shall need below the following explicit description of this splitting. Let ε : 0 ⟶ A_{1} ⟶^{i} A ⟶^{j} A_{2} ⟶ 0. Then ε + (−id)^{∗}ε is the lower row in the following diagram

\xymatrix{0 \ar[r] & A_{1} ∏ A_{1} \ar[d] \ar[r]^{i ∏ i} & A ∏_{A_{2}} A \ar[d] \ar[r]^{(j,j)} & A_{2} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & A_{1} \ar[r] & A_{0} \ar[r] & A_{2} \ar[r] & 0}

where the left vertical arrow is the cokernel of the diagonal embedding ∇ : A_{1} → A_{1} ∏ A_{1}. One can see that the epimorphic map A_{0} → A_{1}, which splits the lower exact sequence, is induced by the morphism p_{1} − p_{2} : A ∏_{A_{2}} A → A. Finally, one can apply Serre’s arguments [30] to obtain for any short exact sequence 0 ⟶ A_{1} ⟶^{i} A ⟶^{j} A_{2} ⟶ 0 and any B ∈ S, the following standard 6-terms exact sequences of abelian groups

0 ⟶ Hom_{S}(B, A_{1}) ⟶^{i_{∗}} Hom_{S}(B, A) ⟶^{j_{∗}} Hom_{S}(B, A_{2})

⟶^{δ} Ext_{S}(B, A_{1}) ⟶^{i_{∗}} Ext_{S}(B, A) ⟶^{j_{∗}} Ext_{S}(B, A_{2})

0 ⟶ Hom_{S}(A_{2}, B) ⟶^{j^{∗}} Hom_{S}(A, B) ⟶^{i^{∗}} Hom_{S}(A_{1}, B)

⟶^{δ} Ext_{S}(A_{2}, B) ⟶^{i^{∗}} Ext_{S}(A, B) ⟶^{j^{∗}} Ext_{S}(A_{1}, B)

A.2. p-divisible groups. In this section S is a special pre-abelian category consisting of group objects. Denote by S_{1} the full subcategory of objects killed by p in S, where p is a fixed prime number. Clearly, S_{1} is again special preabelian.

A.2.1. Basic definitions. Consider an inductive system (C^{(n)}, i^{(n)})_{n⩾0} of objects of S, where C^{(0)} = 0 and all i^{(n)} : C^{(n)} → C^{(n+1)} are strict monomorphisms. Let for all n ⩾ m ⩾ 0, i_{mn} = i^{(n−1)} ∘ … ∘ i^{(m+1)} ∘ i^{(m)} ∈ Hom_{S}(C^{(m)}, C^{(n)}). Then all i_{mn} are strict monomorphisms. Follow Tate’s paper [Ta] to define a p-divisible group in S as an inductive system (C^{(n)}, i^{(n)})_{n⩾0} such that for all 0 ⩽ m ⩽ n,

a) Coker i_{mn} = (C^{(n−m)}, j_{n,n−m}), i.e. there are short exact sequences:

0 ⟶ C^{(m)} ⟶^{i_{mn}} C^{(n)} ⟶^{j_{n,n−m}} C^{(n−m)} ⟶ 0

b) there are commutative diagrams

\xymatrix{C^{(n)} \ar[dr]_{j_{n,n−m}} \ar[rr]^{p^{m}id_{C^{(n)}}} &  & C^{(n)} \\  & C^{(n−m)} \ar[ur]_{i_{n−m,n}}}

The above definition implies the existence of the following commutative diagrams with exact rows (where m ⩽ n ⩽ n_{1}):

\xymatrix{0 \ar[r] & C^{(m)} \ar[d]^{id} \ar[r]^{i_{mn}} & C^{(n)} \ar[d]^{i_{nn_{1}}} \ar[r]^{j_{n,n−m}} & C^{(n−m)} \ar[d]^{i_{n−m,n_{1}−m}} \ar[r] & 0 \\ 0 \ar[r] & C^{(m)} \ar[r]^{i_{mn_{1}}} & C^{(n_{1})} \ar[r]^{j_{n_{1},n_{1}−m}} & C^{(n_{1}−m)} \ar[r] & 0}

\xymatrix{0 \ar[r] & C^{(n)} \ar[d]^{j_{nm}} \ar[r]^{i_{nn_{1}}} & C^{(n_{1})} \ar[d]^{j_{n_{1},m+n_{1}−n}} \ar[r]^{j_{n_{1},n_{1}−n}} & C^{(n_{1}−n)} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & C^{(m)} \ar[r]^{i_{m,m+n_{1}−n}} & C^{(m+n_{1}−n)} \ar[r]^{j_{m+n_{1}−n,n_{1}−n}} & C^{(n_{1}−n)} \ar[r] & 0}

Also, for all n ⩾ m ⩾ 0, one has

• (C^{(m)}, i_{mn}) = Ker(p^{m}id_{C^{(n)}}), (C^{(m)}, j_{nm}) = Coker (p^{n−m}id_{C^{(n)}});

• i_{mn} = i_{n−1,n} ∘ … ∘ i_{m,m+1} and j_{nm} = j_{m+1,m} ∘ … ∘ j_{n,n−1}.

