Good Reduction of Affinoids

on the Lubin-Tate Tower

Jared Weinstein^{1}

Abstract. We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a onedimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their ℓ-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Granting a conjecture on the L-functions of this hypersurface, we find a link between the conjecture of Deligne-Carayol and the theory of Bushnell-Kutzko types, at least for certain class of wildly ramified supercuspidal representations of small conductor.

2010 Mathematics Subject Classification: 14G22, 22E50, 11F70 Keywords and Phrases: Lubin-Tate spaces, local Langlands correspondence, semistable model, rigid analysis

1 Introduction

Good Reduction of Affinoids on the Lubin-Tate Tower 983

Let F be a non-archimedean local field. By the local Langlands correspondence, the irreducible admissible representations of GL_{h}(F) are parametrized in a systematic way by h-dimensional representations of the Weil-Deligne group of F. This is established in [LRS93] for fields of positive characteristic and in [Hen00] and [HT01] for p-adic fields. The local Langlands correspondence appears in a geometric context; namely it is realized in the cohomology of the “Lubin-Tate tower”, a projective system of deformation spaces of a one-dimensional formal O_{F} -module of height h, cf. [Dri74]. We refer to this phenomenon as the conjecture of Deligne-Carayol, after the paper [Car90] which contains the precise statement of the conjecture. The papers [Car83] and [Car86] prove the conjecture for the case h = 2. The complete conjecture of Deligne-Carayol was proved in [Boy99] for fields of positive characteristic and in [HT01] for p-adic fields. Both papers involve embedding F into a global field and appealing to results from the theory of Shimura varieties or Drinfeld modular varieties. In [Har02], Harris identifies some unsettled problems in the study of the local Langlands correspondence, and top among these is the lack of a purely local proof of the correspondence. Bushnell and Kutzko’s theory of types [BK93] parametrizes admissible representations of GL_{h}(F) by finite-dimensional characters of open compact-mod-center subgroups. Naturally one hopes to link the parametrization by types to the parametrization by Weil-Deligne representations, so that one might obtain an “explicit local Langlands correspondence.” There have been some remarkable efforts in this direction, see [Hen92], [BH05a], [BH05b], [BH06], but these do not seem to interface with the geometric interpretation of the local Langlands correspondence afforded by the conjecture of Deligne-Carayol. Harris asks ([Har02], Question 9) whether the Bushnell-Kutzko types can be realized in the cohomology of analytic subspaces of the Lubin-Tate tower. In the present effort we demonstrate progress towards an affirmative answer to this question. We construct a family of open affinoids ℨ of the Lubin-Tate tower which have good reduction equal to a hypersurface \overline{ℨ} whose equation we give explicitly, cf. Thm. 1.1 below. The cohomology of these affinoids appears to contain exactly the Bushnell-Kutzko types for those supercuspidal representations whose Weil parameters are of the form Ind_{E/F} θ, where E/F is the unramified extension of degree h and θ is a character of the Weil group of E of conductor 𝔭_{E}^{2}, where 𝔭_{E} is the maximal ideal of O_{E}. We refer to these as the unramified supercuspidals of level π^{2}. The action of the Weil group on \overline{ℨ} is completely transparent. The question of whether the affinoids ℨ really do realize the local Langlands correspondence for such representations is reduced to the calculation of certain L-functions attached to \overline{ℨ}, see Conj. 1.6. It is hoped that this paper will initiate a systematic study of open affinoids with good reduction in the Lubin-Tate tower. The best outcome would be the construction of a semistable model for the Lubin-Tate spaces, using an appropriate covering by open affinoids. This is precisely what is done in [CM06] for the classical modular curves X_{0}(Np^{3}), and in [Weib] for Lubin-Tate curves with arbitrary level structure. Then the weight spectral sequence of Rapoport-Zink [RZ80] would compute the cohomology of the Lubin-Tate tower in terms of the reduction of the semistable model. A purely local proof of the conjecture of Deligne-Carayol would then be reduced to the computation of the zeta functions associated to the components of the reduction of the semistable model. Before stating our main theorem, we introduce some notation. We write 𝔛(π^{n}), n ≥ 0, for the system of rigid-analytic spaces comprising the Lubin-Tate tower of deformations of a height h one-dimensional formal O_{F} -module with Drinfeld level π^{n} structure; see §2.1 for definitions. Crucial to the analysis are the “canonical points” of 𝔛(π^{n}) arising from the canonical liftings of Gross [Gro86]: these are the deformations with extra endomorphisms by the ring of integers in a separable extension E/F. Such a point is defined over the extension E_{n}/Ê^{nr} obtained by adjoining the π^{n}-division points of a formal Lubin-Tate O_{E}-module of height one. In our analysis we concentrate on those canonical points for which the associated extension E/F is unramified. We refer to these as unramified canonical points. By performing explicit computations with coordinates, we find certain affinoid neighborhoods around each unramified canonical point x which have good reduction. These neighborhoods lie in a space intermediate in the covering 𝔛(π^{2}) → 𝔛(π), which we call 𝔛(K_{x,2}) = 𝔛(π^{2})/K_{x,2}; for details, see §4.2. Briefly put, x determines an embedding of O_{F} -algebras O_{E} ↪ M_{n}(O_{F}), and K_{x,2} is the congruence subgroup defined by

1The author was supported by NSF Postdoctoral Fellowship DMS-0803089.

Documenta Mathematica 15 (2010) 981–1007

K_{x,2} = {g ∈ 1 + πM_{n}(O_{F}) | Tr((g − 1)O_{E}) ⊂ 𝔭_{F}^{2} } .

Our main result is:

Theorem 1.1. Assume that F has positive characteristic, with residue field F_{q}. Let x ∈ 𝔛(π^{2}) be an unramified canonical point. There exists an open affinoid neighborhood ℨ of the image of x in 𝔛(K_{x,2}) whose reduction is the smooth hypersurface \overline{ℨ} in the variables V_{1}, …, V_{h} defined by the equation

det (\begin{matrix}V_{1}^{q^{h}} − V_{1} & V_{2}^{q^{h}} − V_{2} & V_{3}^{q^{h}} − V_{3} & ⋯ & V_{h−1}^{q^{h}} − V_{h−1} & V_{h}^{q^{h}} − V_{h} \\ 1 & V_{1}^{q} & V_{2}^{q} & ⋯ & V_{h−2}^{q} & V_{h−1}^{q} \\ 0 & 1 & V_{1}^{q^{2}} & ⋯ & V_{h−3}^{q^{2}} & V_{h−2}^{q^{2}} \\ ⋮ & & & ... & & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & V_{1}^{q^{h−1}} \end{matrix}) = 0.

Remark 1.2. Let R be the noncommutative polynomial ring F_{q^{h}}[τ]/(τ^{h+1}), whose multiplication law is given by τα = α^{q}τ, α ∈ F_{q^{h}}. Let A = R ⊗_{F_{q^{h}}} F_{q^{h}}[V_{1}, …, V_{h}], and let Φ: A → A be the R-linear endomorphism which sends V_{i} to V_{i}^{q}. Let g = 1 + V_{1}τ + ⋯ + V_{h}τ^{h} ∈ A^{×}; then the coefficient of τ^{n} in Φ^{h}(g)g^{−1} is the determinant appearing in Thm. 1.1. This shows that the hypersurface \overline{ℨ} admits a large group of automorphisms, namely R^{×}. See §5.3 for an interpretation of this automorphism group in terms of the Jacquet-Langlands correspondence.

Remark 1.3. We expect the condition char F > 0 to be unnecessary. This condition enables us to write down explicit models for universal deformations of formal O_{F} -modules with level structure, as in §2.2. It may be possible to remove this condition if one is more careful with error terms.

Remark 1.4. In Yoshida’s paper [Yos10] the space 𝔛(π) is treated, with no condition on the characteristic of F. In that case one finds an affinoid subdomain of 𝔛(π) ⊗ E_{1} whose reduction is the Deligne-Lusztig variety for GL_{h}(k), see §3.5. Based on this calculation, Yoshida proceeds to show that the vanishing cycles of 𝔛(π) realize the local Langlands correspondence for supercuspidal representations “of depth zero”.

Remark 1.5. Thm. 1.1 agrees well with our work in [Weib], which gives a detailed description of a stable reduction of the tower 𝔛(π^{n}) when h = 2. In this case the curve \overline{ℨ} is isomorphic over F_{q} to a disjoint union of copies of the “Hermitian curve” Y + Y^{q} = V^{q+1}. The Hermitian curve also happens to be isomorphic over \overline{F}_{q} to the Deligne-Lusztig curve for SL_{2}(F_{q}), but this seems to be a coincidence which does not persist for h > 2.

In order to apply Thm. 1.1 to the conjecture of Deligne-Carayol, it will be necessary to calculate the compactly supported ℓ-adic cohomology of \overline{ℨ}, ℓ ≠ p, as a module for the action of the stabilizer of ℨ in GL_{2}(O_{F}), which is the group U^{1} = 1 + πM_{h}(O_{F}). This in turn is equivalent to the calculation of the L-functions of some ℓ-adic sheaves on affine (h − 1)-space. To wit, let X be the hypersurface over F_{q^{h}} whose equation is the one appearing in Thm. 1.1. Then X is an Artin-Schreier cover of A^{h−1}/F_{q^{h}} with Galois group F_{q^{h}}. For each character ψ of F_{q^{h}} with values in \overline{Q}_{ℓ}^{×}, let ℒ_{ψ} be the corresponding lisse rank one sheaf on A^{h−1}. Then the zeta function Z(X, t) factors as a product of the L-functions L(A^{h−1}, ℒ_{ψ}, t) as ψ runs over characters of F_{q^{h}}.

Conjecture 1.6. Suppose ψ does not factor through Tr_{F_{q^{h}} /F_{q^{d}}} for any proper divisor d of h. Then

L(A^{h−1}, ℒ_{ψ}, t) = (1 + (−1)^{h}q^{\frac{h(h−1)}{2}} t)^{(−1)^{h}q^{\frac{h(h−1)}{2}}} .

The formula in Conj. 1.6 is striking: it implies that the contribution of the ψ-part of the Euler characteristic H_{c}^{∗}(X ⊗ \overline{F}_{q}, \overline{Q}_{ℓ}) to the quantities #X(F_{q^{h}}), #X(F_{q^{2h}}), … is the maximum possible under the constraints of the Riemann hypothesis for X. In fact we strongly suspect that X has the maximum number of F_{q^{hn}}-rational points relative to its compactly supported Betti numbers. More to the point, Conj. 1.6 would also imply that H_{c}^{h−1}(ℨ, \overline{Q}_{ℓ}) realizes the Bushnell-Kutzko types for the unramified supercuspidals of level π^{2}, and that the action of the Weil group of F on ℨ is in accord with the local Langlands correspondence. We postpone the details of this claim for future work, but see [Weia], §4 and §5 for a comprehensive calculation in the case h = 2. Conj. 1.6 itself can be verified quite easily for h = 2, in which case X is a disjoint union of q copies of the Hermitian curve Y^{q} + Y = X^{q+1}: this curve is “maximal” over F_{q^{2}} in the sense that it attains the Hasse-Weil bound for the maximum number of F_{q^{2}}-rational points. Conj. 1.6 can be verified numerically for small values of q and h > 2, but unfortunately we cannot give a general proof at this time. The polynomial on the right-hand side of the equation in Thm. 1.1 is degenerate in the sense of [AS89], which frustrates efforts to determine even the degree of the rational function L(A^{h−1}, ℒ_{ψ}, t).

The construction of the explicit local Langlands correspondence for unramified supercuspidals appears in [Hen92]. A salient feature of that paper is the discrepancy between two means of passing from a regular character of E^{×} to a supercuspidal representation of GL_{h}(F). The first construction is the local Langlands correspondence, the second construction is induction from a compact-mod-center subgroup, and the discrepancy, which appears exactly when h is even, manifests as the nontrivial unramified quadratic character of E^{×}. Granting Conj. 1.6, we arrive at a geometric explanation for this behavior in terms of the eigenvalue of Frobenius on the middle cohomology of the hypersurface X, for these are positive if and only if h is odd. In the subsequent papers [BH05a] and [BH05b] on the explicit local Langlands correspondence there is a systematic treatment of this discrepancy between the two constructions in the “essentially tame” case; we find it very likely that this discrepancy can always be explained by the behavior of Frobenius eigenvalues acting on the cohomology of an open affinoid in the Lubin-Tate tower having good reduction. We outline our work: In §2, we review the relevant background material from [Dri74] on one-dimensional formal modules and the Lubin-Tate tower. In §3, we impose the condition that char F > 0 and establish a functorial construction of top exterior powers of one-dimensional formal O_{F} -modules which may be of independent interest. The heart of the paper is §4. Given an unramified canonical point x in 𝔛(π^{2}), we construct a coordinate Y on that space which is invariant under K_{x,2}. The coordinate Y is integral on a certain affinoid neighborhood of x in 𝔛(π^{2}), and the reduction of the minimal polynomial for Y over the ring of integral functions on 𝔛(1) gives the equation appearing in Thm. 1.1. We conclude in §5 with some basic observations about the hypersurface \overline{ℨ} which we hope will illuminate the formulas in Conj. 1.6 and motivate future work linking Thm. 1.1 to the local Langlands and Jacquet-Langlands correspondences for GL_{h}(F).

2 Preliminaries on formal modules

2.1 Definitions

Throughout this paper, F is a local non-archimedean field with ring of integers O_{F}, uniformizer π and residue field k having cardinality q, a power of the prime p. Let 𝔭 be the maximal ideal of O_{F}, and let v be the valuation on F, normalized so that v(π) = 1. We also use v for the unique extension of this valuation to finitely ramified extension fields E of F contained in the completion of the separable closure of F.

Definition 2.1. Let R be a commutative O_{F} -algebra, with structure map i: O_{F} → R. A formal one-dimensional O_{F} -module over R is a power series ℱ (X, Y) = X + Y + ⋯ ∈ R⟦X, Y ⟧ which is commutative, associative, admits 0 as an identity, together with a power series [a]_{ℱ} (X) ∈ R⟦X⟧ for each a ∈ O_{F} satisfying [a]_{ℱ} (X) ≡ i(a)X (mod X^{2}) and ℱ ([a]_{ℱ} (X), [a]_{ℱ} (Y)) = [a]_{ℱ} (ℱ (X, Y)).

The addition law on a formal O_{F} -module ℱ will usually be written X +_{ℱ} Y. If ℱ and ℱ′ are two formal O_{F} -modules, there is an evident notion of an isogeny ℱ → ℱ′, and Hom(ℱ , ℱ′) has the structure of an O_{F} -module. If R is a k-algebra, we either have [π]_{ℱ} (X) = 0 or else [π]_{ℱ} (X) = f(X^{q^{h}}) for some power series f(X) with f′(0) ≠ 0. In the latter case, we say ℱ has height h over R. Fix an integer h ≥ 1. Let Σ be a one-dimensional formal O_{F} -module over \overline{k} of height h. The functor of deformations of Σ to complete local Noetherian Ô_{F^{nr}}- algebras is representable by a universal deformation ℱ^{univ} over an algebra A which is isomorphic to the power series ring Ô_{F^{nr}}⟦u_{1}, …, u_{h−1}⟧ in (h − 1) variables, cf. [Dri74]. That is, if A is a complete local Ô_{F}^{nr}-algebra with maximal ideal P, then the isomorphism classes of deformations of Σ to A are given exactly by specializing each u_{i} to an element of P in ℱ^{univ}.