The set of p-divisible groups in S has a natural structure of category. This category is pre-abelian. In particular,

0 ⟶ (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0} ⟶^{(γ_{n})} (C^{(n)}, i^{(n)})_{n⩾0} ⟶^{(δ_{n})} (C_{2}^{(n)}, i_{2}^{(n)})_{n⩾0} ⟶ 0

is a short exact sequence of p-divisible groups iff for all n ⩾ 1, there are the following commutative diagrams with short exact rows in S

\xymatrix{0 \ar[r] & C_{1}^{(n)} \ar[d]^{i_{1}^{(n)}} \ar[r]^{γ_{n}} & C^{(n)} \ar[d]^{i^{(n)}} \ar[r]^{δ_{n}} & C_{2}^{(n)} \ar[d]^{i_{2}^{(n)}} \ar[r] & 0 \\ 0 \ar[r] & C_{1}^{(n+1)} \ar[r]^{γ_{n+1}} & C^{(n+1)} \ar[r]^{δ_{n+1}} & C_{2}^{(n+1)} \ar[r] & 0}

A.2.2. A property of uniqueness of p-divisible groups.

Theorem A.4. Let D be an object of S_{1} such that Ext_{S_{1}}(D, D) = 0. If (C^{(n)}, i^{(n)})_{n⩾0} and (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0} are p-divisible groups in S such that C^{(1)} ≃ C_{1}^{(1)} ≃ D then these p-divisible groups are isomorphic.

Proof. We must prove that for all n ⩾ 1, there are isomorphisms f_{n} : C^{(n)} → C_{1}^{(n)} such that i_{1}^{(n)} ∘ f_{n} = f_{n+1} ∘ i^{(n)}. Suppose n_{0} ⩾ 1 and all such isomorphisms have been constructed for 1 ⩽ n ⩽ n_{0}. Therefore, we can assume that C^{(n)} = C_{1}^{(n)} for 1 ⩽ n ⩽ n_{0}. Consider the following commutative duagrams with exact rows:

(A.5)  \xymatrix{ε_{n_{0}+1} : & 0 \ar[r] & C^{(1)} \ar[r]^{i_{1}} & C^{(n_{0}+1)} \ar[r]^{j_{1}} & C^{(n_{0})} \ar[r] & 0 \\ ε_{n_{0}} : & 0 \ar[r] & C^{(1)} \ar[u]^{id} \ar[r]^{i} & C^{(n_{0})} \ar[u]^{i^{(n_{0})}} \ar[r]^{j} & C^{(n_{0}−1)} \ar[u]^{i^{(n_{0}−1)}} \ar[r] & 0}

(A.6)  \xymatrix{ε′_{n_{0}+1} : & 0 \ar[r] & C^{(1)} \ar[r]^{i′_{1}} & C_{1}^{(n_{0}+1)} \ar[r]^{j′_{1}} & C^{(n_{0})} \ar[r] & 0 \\ ε_{n_{0}} : & 0 \ar[r] & C^{(1)} \ar[u]^{id} \ar[r]^{i} & C^{(n_{0})} \ar[u]^{i_{1}^{(n_{0})}} \ar[r]^{j} & C^{(n_{0}−1)} \ar[u]^{i^{(n_{0}−1)}} \ar[r] & 0}

Here in standard notation of Subsection A.2.1, i_{1} = i_{1,n_{0}+1}, i′_{1} = i′_{1,n_{0}+1}, i = i_{1n_{0}}, j = j_{n_{0},n_{0}−1}, j_{1} = j_{n_{0}+1,n_{0}} and j′_{1} = j′_{n_{0}+1,n_{0}} (the dash means that the corresponding morphism is related to the second p-divisible group). We must construct an isomorphism f_{n_{0}+1} : C^{(n_{0}+1)} → C_{1}^{(n_{0}+1)} such that f_{n_{0}+1} ∘ i^{(n_{0})} = i_{1}^{(n_{0})}. Consider the following commutative diagram obtained from above two diagrams (A.7)

\xymatrix{0 → C^{(1)} ∏ C^{(1)} \ar[r]^{i_{1} ∏ i′_{1}} & C^{(n_{0}+1)} ∏_{C^{(n_{0})}}C_{1}^{(n_{0}+1)} \ar[r]^{(j_{1},j′_{1})} & C^{(n_{0})} → 0 \\ 0 → C^{(1)} ∏ C^{(1)} \ar[u]^{id} \ar[r]^{i ∏ i} & C^{(n_{0})} ∏_{C^{(n_{0}−1)}}C^{(n_{0})} \ar[u]^{i^{(n_{0})} ∏ i_{1}^{(n_{0})}} \ar[r]^{(j,j)} & C^{(n_{0}−1)} → 0 \ar[u]^{i^{(n_{0}−1)}}}

Notice that the morphisms of multiplication by p in C^{(n_{0}+1)} and C_{1}^{(n_{0}+1)} can be factored as follows

\xymatrix{C^{(n_{0}+1)} \ar[dr]_{j_{1}} \ar[rr]^{p} &  & C^{(n_{0}+1)} & C_{1}^{(n_{0}+1)} \ar[dr]_{j′_{1}} \ar[rr]^{p} &  & C_{1}^{(n_{0}+1)} \\  & C^{(n_{0})} \ar[ur]_{i^{(n_{0})}} &  &  & C^{(n_{0})} \ar[ur]_{i_{1}^{(n_{0})}}}