2.2 The universal deformation in the positive characteristic case

The results of the previous paragraph take a very simple form in the equal characteristic case. Assume char F = p, so that F = k((π)) is the field of Laurent series over k in one variable, with O_{F} = k⟦π⟧. Then a model for Σ is given by the simple rules

X +_{Σ} Y = X + Y
[ζ]_{Σ}(X) = ζX, ζ ∈ k
[π]_{Σ}(X) = X^{q^{h}}

The universal deformation ℱ^{univ} also has a simple model over A:

X +_{ℱ^{univ}} Y = X + Y
[ζ]_{ℱ^{univ}} (X) = ζX, ζ ∈ k
[π]_{ℱ^{univ}} (X) = πX + u_{1}X^{q} + ⋯ + u_{h−1}X^{q^{h−1}} + X^{q^{h}}.  (2.2.1)

Let O_{B} = End Σ, and let B = O_{B} ⊗_{O_{F}} F. Then B is the central division algebra over F of invariant 1/h. Let k_{h}/k be the field extension of degree h: then O_{B} is generated by the unramified extension O_{E} = k_{h}⟦π⟧ of O_{K} of degree h, which acts on Σ in an evident way, together with the endomorphism Φ(X) = X^{q}. (The relations are Φ^{h} = π and Φζ = ζ^{q}Φ, ζ ∈ k_{h}.) Inasmuch as A = Ô_{F}^{nr}⟦u_{1}, …, u_{h−1}⟧ is the moduli space of deformations of Σ, the automorphism group Aut Σ = O_{B}^{×} acts naturally on A. It is natural to ask how O_{B}^{×} acts on the level of coordinates. The action of an element ζ ∈ k_{n}^{×} is simple enough: ζ(u_{i}) = ζ^{q^{i}−1}u_{i}, i = 1, …, h − 1. On the other hand the action of an element such as 1 + Φ ∈ O_{B}^{×} seems difficult to give explicitly.

2.3 Moduli of deformations with level structure

Let A be a complete local O_{F} -algebra with maximal ideal M, and let ℱ be a one-dimensional formal O_{F} -module over A, and let h ≥ 1 be the height of ℱ ⊗ A/M.

Definition 2.2. Let n ≥ 1. A Drinfeld level π^{n} structure on ℱ is an O_{F} - module homomorphism ϕ: (π^{−n}O_{F} /O_{F})^{⊕h} → M for which the relation

∏_{x∈(𝔭^{−1}/O_{F})^{⊕h}} (X − ϕ(x)) | [π]_{ℱ} (X)

holds in A⟦X⟧. If ϕ is a Drinfeld level π^{n} structure, the images under ϕ of the standard basis elements (π^{−n}, 0, …, 0), …, (0, 0, …, π^{−n}) of (𝔭^{−n}/O_{F})^{⊕h} form a Drinfeld basis of ℱ [π^{n}].

Fix a formal O_{F} -module Σ of height h over \overline{k}. Let A be a noetherian local Ô_{F}^{nr}- algebra such that the structure morphism Ô_{F}^{nr} → A induces an isomorphism between residue fields. A deformation of Σ with level π^{n} structure over A is a triple (ℱ , ι, ϕ), where ι: ℱ ⊗ \overline{k} → Σ is an isomorphism of O_{F} -modules over \overline{k} and ϕ is a Drinfeld level π^{n} structure on ℱ.

Proposition 2.3. [Dri74] The functor which assigns to each A as above the set of deformations of Σ with Drinfeld level π^{n} structure over A is representable by a regular local ring A(π^{n}) of relative dimension h − 1 over Ô_{F}^{nr}. Let X_{1}^{(n)}, …, X_{h}^{(n)} ∈ A(π^{n}) be the corresponding Drinfeld basis for ℱ^{univ}[π^{n}]; then these elements form a set of regular parameters for A(π^{n}).

There is a finite injection of Ô_{F}^{nr}-algebras A(π^{n}) → A(π^{n+1}) corresponding to the obvious degeneration map of functors. We therefore may consider A(π^{n}) as a subalgebra of A(π^{n+1}), with the equation [π]_{u} (X_{i}^{(n+1)}) = X_{i}^{(n)} holding

in A(π^{n+1}). Let X(π^{n}) = Spf A(π^{n}), so that X(π^{n}) is a formal scheme of relative dimension h − 1 over Spf Ô_{F}^{nr}. Let 𝔛(π^{n}) be the generic fiber of X(π^{n}); then 𝔛(π^{n}) is a rigid analytic variety. The coordinates X_{i}^{(n)} are then analytic functions on 𝔛(π^{n}) with values in the open unit disc. We have that 𝔛(1) is the rigid-analytic open unit polydisc of dimension h − 1. The group GL_{h}(O_{F} /π^{n}O_{F}) acts on the right on 𝔛(π^{n}) and on the left on A(π^{n}). The degeneration map 𝔛(π^{n}) → 𝔛(1) is Galois with group GL_{h}(O_{F} /π^{n}O_{F}). For an element M ∈ GL_{h}(O_{F} /π^{n}O_{F}) and an analytic function f on 𝔛(π^{n}), we write M(f) for the translated function z ↦ f(zM). When f happens to be one of the parameters X_{i}^{(n)}, there is a natural definition of M (X_{i}^{(n)}) when M ∈ M_{h}(O_{F} /π^{n}O_{F}) is an arbitrary matrix: if M = (a_{ij}), then

M (X_{i}^{(n)}) = [a_{i1}]_{ℱ^{univ}} (X_{1}^{(n)}) +_{ℱ^{univ}} ⋯ +_{ℱ^{univ}} [a_{ih}]_{ℱ^{univ}} (X_{h}^{(n)}) . (2.3.1)

3 Determinants

A natural first question in the study of the Lubin-Tate tower 𝔛(π^{n}) is to compute its zeroth cohomology; i.e. to determine its geometrically connected components along with the appropriate group actions. This question is answered completely by Strauch in [Str08b]. Let LT be a one-dimensional formal O_{F} -module over Ô_{F^{nr}} for which LT ⊗\overline{k} has height one. Let F_{0} = F̂^{nr}, and for n ≥ 1, let F_{n} = F_{0}(LT[π^{n}]) be the classical Lubin-Tate extension. Let χ: Gal(F_{n}/F_{0}) → (O_{F} /π^{n}O_{F})^{×} be the isomorphism of local class field theory, so that Gal(F_{n}/F_{0}) acts on LT[π^{n}] through χ. Finally, let 𝔛_{LT}(π^{n}) be the (zero-dimensional) space of deformations of LT ⊗\overline{k} with Drinfeld π^{n} structure, so that 𝔛_{LT}(π^{n})(F_{n}) is the set of bases for LT[π^{n}](F_{n}) as a free (O_{F} /π^{n}O_{F})- module of rank one. We now paraphrase [Str08b], Thm. 4.4 in the context of the rigid-analytic spaces 𝔛(π^{n}).

Theorem 3.1. The geometrically connected components of 𝔛(π^{n}) are defined over F_{n}, and there is a bijection

π_{0}(𝔛(π^{n}) ⊗ F_{n}) ˜ ⟶𝔛_{LT}(π^{n})(F_{n}).

Under this bijection, the action of an element (g, b, τ) in GL_{h}(O_{F}) × O_{B}^{×} × Gal(F_{n}/F_{0}) on 𝔛_{LT}(π^{n})(F_{n}) is through the character

(g, b, τ) ↦ det(g) N_{B/F} (b)^{−1}χ(τ)^{−1} ∈ (O_{F} /π^{n}O_{F})^{×}.  (3.0.2)

(In [Str08b], π_{0}(𝔛(π^{n}) ⊗ C_{π}) is identified with π_{0}(Spec(F_{n} ⊗_{F_{0}} C_{π})), where C_{π} is the completion of a separable closure of F. But this latter π_{0}, being the set of F_{0}-linear embeddings of F_{n} into C_{π}, is the same as the set of bases for LT[π^{n}](C_{π}). Thus Thm. 3.1 carries the same content as the theorem cited in [Str08b].) As noted in the introduction to [Str08b], Thm. 3.1 suggests a determinant functor ℱ ↦ Λ^{h}ℱ assigning to each deformation ℱ of Σ a deformation Λ^{h}ℱ of LT ⊗\overline{k}. This functor would of course identify the top exterior power of the Tate module T (ℱ) with T (Λ^{h}ℱ). In this section we provide just such a determinant functor in the case of equal characteristic, taking advantage of the explicit model of the universal deformation ℱ^{univ} described in §2.2. More precisely we prove:

Theorem 3.2. Assume char F > 0. For each n ≥ 1 there exists a morphism

μ_{n}: ℱ^{univ}[π^{n}] × ⋯ × ℱ^{univ}[π^{n}] → LT[π^{n}] ⊗ A

of group schemes over A = Ô_{F^{nr}}⟦u_{1}, …, u_{h−1}⟧ which is O_{F} -multilinear and alternating, and which satisfies the following properties:

1. The maps μ_{n} are compatible in the sense that

μ_{n}([π]_{ℱ^{univ}} (X_{1}), …, [π]_{ℱ^{univ}} (X_{h})) = μ_{n−1}(X_{1}, …, X_{h})

for n ≥ 2.

2. If X_{1}, …, X_{h} are sections of ℱ^{univ}[π^{n}] over an A-algebra R which form a Drinfeld level π^{n} structure, then μ_{n}(X_{1}, …, X_{h}) is a Drinfeld level π^{n} structure for LT[π^{n}] ⊗ R.

Remark 3.3. It is also possible to show that μ_{n} transforms the action of GL_{h}(O_{F})×O_{B}^{×}×Gal(F_{n}/F̂^{nr}) on ℱ^{univ}[π^{n}]×⋯×ℱ^{univ}[π^{n}] into the character defined in Eq. (3.0.2), but we will not be needing this.

The proof of Thm. 3.2 will occupy §3.1 and §3.3. Up to isomorphism there is only one formal O_{F} -module LT whose reduction has height one, so we are free to choose a model for it. For the remainder of the paper, LT will denote the formal O_{F} -module over Ô_{F^{nr}} with operations

X +_{LT} Y = X + Y
[α]_{LT}(X) = αX, α ∈ k
[π]_{LT}(X) = πX + (−1)^{h−1}X^{q}.

3.1 Determinants of level π structures

First define the polynomial in h variables

μ(X_{1}, …, X_{h}) = det (X_{i}^{q^{j}}) ∈ k[X_{1}, …, X_{h}]

(the exponent j ranges from 0 to h − 1). Then μ is a k-linear alternating form, known as the Moore determinant, cf. [Gos96], Ch. 1. We will need two simple identities involving μ. The first is

∏_{0≠a∈k^{h}}(a_{1}X_{1} + ⋯ + a_{h}X_{h}) = (−1)^{h}μ(X_{1}, …, X_{h})^{q−1},  (3.1.1)

in which the product runs over nonzero vectors a = (a_{1}, …, a_{h}) in k^{h}. Second, there is the identity

[π]_{LT}(μ(X_{1}, …, X_{n})) = det ([π]_{ℱ^{univ}} (X_{i}) | X_{i}^{q} | ⋯ | X_{i}^{q^{h−1}})_{1≤i≤h} , (3.1.2)

valid in A[X_{1}, …, X_{n}]. This is easily seen by expanding the first column of the matrix according to Eq. (2.2.1).

Lemma 3.4. If X_{1}, …, X_{h} are sections of ℱ^{univ}[π], then μ(X_{1}, …, X_{h}) is a section of LT[π]. If the X_{i} form a Drinfeld basis for ℱ^{univ}[π], then μ(X_{1}, …, X_{h}) constitutes a Drinfeld basis for LT[π].

Proof. Suppose X_{1}, …, X_{h} are sections of ℱ^{univ}[π] over an A-algebra R. Then the claim that μ(X_{1}, …, X_{h}) is annihilated by [π]_{LT} follows from Eq. (3.1.2). Now assume that X_{1}, …, X_{h} is a Drinfeld basis for ℱ^{univ}[π]. This means that

∏_{a∈k^{h}} (T − (a_{1}X_{1} + ⋯ + a_{h}X_{h})) divides [π]_{ℱ^{univ}} (T)

in R⟦T ⟧, hence in R[T]. Since [π]_{ℱ^{univ}} (T) is monic, these polynomials are equal:

∏_{a∈k^{h}} (T − (a_{1}X_{1} + ⋯ + a_{h}X_{h})) = πT +u_{1}T^{q} +⋯+u_{h−1}T^{q^{h−1}} +T^{q^{h}} (3.1.3)

Equating coefficients of T and using Eq. (3.1.1) shows that

μ(X_{1}, …, X_{h})^{q−1} = (−1)^{h}π.

On the other hand,

∏_{a∈k}(T − aμ(X_{1}, …, X_{h})) = T^{q} − μ(X_{1}, …, X_{h})^{q−1}T = (−1)^{h−1}[π]_{LT}(T),

which shows that μ(X_{1}, …, X_{h}) forms a Drinfeld basis for LT[π] ⊗ R.

3.2 Good reduction of an affinoid in 𝔛(π)

In this interlude we find an affinoid in 𝔛(π) whose reduction is the Deligne-Lusztig variety for GL_{h}(k). This is nothing new in light of [Yos10], Prop. 6.15, but it will give a flavor of the corresponding calculation for 𝔛(π^{2}).

Proposition 3.5. There is an isomorphism of local Ô_{F^{nr}}-algebras

\frac{Ô_{F^{nr}}⟦X_{1}, …, X_{h}⟧}{μ(X_{1}, …, X_{h})^{q−1} − (−1)^{h}π} ˜ ⟶A(π)

carrying X_{i} onto X_{i}^{(1)}.

Proof. Let A(π)′ = Ô_{F^{nr}}⟦X_{1}, …, X_{h}⟧/(μ(X_{1}, …, X_{h})^{q−1} − (−1)^{h}π). By Lemma 3.4 there is unique homomorphism A(π)′ → A(π) of Ô_{F^{nr}}-algebras carrying X_{i} onto X_{i}^{(1)}. Since the X_{i}^{(1)} form a system of regular local parameters of A(π), this homomorphism is surjective. The algebra A(π) is a Galois extension of A with group GL_{h}(k). But we can also furnish A(π)′ with the structure of an A-algebra, by identifying u_{i} ∈ A with the coefficient of T^{q^{i}} on the left-hand side of Eq. (3.1.3). Then A(π)′ becomes a Galois extension of A with group GL_{h}(k) as well, and the homomorphism A(π)′ → A(π) respects the A-algebra structure. We conclude that A(π)′ → A(π) is an isomorphism.