Documenta Mathematica 18 (2013) 547–619

Therefore, we obtain the following commutative diagram

(A.8)  \xymatrix{C_{1}^{(n_{0}+1)} ∏_{C^{(n_{0})}} C^{(n_{0}+1)} \ar[d]^{(j′_{1},j_{1})} \ar[r]^{p} & C_{1}^{(n_{0}+1)} ∏_{C^{(n_{0})}} C^{(n_{0}+1)} \\ C^{(n_{0})} \ar[r]^{∇} & C^{(n_{0})} ∏_{C^{(n_{0}−1)}} C^{(n_{0})} \ar[u]^{i_{1}^{(n_{0})} ∏ i^{(n_{0})}}}

(here ∇ is the diagonal morphism). Let α : C^{(1)} ∏ C^{(1)} → C^{(1)} be the cokernel of the diagonal morphism ∇ : C^{(1)} → C^{(1)} ∏ C^{(1)}. Clearly, ∇ and α are, resp., strict monomorphism and strict epimorphism. Set (D_{n_{0}+1}, α_{1}) = Coker ((i_{1} ∏ i′_{1}) ∘ ∇) and (D_{n_{0}}, α_{0}) = Coker ((i ∏ i) ∘ ∇). Applying α_{∗} to diagram (A.7) obtain the two lower rows of the following diagram

(A.9)  \xymatrix{0 \ar[r] & C^{(1)} \ar[r] & D_{0} \ar[r] & C^{(1)} \ar[r] & 0 \\ 0 \ar[r] & C^{(1)} \ar[u]^{id} \ar[r] & D_{n_{0}+1} \ar[r] \ar[u]^{s} & C^{(n_{0})} \ar[u]^{j_{n_{0}1}} \ar[r] & 0 \\ 0 \ar[r] & C^{(1)} \ar[u]^{id} \ar[r] & D_{n_{0}} \ar[r] \ar[u]^{u} & C^{(n_{0}−1)} \ar[r] \ar[u]^{i_{n_{0}−1,n_{0}}} & 0}

Note that the middle line of this diagram equals ε_{n_{0}+1} − ε′_{n_{0}+1} ∈ Ext (C^{(n_{0})}, C^{(1)}), and at the third row we have a trivial extension. This implies the existence of the first row of our diagram. As it was pointed out earlier, a splitting of the third line can be done via the morphism f from the commutative diagram

(A.10)  \xymatrix{ & C^{(n_{0})} ∏_{C^{(n_{0}−1)}} C^{(n_{0})} \ar[dr]^{α_{0}} \ar[dl]_{p_{1}−p_{2}} \\ C^{(1)} &  & D_{n_{0}} \ar[ll]_{f}}

(Notice that the morphism s : D_{n_{0}+1} → D_{0} is the cokernel of the composition

Kerf → D_{n_{0}} →^{u} D_{n_{0}+1}. )

Above diagram (A.8) means that the morphism of multiplication by p on C_{1}^{(n_{0}+1)} ∏_{C^{(n_{0})}} C^{(n_{0}+1)} factors through the diagonal embedding of C^{(n_{0})} into C^{(n_{0})} ∏_{C^{(n_{0}−1)}} C^{(n_{0})}. From diagram (A.10) it follows that p id_{D_{n_{0}+1}} factors through the embedding Kerf → D_{n_{0}} →^{u} D_{n_{0}+1}. Therefore, pD_{0} = 0 i.e. the first line in diagram (A.9) is an element of the trivial group Ext_{S_{1}}(C^{(1)}, C^{(1)}) = 0. So, the second row in (A.9) is a trivial extension, i.e. the extensions ε_{n_{0}+1} and ε′_{n_{0}+1} from diagrams (A.5) and (A.6) are equivalent. This implies the existence of isomorphism f_{n_{0}+1}. □

A.2.3. Splitting of extensions of p-divisible groups.

Theorem A.5. Suppose (C^{(n)}, i^{(n)})_{n≥0} is a p-divisible group in the category S and there are D_{1}, D_{2} ∈ S_{1} such that C^{(1)} ∈ Ext_{S_{1}}(D_{2}, D_{1}) and Ext_{S_{1}}(D_{1}, D_{2}) = 0. Then there is an exact sequence of p-divisible groups

0 ⟶ (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0} ⟶ (C^{(n)}, i^{(n)})_{n⩾0} ⟶ (C_{2}^{(n)}, i_{2}^{(n)})_{n⩾0} ⟶ 0

in S such that C_{1}^{(1)} = D_{1} and C_{2}^{(1)} = D_{2}.

Proof. We have the exact sequence 0 ⟶ D_{1} ⟶^{i} C^{(1)} ⟶^{j} D_{2} ⟶ 0. We must show for all n ⩾ 1, the existence of objects C_{1}^{(n)}, strict monomorphisms γ_{n} : C_{1}^{(n)} → C^{(n)} and i_{1}^{(n)} : C_{1}^{(n)} → C_{1}^{(n+1)} such that (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0} is a pdivisible group, the system (γ_{n})_{n⩾0} defines an embedding of this p-divisible group into the original p-divisible group (C^{(n)}, i^{(n)})_{n⩾0}, C_{1}^{(1)} = D_{1} and γ_{1} = i. Agree to use for all 0 ⩽ m ⩽ n, the notation i_{mn} and j_{nm} from Subsection A.2.1 for the original p-divisible group and set C^{(n)} = C_{n0}. Illustrate the idea of proof by considering the case n = 2. Set C_{11} = D_{1} and consider the following commutative diagram with exact rows ε_{2} and ε_{2}^{(1)} = i^{∗}ε_{2}:

(A.11)  \xymatrix{ε_{2} : 0 \ar[r] & C_{10} \ar[r]^{i_{12}} & C_{20} \ar[r]^{j_{21}} & C_{10} \ar[r] & 0 \\ ε_{2}^{(1)} : 0 \ar[r] & C_{10} \ar[u]^{id} \ar[r]^{i_{12}^{(1)}} & C_{21} \ar[u]^{γ_{2}^{(1)}} \ar[r]^{j_{21}^{(1)}} & C_{11} \ar[u]^{i} \ar[r] & 0}

Then γ_{2}^{(1)}∘p id_{C_{21}} = p id_{C_{20}}∘γ_{2}^{(1)} = i_{12}∘j_{21}∘γ_{2}^{(1)} = i_{12}∘i∘j_{21}^{(1)} = γ_{2}^{(1)}∘i_{12}^{(1)}∘i∘j_{21}^{(1)}. By Lemma A.2, γ_{2}^{(1)} is (strict) monomorphism. Therefore, p id_{C_{21}} = i_{21}^{(1)}∘i∘j_{21}^{(1)}. Then the morphism j_{∗} : Ext_{S}(C_{11}, C_{10}) → Ext_{S}(C_{11}, D_{2}) induces the following commutative diagram

\xymatrix{0 \ar[r] & C_{10} \ar[d]^{j} \ar[r]^{i_{12}^{(1)}} & C_{21} \ar[d]^{f} \ar[r]^{j_{21}^{(1)}} & C_{11} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & D_{2} \ar[r]^{α} & D_{21} \ar[r] & C_{11} \ar[r] & 0}

Here p id_{D_{21}} ∘f = f ∘p id_{C_{21}} = f ∘i_{12}^{(1)}∘i∘j_{21}^{(1)} = α∘j∘i∘j_{21}^{(1)} = 0. By Lemma A.1, f is (strictly) epimorphic. Therefore, pid_{D_{21}} = 0, i.e. D_{21} ∈ Ext_{S_{1}}(C_{11}, D_{2}) = 0. Then the exact sequence Hom_{S} − Ext_{S} implies the commutative diagram

\xymatrix{0 \ar[r] & C_{10} \ar[r]^{i_{12}^{(1)}} & C_{21} \ar[r]^{j_{21}^{(1)}} & C_{11} \ar[r] & 0 \\ 0 \ar[r] & C_{11} \ar[u]^{i} \ar[r]^{i_{12}^{(2)}} & C_{22} \ar[r]^{j_{21}^{(2)}} \ar[u]^{γ_{2}^{(2)}} & C_{11} \ar[u]^{id} \ar[r] & 0}

Verify that one can set C_{1}^{(2)} = C_{22} and i_{1}^{(1)} = i_{12}^{(2)}. Indeed,

γ_{2}^{(2)} ∘ p id_{C_{22}} = p id_{C_{21}} ∘ γ_{2}^{(2)} = (i_{12}^{(1)} ∘ i) ∘ (j_{21}^{(1)} ∘ γ_{2}^{(2)}) = γ_{2}^{(2)} ∘ i_{12}^{(2)} ∘ j_{21}^{(2)}

and because γ_{2}^{(2)} is monomorphism (use axiom SP1), p id_{C_{22}} = i_{12}^{(2)} ∘ j_{21}. Thus, we constructed a segment of length 2 of the p-divisible group (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0}. Consider the general case.

Lemma A.6. In the category S there are the following commutative diagrams with exact lines:

• for k ⩾ 1,

\xymatrix{E_{k}^{1} : 0 \ar[r] & C_{k−1,0} \ar[r]^{i_{k−1,k}^{(0)}} & C_{k0} \ar[r]^{j_{k1}^{(0)}} & C_{10} \ar[r] & 0 \\ 0 \ar[r] & C_{k−1,0} \ar[u]^{id} \ar[r]^{i_{k−1,k}^{(1)}} & C_{k1} \ar[r]^{j_{k1}^{(1)}} \ar[u]^{γ_{k}^{(1)}} & C_{11} \ar[u]^{i} \ar[r] & 0}

• for 2 ⩽ t ⩽ k,

\xymatrix{E_{k}^{t} : 0 \ar[r] & C_{k−1,t−2} \ar[r]^{i_{k−1,k}^{(t−1)}} & C_{k,t−1} \ar[r]^{j_{k1}^{(t−1)}} & C_{11} \ar[r] & 0 \\ 0 \ar[r] & C_{k−1,t−1} \ar[u]^{γ_{k−1}^{(t−1)}} \ar[r]^{i_{k−1,k}^{(t)}} & C_{kt} \ar[r]^{j_{k1}^{(t)}} \ar[u]^{γ_{k}^{(t)}} & C_{11} \ar[u]^{id} \ar[r] & 0}

• for 1 ⩽ t < k,

\xymatrix{Δ_{k}^{t} : 0 \ar[r] & C_{k−1,t−1} \ar[d]_{j_{k−1,k−2}^{(t−1)}} \ar[r]^{i_{k−1,k}^{(t)}} & C_{kt} \ar[d]_{j_{k,k−1}^{(t)}} \ar[r]^{j_{k1}^{(t)}} & C_{11} \ar[d]_{id} \ar[r] & 0 \\ 0 \ar[r] & C_{k−2,t−1} \ar[r]^{i_{k−2,k−1}^{(t)}} & C_{k−1,t} \ar[r]^{j_{k−1,1}^{(t)}} & C_{11} \ar[r] & 0}