Now let E/F be the unramified extension of degree h, and let E_{1}/E^{nr} be the extension obtained by adjoining a root ϖ of X^{q^{h}−1} − (−1)^{h}π. Then E_{1}/E^{nr} is totally tamely ramified of degree q^{h} − 1. Let 𝔛(1)^{ts} ⊂ 𝔛(1) ⊗ E_{1} be the affinoid polydisc defined by the conditions

v(u_{i}) ≥ v(ϖ^{q^{h}−q^{i}}) = \frac{q^{h} − q^{i}}{q^{h} − 1}

The notation is borrowed from [CM06]: This is exactly the domain on which ℱ^{univ}[π] admits no canonical subgroups; i.e. where ℱ^{univ} is “too supersingular”. Whenever ℱ is a deformation of Σ lying in 𝔛(1)^{ts}, all nonzero roots of ℱ [π] have valuation equal to v(ϖ). By applying the change of variables X_{i} = ϖV_{i} to Prop. 3.5 we find:

Theorem 3.6. The preimage of 𝔛(1)^{ts} in 𝔛(π) ⊗ E_{1} has reduction isomorphic to the smooth affine hypersurface over \overline{k} with equation μ(V_{1}, …, V_{h})^{q−1} = 1.

3.3 Determinants of structures of higher level.

Now let n ≥ 1, and suppose X_{1}, …, X_{h} are sections of ℱ^{univ}[π^{n}]. We write [π^{a}]_{u}(X) as an abbreviation for [π^{a}]_{ℱ^{univ}} (X). We define the form μ_{n} by

μ_{n}(X_{1}, …, X_{h}) = ∑_{(a_{1},...,a_{h})} μ ([π^{a_{1}}]_{u}(X_{1}), …, [π^{a_{h}}]_{u}(X_{h})) ,

where the sum runs over tuples of integers (a_{1}, …, a_{h}) with 0 ≤ a_{i} ≤ n − 1 whose sum is (h − 1)(n − 1). It is clear that μ_{n} is k-multilinear and alternating in X_{1}, …, X_{h}. Before proving that μ_{n} is O_{F} -linear, we will show:

Proposition 3.7. For sections X_{1}, …, X_{h} of ℱ^{univ}[π^{n}], we have

[π]_{LT}(μ_{n}(X_{1}, …, X_{h})) = μ_{n−1}([π]_{u}(X_{1}), …, [π]_{u}(X_{h})).

In particular μ_{n}(X_{1}, …, X_{h}) is a section of the group scheme LT[π^{n}].

Proof. Let a = (a_{1}, …, a_{h}) be a tuple of nonnegative integers. Write [π^{a}](X) for the tuple ([π^{a_{1}}]_{u}(X_{1}), …, [π^{a_{h}}]_{u}(X_{h})). Applying Eq. (3.1.2) we find

[π]_{LT}(μ ([π^{a}](X))) = det ([π^{a_{i}+1}]_{u}(X_{i}) | [π^{a_{i}}]_{u}(X_{i})^{q} | ⋯ | [π^{a_{i}}]_{u}(X_{i})^{q^{h−1}})

= ∑_{σ∈S_{h}} sgn(σ)[π^{a_{σ(1)}+1}]_{u} (X_{σ(1)}) ∏_{j=1}^{h−1}[π^{a_{σ(j+1)}}]_{u} (X_{σ(j+1)})^{q^{j}}

Now assume the X_{i} are sections of ℱ^{univ}[π^{n}]: this means that the terms in the sum with a_{σ(1)} = n − 1 vanish. The expression [π]_{LT}(μ_{n}(X_{1}, …, X_{n})) is thus a sum over pairs (a, σ), where σ ∈ S_{h} is a permutation and a = (a_{1}, …, a_{h}) is a tuple of integers satisfying the conditions

1. 0 ≤ a_{i} ≤ n − 1

2. a_{σ(1)} < n − 1

3. ∑_{i} a_{i} = (n − 1)(h − 1)

Let b = (b_{1}, …, b_{h}) be the tuple defined by

b_{j} = \begin{cases}a_{j}, & j = σ(1) \\ a_{j} − 1, & j ≠ σ(1) \end{cases}

Note that each b_{i} is nonnegative: If a_{j} = 0 for some j ≠ σ(1), the condition ∑_{i} a_{i} = (n − 1)(h − 1) forces a_{k} = n − 1 for all k ≠ j, which implies that a_{σ(1)} = n − 1, contradicting condition (ii) above. As (a, σ) runs over all pairs of tuples and permutations satisfying (1)–(3), the pair (b, σ) runs over all pairs of tuples and permutations satisfying 0 ≤ b_{i} ≤ n − 2 and ∑_{i} b_{i} = (n − 1)(h − 1) − (h − 1) = (n − 2)(h − 1). We find

[π]_{LT} (μ_{n}(X_{1}, …, X_{h})) = ∑_{(b,σ)} sgn(σ) ∏_{j=1}^{h−1}[π^{b_{σ(j)+1}}]_{u} (X_{σ(j)})^{q^{j}}

= ∑_{b} μ([π^{b_{1}+1}]_{u}(X_{1}), …, [π^{b_{h}+1}]_{u}(X_{h}))

= μ_{n−1} ([π]_{u}(X_{1}), …, [π]_{u}(X_{h}))

as required.

Now we can establish the O_{F} -linearity of μ_{n}. For this it suffices to show that μ_{n}([π]_{u}(X_{1}), X_{2}, …, X_{h−1}) = [π]_{LT}(μ_{n}(X_{1}, …, X_{h})). We have

μ_{n}([π]_{u}(X_{1}), X_{2}, …, X_{h−1}) = ∑_{a} μ([π^{a}](X)),

where a = (a_{1}, …, a_{h−1}) runs over tuples satisfying 1 ≤ a_{1} ≤ n − 1, 0 ≤ a_{i} ≤ n − 1 for i > 1, and ∑_{i} a_{i} = (h − 1)(n − 1) + 1. But these conditions force a_{i} ≥ 1 for i = 1, …, h. Write a_{i} = b_{i} + 1, so that 0 ≤ b_{i} ≤ n − 2 and ∑_{i} b_{i} = (h − 1)(n − 1). Then

μ_{n}([π]_{u}(X_{1}), X_{2}, …, X_{h−1}) = ∑_{b} μ([π^{b_{1}+1}]_{u}(X_{1}), …, [π^{b_{h}+1}]_{u}(X_{h}))
= μ_{n−1}([π]_{u}(X_{1}), …, [π]_{u}(X_{h}))
= [π]_{LT}(μ_{n}(X_{1}, …, X_{h}))

by Prop. 3.7. We have established part (1) of Thm. 3.2. Part (1) allows us to reduce part (2) to the case of n = 1, which has already been treated in Prop. 3.4. Recall that X_{1}^{(n)}, …, X_{h}^{(n)} are the canonical coordinates on 𝔛(π^{n}). Thm. 3.2 shows that the function Δ^{(n)} = μ_{n}(X_{1}^{(n)}, …, X_{h}^{(n)}) is a nonzero root of [π^{n}]_{LT}(T). The following simple lemma will be useful in the next section.

Lemma 3.8. Let M ∈ M_{h}(O_{F} /π^{n}O_{F}) be a matrix. Then

μ_{n}(M(X_{1}^{(n)}), …, X_{h}^{(n)}) + ⋯ + μ_{n}(X_{1}^{(n)}, …, M(X_{h}^{(n)})) = [Tr M]_{LT}(Δ^{(n)}).