• for 1 ⩽ t < k,

\xymatrix{Ω_{k}^{t} : 0 \ar[r] & C_{kt} \ar[d]_{j_{k,k−1}^{(t)}} \ar[r]^{γ_{k}^{(t)}} & C_{k,t−1} \ar[d]_{j_{k,k−1}^{(t−1)}} \ar[r]^{f_{k}^{(t)}} & D_{2} \ar[d]_{id} \ar[r] & 0 \\ 0 \ar[r] & C_{k−1,t} \ar[r]^{γ_{k−1}^{(t)}} & C_{k−1,t−1} \ar[r]^{f_{k−1}^{(t)}} & D_{2} \ar[r] & 0}

where for all indices k, C_{k0} = C^{(k)}, i_{k,k+1}^{(0)} = i^{(k)}, j_{k+1,1}^{(0)} = j_{k+1,1} and j_{k+1,k}^{(0)} = j_{k+1,k}.

Proof. Construct the diagram E_{1}^{1} by setting γ_{1}^{(1)} = i, j_{11}^{(0)} = id_{C_{10}}, j_{11}^{(1)} = id_{C_{11}}. Then for any k ⩾ 2, the upper row of E_{k}^{1} is the short exact sequence ε_{k} ∈ Ext_{S}(C_{10}, C_{k−1,0}) from the original p-divisible group (C_{k0}, i^{(k)})_{k≥0}. Then E_{k}^{1} is just a standard diagram relating ε_{k} and i^{∗}ε_{k}. For any k ≥ 2, we have (j_{k,k−1})_{∗}ε_{k} = ε_{k−1}, therefore, (j_{k,k−1})_{∗}(i^{∗}ε_{k}) = i^{∗}ε_{k−1} and we obtain Δ_{k}^{1}. The upper row of Ω_{k}^{1} is obtained from the middle column of E_{k}^{1} because Coker γ_{k}^{(1)} ≃ Coker i = (D_{2}, j). Similarly, the lower row of Ω_{k}^{1} is obtained from E_{k−1}^{1}. The left square of Ω_{k}^{1} is commutative by the definition of j_{k,k−1}^{(1)}. The right square is commutative because Ω_{k}^{1} relates diagrams E_{k}^{1} and E_{k−1}^{1}. Suppose now we are given integers k_{0} ⩾ 2 and t_{0} < k_{0} such that the required diagrams E_{k}^{t}, Δ_{k}^{t} and Ω_{k}^{t} have been already constructed for all k < k_{0} with all relevant t and for k = k_{0} with 1 ⩽ t ⩽ t_{0}. Clearly, all i_{k,k−1}^{(t)} and γ_{k}^{(t)} are strict

monomorphisms and all j_{k1}^{(t)}, j_{k,k−1}^{(t)} and f_{k}^{(t)} are strict epimorphisms.

Constructing E_{k_{0}}^{t_{0}+1}. Apply (f_{k_{0}−1}^{(t_{0})})_{∗} to E_{k_{0}}^{t_{0}}:

\xymatrix{ε_{k_{0}}^{(t_{0})} : 0 \ar[r] & C_{k_{0}−1,t_{0}−1} \ar[d]_{f_{k_{0}−1}^{(t_{0})}} \ar[r]^{i_{k_{0}−1,k_{0}}^{(t_{0})}} & C_{k_{0},t_{0}} \ar[d] \ar[r]^{j_{k_{0}1}^{(t_{0})}} & C_{11} \ar[d]_{id} \ar[r] & 0 \\ 0 \ar[r] & D_{2} \ar[r] & D^{∗} \ar[r] & C_{11} \ar[r] & 0}

Then Ker (C_{k_{0}t_{0}} → D^{∗}) = (C_{k_{0}−1,t_{0}}, i_{k_{0}−1,k_{0}}^{(t_{0})} ∘ γ_{k_{0}−1}^{(t_{0})}). Consider the strict

monomorphism γ_{k_{0}t_{0}} := γ_{k_{0}}^{(1)} ∘ … ∘ γ_{k_{0}}^{(t_{0})} : C_{k_{0}t_{0}} → C_{k_{0}0} and its analogue γ_{k_{0}−1,t_{0}−1} : C_{k_{0}−1,t_{0}−1} → C_{k_{0}−1,0}. Because t_{0} ≠ k_{0}, the diagrams Ω_{k_{0}}^{t_{0}} and E_{k_{0}}^{t} give the commutative diagram

\xymatrix{C_{k_{0}t_{0}} \ar[d]_{j_{k_{0},k_{0}−1}^{(t_{0})}} \ar[rr]^{γ_{k_{0}t_{0}}} &  & C_{k_{0}0} \ar[d]_{j_{k_{0},k_{0}−1}^{(0)}} \\ C_{k_{0}−1,t_{0}} \ar[r]^{γ_{k_{0}−1}^{(t_{0})}} & C_{k_{0}−1,t_{0}−1} \ar[d]_{i_{k_{0}−1,k_{0}}^{(t_{0})}} \ar[r]^{γ_{k_{0}−1,t_{0}−1}} & C_{k_{0}−1,0} \ar[d]_{i_{k_{0}−1,k_{0}}^{(0)}} \\  & C_{k_{0}t_{0}} \ar[r]^{γ_{k_{0}t_{0}}} & C_{k_{0}0}}

Then pid_{C_{k_{0}0}} = i_{k_{0}−1,k_{0}}^{(0)} ∘j_{k_{0},k_{0}−1}^{(0)} implies pid_{C_{k_{0}t_{0}}} = (i_{k_{0}−1,k_{0}}^{(t_{0})} ∘γ_{k_{0}−1}^{(t_{0})})∘j_{k_{0},k_{0}−1}^{(0)}, i.e. pid_{C_{k_{0}t_{0}}} factors through Ker(C_{k_{0}t_{0}} → D^{∗}) and pid_{D^{∗}} = 0. Then Ext_{S_{1}}(C_{11}, D_{2}) = 0 implies (f_{k_{0}−1}^{(t_{0})})_{∗}ε_{k_{0}}^{(t_{0})} = 0, and 6-terms Hom_{S} − Ext_{S} exact sequence gives E_{k_{0}}^{t_{0}+1}:

\xymatrix{0 \ar[r] & C_{k_{0}−1,t_{0}−1} \ar[r]^{i_{k_{0}−1,k_{0}}^{(t_{0})}} & C_{k_{0}t_{0}} \ar[r]^{j_{k_{0}1}^{(t_{0})}} & C_{11} \ar[r] & 0 \\ 0 \ar[r] & C_{k_{0}−1,t_{0}} \ar[u]_{γ_{k_{0}−1}^{(t_{0})}} \ar[r]^{i_{k_{0}−1,k_{0}}^{(t_{0}+1)}} & C_{k_{0},t_{0}+1} \ar[u]_{γ_{k_{0}}^{(t_{0}+1)}} \ar[r]^{j_{k_{0}1}^{(t_{0}+1)}} & C_{11} \ar[u]^{id} \ar[r] & 0}

We shall denote the rows of this diagram by ε_{k_{0}}^{(t_{0})} and ε_{k_{0}}^{(t_{0}+1)}.

Documenta Mathematica 18 (2013) 547–619

Constructing Δ_{k_{0}}^{t_{0}+1}. Assume that t_{0} + 1 < k_{0}. The above extension ε_{k_{0}}^{(t_{0}+1)} is not uniquely defined by ε_{k_{0}}^{(t_{0})}. Show that its choice can be done in such a way that the diagram Δ_{k_{0}}^{t_{0}+1} commutes. Consider the short exact sequences from Ω_{k_{0}−1}^{t_{0}}. They give rise to the following exact sequences of abelian groups, where H := Hom(C_{11}, D_{2}) and E := Ext(C_{11}, D_{2})

(A.12)

\xymatrix{H \ar[d]^{id} \ar[r] & Ext(C_{11}, C_{k_{0}−1,t_{0}}) \ar[d]_{j_{k_{0}−1,k_{0}−2∗}^{(t_{0})}} \ar[r]^{γ_{k_{0}−1∗}^{(t_{0})}} & Ext(C_{11}, C_{k_{0}−1,t_{0}−1}) \ar[d]^{j_{k_{0}−1,k_{0}−2∗}^{(t_{0}−1)}} \ar[r] & E \ar[d]^{id} \\ H \ar[r] & Ext(C_{11}, C_{k_{0}−2,t_{0}}) \ar[r]^{γ_{k_{0}−2∗}^{(t_{0})}} & Ext(C_{11}, C_{k_{0}−2,t_{0}−1}) \ar[r] & E}

As we saw earlier, the commutativity of E_{k_{0}}^{t_{0}+1} is equivalent to the following relation

(A.13)  (γ_{k_{0}−1}^{(t_{0})})_{∗}ε_{k_{0}}^{(t_{0}+1)} = ε_{k_{0}}^{(t_{0})}

From Δ_{k_{0}}^{t_{0}} it follows that ε_{k_{0}−1}^{(t_{0})} = (j_{k_{0}−1,k_{0}−2}^{(t_{0}−1)})_{∗}ε_{k_{0}}^{(t_{0})}, and from E_{k_{0}−1}^{t_{0}+1} it follows that (γ_{k_{0}−2}^{(t_{0})})_{∗}ε_{k_{0}−1}^{(t_{0}+1)} = ε_{k_{0}−1}^{(t_{0})}. Then (A.12) implies that ε_{k_{0}}^{(t_{0}+1)} from (A.13) can be chosen in such a way that (j_{k_{0}−1,k_{0}−2}^{(t_{0})})_{∗}ε_{k_{0}}^{(t_{0}+1)} = ε_{k_{0}−1}^{(t_{0}+1)}. This gives the diagram Δ_{k_{0}}^{t_{0}+1}.