4 An affinoid with good reduction

We now reach the technical heart of the paper. In this section we will construct an open affinoid neighborhood ℨ around an unramified canonical point x whose reduction is as in Thm. 1.1. These affinoids appear as connected components of the preimage of a subdisc 𝔛(1)^{1} inside of the polydisc 𝔛(1). The polydisc 𝔛(1)^{1} is small enough so that the local system ℱ^{univ}[π] may be trivialized over 𝔛(1)^{1}, which is to say that the quotient map 𝔛(π) → 𝔛(1) admits a section over 𝔛(1)^{1}. An approximation to this section is computed explicitly in §4.1. A consequence is that the preimage of 𝔛(1)^{1} in 𝔛(π) is a disjoint union of polydiscs 𝔛(π)^{1,x} indexed by the canonical points of 𝔛(π). In §4.2 we turn to the space 𝔛(π^{2}). An unramified canonical point x ∈ 𝔛(π^{2}) determines a subgroup K_{x,2} of GL_{h}(O_{F}) lying properly between 1 + πM_{h}(O_{F}) and 1 + π^{2}M_{h}(O_{F}). Let

𝔛(K_{x,2}) = 𝔛(π^{2})/K_{x,2}.

Then the affinoid ℨ of Thm. 1.1 is the preimage of 𝔛(π)^{1,x} in 𝔛(K_{x,2}). We introduce a family of coordinates Y (ζ) on 𝔛(π^{2}) which are invariant under K_{x,2}, one for each ζ in O_{E}. (The formation of the Y (ζ) is modeled on the determinant functor μ_{2} from §3.) Thus the Y (ζ) are analytic functions on 𝔛(K_{x,2}); it turns out (Prop. 4.2) that the Y (ζ) are integral functions on ℨ. A simple linear combination Y of the coordinates Y (ζ) generates the ring of integral analytic functions on ℨ as an algebra over the ring of integral analytic functions on the polydisc 𝔛(π)^{x,1}. The equation for the reduction \overline{ℨ} follows from the congruence calculated in Prop. 4.3. We often work with affinoid algebras B over a field E, where E/F is a finitely ramified extension contained in the completion of the separable closure of F. For f ∈ B we write v(f) for the infimum of v(f(z)) as z runs though Spm B.

4.1 Analytic sections of ℱ^{univ}[π]

Let E/F be the unramified extension of degree h, so that O_{E} = k_{h}⟦π⟧. Let ℱ_{0} be the deformation obtained by specializing the variables u_{i} to 0 in ℱ^{univ}, so that [π]_{ℱ_{0}} (X) = πX + X^{q^{h}}. Then ℱ_{0} admits endomorphisms by O_{E}. As a formal O_{E}-module, ℱ_{0} has height 1. We will denote by x^{(0)} the unramified canonical point in 𝔛(1) corresponding to ℱ_{0}. For n ≥ 1, let E_{n} be the extension of Ê^{nr} given by adjoining the roots of [π^{n}]_{ℱ_{0}} (X). Thus the preimages of x^{(0)} in 𝔛(π) are the points x = x^{(1)} ∈ 𝔛(π) corresponding to Drinfeld bases x_{1}, …, x_{h} ∈ 𝔭_{E_{1}} for ℱ_{0}[π]. Let 𝔛(1)^{1} ⊂ 𝔛(1) be the affinoid neighborhood defined by the conditions v(u_{i}) ≥ 1, i = 1, …, h− 1. Let V_{i} = π^{−1}u_{i}, so that the V_{i} are a chart of integral coordinates on 𝔛(1)^{1}. The ring of integral analytic functions on 𝔛(1)^{1} is therefore Ô_{F^{nr}}⟨V_{1}, …, V_{h−1}⟩. We claim that over 𝔛(1)^{1} ⊗ E_{1}, the local system ℱ^{univ}[π] may be trivialized. This means that every nonzero torsion point of ℱ_{0}[π] can be “spread out” to a unique section of ℱ^{univ}[π] over 𝔛(1)^{1} ⊗ E_{1}. To be precise:

Proposition 4.1. The preimage of 𝔛(1)^{1}⊗E_{1} in 𝔛(π)⊗E_{1} is the disjoint union of polydiscs 𝔛(π)^{1,x} over E_{1}, each containing a unique unramified canonical point x. For such a point x, corresponding to the basis x_{1}, …, x_{h} of ℱ_{0}[π], we have the following congruence, valid in the ring of integral analytic functions on 𝔛(π)^{1,x}:

X_{r}^{(1)} ≡ (−1)^{h−1} det (\begin{matrix}V_{1} & V_{2} & ⋯ & V_{h−1} & x_{r} \\ 1 & V_{1}^{q} & ⋯ & V_{h−2}^{q} & x_{r}^{q} + πx_{r}V_{h−1}^{q} \\ 0 & 1 & ⋯ & V_{h−3}^{q^{2}} & x_{r}^{q^{2}} + πx_{r}V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & x_{r}^{q^{h−1}} + πx_{r}V_{1}^{q^{h−1}} \end{matrix} ) (4.1.1)

modulo π^{q−1+ \frac{q}{q^{h}−1}} .

Proof. Let x_{1}, …, x_{h} be a basis of ℱ_{0}[π]. Consider the polynomial [π]_{ℱ^{univ}} (X) = πX + πV_{1}X^{q} + ⋯ + πV_{h−1}X^{q^{h−1}} + X^{q^{h}} ∈ O_{F} ⟨V_{1}, …, V_{h−1}⟩[X]. By studying the Newton polygon of the translate [π]_{ℱ^{univ}} (X − x_{r}), we find that there is a unique root X_{r} ∈ O_{E_{1}}⟨V_{1}, …, V_{h−1}⟩ of [π]_{ℱ^{univ}} (X) for which v(X_{r} − x_{r}) > v(x_{r}) = 1/(q^{h} − 1). This root satisfies v(X_{r} − x_{r}) = v(x_{r}^{q}) = q/(q^{h} − 1). Then v(X_{r} − x_{s}) = 1/(q^{h} − 1) for r ≠ s. This already implies that the preimage of 𝔛(1)^{1} ⊗ E_{1} in 𝔛(π) ⊗ E_{1} is the union of polydiscs 𝔛(π)^{1,x}, where 𝔛(π)^{1,x} is the affinoid described by the inequalities v(X_{r}^{(1)} −x_{r}) ≥ v(x_{r}^{q}), r = 1, …, h. Now let D ∈ O_{E_{1}}[V_{1}, …, V_{h−1}] be the expression on the right hand side of Eq. (4.1.1). Expand the determinant in Eq. (4.1.1) along its first row and label the minors A_{1}, … A_{h}, signed appropriately so that

D = ∑_{i=1}^{h−1} V_{i}A_{i} + x_{r}A_{h}.  (4.1.2)

That is,

A_{i} = (−1)^{h−i} det (\begin{matrix}V_{1}^{q^{i}} & V_{2}^{q^{i}} & ⋯ & V_{h−i−1}^{q^{i}} & x_{r}^{q^{i}} + πx_{r}V_{h−i}^{q^{i}} \\ 1 & V_{1}^{q^{i}} & ⋯ & V_{h−i−2}^{q^{i+1}} & x_{r}^{q^{i+1}} + πx_{r}V_{h−i−1}^{q^{i+1}} \\ 0 & 1 & ⋯ & V_{h−i−3}^{q^{i+2}} & x_{r}^{q^{i+2}} + πx_{r}V_{h−i−2}^{q^{i+2}} \\ ⋮ & & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & x_{r}^{q^{h−1}} + πx_{r}V_{1}^{q^{h−1}} \end{matrix}) (4.1.3)

for i = 1, …, h − 1, and A_{h} = 1. In order to complete the proof of Prop. 4.1, we will show that [π]_{ℱ^{univ}} (D) is sufficiently close to 0 to ensure the congruence in Eq. (4.1.1). Observe that for i = 1, …, h − 1 we have the following congruence modulo π^{q+ \frac{q}{q^{h}−1}} :