Constructing Ω_{k_{0}}^{t_{0}+1}. The above arguments imply that the left squares of diagrams E_{k_{0}}^{t_{0}+1} and E_{k_{0}−1}^{t_{0}} are related via the following commutative diagram

\xymatrix{C_{k_{0}−1,t_{0}−1} \ar[rrr]^{i_{k_{0}−1,k_{0}}^{(t_{0})}} \ar[dr]^{j_{k_{0}−1,k_{0}−2}^{(t_{0}−1)}} &  &  & C_{k_{0}t_{0}} \ar[dl]^{j_{k_{0},k_{0}−1}^{(t_{0})}} \\  & C_{k_{0}−1,t_{0}−2} \ar[r]^{i_{k_{0}−2,k_{0}−1}^{(t_{0}−1)}} & C_{k_{0}−1,t_{0}−1} \\  & C_{k_{0}−2,t_{0}−1} \ar[u]_{γ_{k_{0}−2}^{(t_{0}−1)}} \ar[r]^{i_{k_{0}−2,k_{0}−1}^{(t_{0})}} & C_{k_{0}−1,t_{0}} \ar[u]^{γ_{k_{0}−1}^{(t_{0})}} \\ C_{k_{0}−1,t_{0}} \ar[rrr]^{i_{k_{0}−2,k_{0}−1}^{(t_{0}+1)}} \ar[ur]_{j_{k_{0}−1,k_{0}−2}^{(t_{0})}} \ar[uuu]_{γ_{k_{0}−1}^{(t_{0}−1)}} &  &  & C_{k_{0},t_{0}+1} \ar[uuu]^{γ_{k_{0}}^{(t_{0}+1)}} \ar[ul]^{j_{k_{0},k_{0}−1}^{(t_{0}+1)}}}

From diagrams Ω_{k_{0}−1}^{t_{0}}, E_{k_{0}}^{t_{0}+1} and E_{k_{0}−1}^{t_{0}} it follows that the induced map Cokerγ_{k_{0}}^{(t_{0}+1)} → Cokerγ_{k_{0}−1}^{(t_{0})} ≃ D_{2} is isomorphism. This is equivalent to the existence of diagram Ω_{k_{0}}^{t_{0}+1}. The lemma is proved. □

For any k ⩾ 1, set C_{kk} = C_{1}^{(k)}, i_{k−1,k}^{(k)} = i_{1}^{(k)}. Then use diagrams E_{k}^{k} to define the inductive system (C_{1}^{(k)}, i_{1}^{(k)})_{k⩾0}. Denote by γ_{k} the strict monomorphism γ_{k}^{(1)} ∘ … ∘ γ_{k}^{(k)} : C_{1}^{(k)} → C^{(k)}. From diagrams E_{k}^{t}, 1 ≤ t ≤ k, obtain the following commutative diagrams:

(A.14)  \xymatrix{0 \ar[r] & C^{(k−1)} \ar[r]^{i^{(k)}} & C^{(k)} \ar[r]^{j_{k1}} & C^{(1)} \ar[r] & 0 \\ 0 \ar[r] & C_{1}^{(k−1)} \ar[u]^{γ_{k−1}} \ar[r]^{i_{1}^{(k)}} & C_{1}^{(k)} \ar[u]^{γ_{k}} \ar[r]^{j_{k1}^{(k)}} & C_{1}^{(1)} \ar[u]^{γ_{1}} \ar[r] & 0}

It remains only to prove that the inductive system (C_{1}^{(n)}, i_{1}^{(n)})_{n≥0} is a p-divisible group in S. From diagrams E_{k}^{k} and Δ_{k}^{k−1} obtain the following commutative

Varieties with Bad Reduction at 3 Only 615

diagrams with exact rows

(A.15) \xymatrix{0 \ar[r] & C_{k−1,k−1} \ar[d]_{j_{k−1,k−2}^{(k−2)}∘γ_{k−1}^{(k−1)}} \ar[r]^{i_{k−1,k}^{(k)}} & C_{kk} \ar[d]^{j_{k,k−1}^{(k−1)}∘γ_{k}^{(k)}} \ar[r]^{j_{k1}^{(k)}} & C_{11} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & C_{k−2,k−2} \ar[r]^{i_{k−2,k−1}^{(k−1)}} & C_{k−1,k−1} \ar[r]^{j_{k−1,1}^{(k−1)}} & C_{11} \ar[r] & 0}

If k = 3 then the left vertical morphism of this diagram is equal to j_{21}^{(1)} ∘ γ_{2}^{(2)} = j_{21}^{(2)} and is a strict monomorphism. By induction all morphisms j′_{k,k−1} := j_{k,k−1}^{(k−1)} ∘ γ_{k}^{(k)} are strict epimorphisms and are included in the following commutative diagrams

(A.16)  \xymatrix{C^{(k)} \ar[r]^{j_{k,k−1}} & C^{(k−1)} \\ C_{1}^{(k)} \ar[u]^{γ_{k}} \ar[r]^{j′_{k,k−1}} & C_{1}^{(k−1)} \ar[u]^{γ_{k−1}}}

For 0 ⩽ m ⩽ n, set j′_{nm} = j′_{m+1,m} ∘ … ∘ j′_{n,n−1} and i′_{mn} = i′_{n−1,n} ∘ … ∘ i′_{m,m+1}. Composing diagrams (A.15) obtain the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & C_{1}^{(n−1)} \ar[d]^{j′_{n−1,m−1}} \ar[r]^{i′_{n−1,n}} & C_{1}^{(n)} \ar[d]^{j′_{nm}} \ar[r]^{j_{n1}^{(n)}} & C_{1}^{(1)} \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & C_{1}^{(m−1)} \ar[r]^{i′_{m−1,m}} & C_{1}^{(m)} \ar[r]^{j_{m1}^{(m)}} & C_{1}^{(1)} \ar[r] & 0}

Thus, i′_{n−1,n} induces the isomorphism Ker j′_{n−1,m−1} ≃ Ker j′_{nm}. Therefore, Ker j′_{nm} = (C_{1}^{(n−m)}, i′_{n−m,n}) if we prove that

(A.17)  Ker j′_{k1} = (C_{1}^{k−1}, i′_{k−1,k}).