D^{q^{i}} ≡ (−1)^{h−1} det (\begin{matrix}V_{1}^{q^{i}} & V_{2}^{q^{i}} & ⋯ & V_{h−i}^{q^{i}} & V_{h−i+1}^{q^{i}} & ⋯ & V_{h−1}^{q^{i}} & x_{r}^{q^{i}} \\ 1 & V_{1}^{q^{i+1}} & … & V_{h−i−1}^{q^{i+1}} & V_{h−i}^{q^{i+1}} & ⋯ & V_{h−2}^{q^{i+1}} & x_{r}^{q^{i+1}} \\ ⋮ & & ... & ⋮ & ⋮ & & & ⋮ \\ 0 & 0 & … & V_{1}^{q^{h−1}} & V_{2}^{q^{h−1}} & … & V_{i}^{q^{h−1}} & x_{r}^{q^{h−1}} \\ 0 & 0 & … & 1 & V_{1}^{q^{h}} & … & V_{i−1}^{q^{h}} & −πx_{r} \\ 0 & 0 & … & 0 & 1 & … & V_{i−2}^{q^{h+1}} & 0 \\ ⋮ & & & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 0 & 0 & … & 1 & 0 \end{matrix})

Placing the final column of this matrix into position (h − i + 1) transforms the above matrix into one of the form (\begin{matrix}A & B \\ 0 & C \end{matrix}), where A is a matrix with dimensions (h − i + 1) × (h − i + 1) and C is an upper triangular matrix with 1s along the diagonal. We find

D^{q^{i}} ≡ (−1)^{h+i} det (\begin{matrix}V_{1}^{q^{i}} & V_{2}^{q^{i}} & ⋯ & V_{h−i−1}^{q^{i}} & V_{h−i}^{q^{i}} & x_{r}^{q^{i}} \\ 1 & V_{1}^{q^{i+1}} & ⋯ & V_{h−i−2}^{q^{i+1}} & V_{h−i−1}^{q^{i+1}} & x_{r}^{q^{i+1}} \\ 0 & 1 & ⋯ & V_{h−i−3}^{q^{i+2}} & V_{h−i−2}^{q^{i+2}} & x_{r}^{q^{i+2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & V_{1}^{q^{h−1}} & x_{r}^{q^{h−1}} \\ 0 & 0 & ⋯ & 0 & 1 & −πx_{r} \end{matrix} ) (mod π^{q+ \frac{q}{q^{h}−1}} ).

(4.1.4)

We can apply elementary row operations to use the 1 in column h − i of this matrix to cancel the entries above it. When this is done, we find

D^{q^{i}} ≡ −A_{i} (mod π^{q+ \frac{q}{q^{h}−1}} ), i = 1, …, h − 1  (4.1.5)

where A_{1}, …, A_{h−1} are the minors from Eq. (4.1.3). We also have

D^{q^{h}} ≡ −πx_{r} ≡ −πx_{r}A_{h} (mod π^{q+ \frac{q}{q^{h}−1}} ).  (4.1.6)

Combining Eqs. (4.1.2), (4.1.5) and (4.1.6) gives

[π]_{ℱ^{univ}} (D) = πD + πV_{1}D^{q} + ⋯ + πV_{h−1}D^{q^{h−1}} + D^{q^{h}}
≡ πD − π(V_{1}A_{1} + ⋯ + V_{h−1}A_{h−1} + x_{r}A_{h})
≡ 0 (mod π^{q+ \frac{q}{q^{h}−1}} ).

The ring of integral analytic functions on the polydisc 𝔛(π)^{1,x} is O_{E_{1}}⟨V_{1}, …, V_{h}⟩. In this ring we have the congruences D ≡ X_{r}^{(1)} ≡ x_{r} (mod x_{r}^{q}). Let Y = D − X_{r}^{(1)}. Then Y ≡ 0 (mod x_{r}^{q}) and [π]_{ℱ^{univ}} (Y) ≡ 0 (mod π^{q+1/(q^{h}−1)}). Examining the Newton polygon of [π]_{ℱ^{univ}} (X) shows that Y ≡ 0 (mod π^{q−1+1/(q^{h}−1)}).

4.2 Some invariant coordinates on 𝔛(π^{2}).

Choose a compatible system of bases x_{1}^{(n)}, …, x_{h}^{(n)} for ℱ_{0}[π^{n}], n ≥ 1. This is tantamount to choosing a compatible system of unramified canonical points x^{(n)} ∈ 𝔛(π^{n}) lying above the point x^{(0)} ∈ 𝔛(1) corresponding to the deformation ℱ_{0}. Since ℱ_{0} admits O_{F} -linear endomorphisms by O_{E}, our choice of compatible system induces an embedding of O_{E} into 𝔄 = M_{h}(O_{F}), and we identify O_{E} with its image. For M ∈ 𝔄, recall the definition of M(X_{i}^{(n)}) from Eq. (2.3.1). We have ζ(X_{i}^{(n)})(x^{(n)}) = ζx_{i}^{(n)} for i = 1, …, h, ζ ∈ k_{h}. The unit group 𝔄^{×} = GL_{h}(O_{F}) has the usual filtration U_{𝔄}^{n} = 1 + 𝔭^{n}𝔄, n ≥ 1. Let C ⊂ 𝔄 be the orthogonal complement of O_{E} under the standard trace pairing, and let 𝔭_{E} be the maximal ideal of O_{E}. Define a subgroup K_{x,2} of 𝔄^{×} by

K_{x,2} = 1 + 𝔭_{E}^{2} + 𝔭_{E}C,

so that K_{x,2} lies between U_{𝔄}^{1} and U_{𝔄}^{2}. In what follows we will assume the choice of x is fixed and write simply K_{2}. Write 𝔛(K_{2}) for the quotient of 𝔛(π^{2}) by K_{2}. We shall construct an alternating k-linear expression Y in the canonical coordinates X_{1}^{(2)}, …, X_{h}^{(2)} which is fixed by K_{2}, so that it descends to an analytic function on 𝔛(K_{2}). It happens that Y satisfies a polynomial equation with coefficients in O_{E_{2}}⟨V_{1}, …, V_{h}⟩ whose reduction modulo the maximal ideal of O_{E_{2}} gives the smooth hypersurface of Thm. 1.1. We continue using the shorthand X_{r} = X_{r}^{(1)}. We introduce the new shorthand Y_{r} = X_{r}^{(2)}, so that [π]_{ℱ^{univ}} (Y_{r}) = X_{r}. Also we let Δ = Δ^{(1)} = μ(X_{1}, …, X_{h}); this is a locally constant function satisfying Δ^{q−1} = (−1)^{h}π. For ζ ∈ O_{E}, let

W (ζ) = μ(ζ(Y_{1}), X_{2}, …, X_{h}) + ⋯ + μ(X_{1}, X_{2}, …, ζ(Y_{h})).