As we noticed earlier, j′_{k1} = j_{21} ∘ … ∘ j_{k,k−1}. Therefore, diagrams (A.16) imply that j_{k1} ∘ γ_{k} = γ_{1} ∘ j′_{k1}. Now diagram (A.14) implies that γ_{1} ∘ j_{k1}^{(k)} = γ_{1} ∘j′_{k1} and, therefore, j_{k1}^{(k)} = j′_{k1} because γ_{1} is monomorphism. Hence equality (A.17) folows from diagram (A.14) and (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0} satisfies the part a) of the definition of p-divisible groups.

Documenta Mathematica 18 (2013) 547–619

From diagrams (A.14) and (A.16) one can easily obtain for all indices 0 ⩽ m ⩽ n, the commutativity of the following diagrams

\xymatrix{C^{(n)} \ar[r]^{j_{n,n−m}} & C^{(n−m)} \ar[r]^{i_{n−m,n}} & C^{(n)} \\ C_{1}^{(n)} \ar[u]^{γ_{n}} \ar[r]^{j′_{n,n−m}} & C_{1}^{(n−m)} \ar[u]^{γ_{n−m}} \ar[r]^{i′_{n−m,n}} & C_{1}^{(n)} \ar[u]^{γ_{n}}}

Because γ_{n} is monomorphism, the equality i_{n−m,n} ∘ j_{n,n−m} = p^{m}id_{C^{(n)}} implies the equality i′_{n−m,n}∘j′_{n,n−m} = p^{m}id_{C_{1}^{(n)}}. This gives the part b) of the definition

of p-divisible groups for (C_{1}^{(n)}, i_{1}^{(n)})_{n⩾0}. The proposition is proved. □

Appendix B. SAGE program

This program computes the class number of the field ℚ(\sqrt[3]{3}, ζ_{9}) and finds the basis f_{1}, f_{2}, …, f_{9} of the 3-subgroup of units in this field such that for the normalized 3-adic valuation v_{3} and all 1 ⩽ i ⩽ 9, the natural numbers a_{i} = 18v_{3}(f_{i} ± 1) are prime to 3 and 1 ⩽ a_{1} < a_{2} < ⋯ < a_{9}. The result appears as the vector af = (a_{1}, a_{2}, …, a_{9}) = (1, 2, 4, 5, 7, 8, 10, 13, 16).

sage: L.<b>=NumberField(x^3-3);
sage: R.<t>=L[]
sage: M.<c>=L.extension(t^6+t^3+1);
sage: X.<d>=M.absolute_field();
sage: h=X.class_number();
sage: e=list(X.unit_group().gens())
sage: def p(x):
...       for i in range(1,3):
...          if valuation(norm(X(x+2*i-3)),3)!=0:
...               break
...       return valuation(norm(X(x+2*i-3)),3)
...
...
sage: a=[p(x) for x in e]
sage: f=[e.pop(a.index(min(a)))]
sage: while len(e)!=0:
...       a=[p(x) for x in e]
...       i0=a.index(min(a))
...
...       for j in range(len(f)):
...           for k in range(5):
...               s=0
...               if p(f[j]^(3^k))>p(e[i0]):
...                   break

...
...               if p(e[i0])==p(f[j]^(3^k)):
...                   s=1
...                   break
...
...           if s==1:
...               for i in range(1,3):
...                   if p(e[i0])<p(e[i0]/(f[j]^(i*3^k))):
...                       e[i0]=e[i0]/(f[j]^(i*3^k))
...                       break
...
...               break
...           if j+1==len(f) and s==0:
...                     f.append(e.pop(i0))
...
sage: af=[p(x) for x in f];
sage: print h
sage: print af
1
[1, 2, 4, 5, 7, 8, 10, 13, 16]

Remark. First 4 lines introduce the field X = ℚ(\sqrt[3]{3}, ζ_{9}); its elements appear as polynomials in variable d of degree ⩽ 17. Then we find the class number of X and form the array e = (e[1], …, e[9]) of minimal generators of the group U/U^{3}, where U is the group of units in X. Next block gives a standard procedure to determine for any x ∈ U the maximal natural number p(x) such that x ± 1 is divisible precisely by π^{p(x)}, where π ∈ X, (π^{18}) = (3). The remaining part of the program is based on a standard technique from Linear algebra to rearrange the given system of generators e into a new system f with required properties. As a matter of fact, we use that the class number of X is prime to 3 (it equals 1) by allowing k < 5 on line 21. (Any unit x ≡ 1 mod π^{28} is a cube in the 3-completion of X by Hensel’s Lemma and, therefore, is a cube in X.) The last two lines contain the values of the class number of X and the exponents (a(f[1]), …, a(f[9])).

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Victor Abrashkin

Department of Mathematical Sciences Durham University Science Laboratories, South Rd, Durham DH1 3LE United Kingdom victor.abrashkin@durham.ac.uk and Steklov Mathematical Institute Gubkina str. 8, 119991 Moscow Russia 620

Documenta Mathematica 18 (2013)