Note that W (1) = μ_{2}(X_{1}, …, X_{h}) = Δ^{(2)}. We record the action of U_{𝔄}^{1} on the functions W (ζ): For g = 1 + πM ∈ U_{𝔄}^{1}, we have

g(W (ζ)) = W (ζ) + [Tr(Mζ)]_{LT}(Δ)  (4.2.1)

by Lemma 3.8. It follows that W (ζ) is invariant under K_{2}, and that [π]_{LT}(W (ζ)) is invariant under U_{𝔄}^{1}, so that [π]_{LT}(W (ζ)) belongs to A(π). We can see this directly: by Eq. (3.1.2) we have

[π]_{LT}(W (ζ)) = det (\begin{matrix}ζ(X_{1}) & X_{1}^{q} & ⋯ & X_{1}^{q^{h−1}} \\ ⋮ & ⋮ & ... & ⋮ \\ ζ(X_{h}) & X_{h}^{q} & ⋯ & X_{h}^{q^{h−1}} \end{matrix} ) ,  (4.2.2)

which visibly belongs to A(π). We will use the symbol x to denote our compatible system of canonical points x^{(n)} ∈ 𝔛(π^{n}). Then f(x) is well-defined when f is an analytic function on any of the spaces 𝔛(π^{n}). We will use 𝔛(π)^{1,x} to refer to the polydisc constructed in §4.1 using the canonical point x^{(1)}. By Prop. 4.1, the restriction of the function [π]_{LT}(W (ζ)) to 𝔛(π)^{1,x} lies in O_{E_{1}}⟨V_{1}, …, V_{h}⟩, where we recall that the variables V_{r} = π^{−1}u_{r} form our chart of integral coordinates on 𝔛(1)^{1}. Let ℨ be the preimage of the polydisc 𝔛(π)^{1,x} in 𝔛(K_{2}) ⊗ E_{2}. It will be useful to transform the functions W (ζ) into integral functions Y (ζ) on ℨ for which |Y (ζ)|_{ℨ} = 1. Let w(ζ) = W (ζ)(x), and let

Y (ζ) = (−1)^{h−1} \frac{W (ζ) − w(ζ)}{Δ}.  (4.2.3)

Proposition 4.2. There exists ε > 0 for which the congruence

Y (ζ)^{q} − Y (ζ) ≡ (\begin{matrix}V_{1} & V_{2} & … & V_{h−1} & 0 \\ 1 & V_{1}^{q} & … & V_{h−2}^{q} & (ζ^{q} − ζ)V_{h−1}^{q} \\ 0 & 1 & … & V_{h−3}^{q^{2}} & (ζ^{q^{2}} − ζ)V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & (ζ^{q^{h−1}} − ζ)V_{1}^{q^{h−1}} \end{matrix} ) (mod π^{ε})

is valid in the ring of integral analytic functions on ℨ.

Proof. The idea is to apply Prop. 4.1 to Eq. (4.2.2). In preparation for this, we need some determinant identities. For i = 1, …, h, let B_{i} ∈ k[V_{1}, …, V_{h−1}] be (−1)^{i} times the determinant of the top left i × i submatrix of

(\begin{matrix}V_{1} & V_{2} & ⋯ & V_{h−1} & 0 \\ 1 & V_{1}^{q} & ⋯ & V_{h−2}^{q} & V_{h−1}^{q} \\ 0 & 1 & ⋯ & V_{h−3}^{q^{2}} & V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & V_{1}^{q^{h−1}} \end{matrix} )

Curiously, the transformation (V_{1}, …, V_{h−1}) ↦ (B_{1}, …, B_{h−1}) is an involution. That is, the determinant of the top left i × i submatrix of

(\begin{matrix}B_{1} & B_{2} & ⋯ & B_{h−1} & 0 \\ 1 & B_{1}^{q} & ⋯ & B_{h−2}^{q} & B_{h−1}^{q} \\ 0 & 1 & ⋯ & B_{h−3}^{q^{2}} & B_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & B_{1}^{q^{h−1}} \end{matrix} )

is (−1)^{i}V_{i}: this can be proven by induction on i. This implies the following identity, valid in the polynomial ring k[V_{1}, …, V_{h−1}, z_{1}, …, z_{h−1}]:

det (\begin{matrix}z_{1}B_{1} & z_{2}B_{2} & ⋯ & z_{h−1}B_{h−1} & 0 \\ 1 & B_{1}^{q} & ⋯ & B_{h−2}^{q} & B_{h−1}^{q} \\ 0 & 1 & ⋯ & B_{h−3}^{q^{2}} & B_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ... & 1 & B_{1}^{q^{h−1}} \end{matrix} )

= det (\begin{matrix}V_{1} & V_{2} & ⋯ & V_{h−1} & 0 \\ 1 & V_{1}^{q} & ⋯ & V_{h−2}^{q} & z_{1}V_{h−1}^{q} \\ 0 & 1 & ⋯ & V_{h−3}^{q^{2}} & z_{2}V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & z_{h−1}V_{1}^{q^{h−1}} \end{matrix} )  (4.2.4)

This is because both expressions equal

z_{1}B_{1}V_{h−1}^{q} + z_{2}B_{2}V_{h−2}^{q^{2}} + ⋯ + z_{h−1}B_{h−1}V_{1}^{q^{h−1}}.

According to Prop. 4.1, the coordinate X_{r} may be expressed modulo π^{q−1+ \frac{q}{q^{h}−1}} as a linear combination of the powers x_{r}, …, x_{r}^{q^{h−1}}:

X_{r} ≡ (1−πB_{h})x_{r}+B_{1}x_{r}^{q}+B_{2}x_{r}^{q^{2}}+⋯+B_{h−1}x_{r}^{q^{h−1}} (mod π^{q−1+ \frac{q}{q^{h}−1}} ). (4.2.5)

For ζ ∈ k_{h} we have

ζ(X_{r}) ≡ ζ(1−πB_{h})x_{r}+ζ^{q}B_{1}x_{r}^{q}+ζ^{q^{2}}B_{2}x_{r}^{q^{2}}+⋯+ζ^{q^{h−1}}B_{h−1}x_{r}^{q^{h−1}} (mod π^{q−1+ \frac{q}{q^{h}−1}} )

(4.2.6)

Also, for i = 1, …, h − 1 we have

X_{r}^{q^{i}} ≡ −πB_{h−i}^{q^{i}}x_{r} + x_{r}^{q^{i}} + B_{1}^{q^{i}}x_{r}^{q^{i+1}} + ⋯ + B_{h−1−i}^{q^{i}}x_{r}^{q^{h−1}} (mod π^{N}), (4.2.7)

where N ≥ q + \frac{q}{q^{h}−1}. Eqs. (4.2.6) and (4.2.7) may be combined into the congruence of matrices

(\begin{matrix}ζ(X_{1}) + E_{1} & ⋯ & ζ(X_{h}) + E_{h} \\ X_{1}^{q} & ⋯ & X_{h}^{q} \\ ⋮ & ... & ⋮ \\ X_{1}^{q^{h−1}} & ⋯ & X_{h}^{q^{h−1}} \end{matrix})

≡ (\begin{matrix}ζ(1 − πB_{h}) & ζ^{q}B_{1} & ζ^{q^{2}}B_{2} & … & ζ^{q^{h−1}}B_{h−1} \\ −πB_{h−1}^{q} & 1 & B_{1}^{q} & ⋯ & B_{h−2}^{q} \\ −πB_{h−2}^{q^{2}} & 0 & 1 & ⋯ & B_{h−3}^{q^{2}} \\ ⋮ & & & ... & ⋮ \\ −πB_{1}^{q^{h−1}} & 0 & 0 & ⋯ & 1 \end{matrix})

× (\begin{matrix}x_{1} & ⋯ & x_{h} \\ ⋮ & ... & ⋮ \\ x_{1}^{q^{h−1}} & ⋯ & x_{h}^{q^{h−1}} \end{matrix} )  (4.2.8)

modulo π^{N}, where v(E_{i}) ≥ q − 1 + q/(q^{h} − 1). We take determinants of both sides of Eq. (4.2.8). On the left hand side, we apply Eq. (4.2.2): the determinant is congruent to [π]_{LT}(W (ζ)) modulo an error term π^{δ}, of valuation

δ ≥ q − 1 + \frac{q}{q^{h} − 1} + \frac{q + q^{2} + ⋯ + q^{h−1}}{q^{h} − 1} = q − 1 + \frac{q − 1}{q^{h} − 1} + \frac{1}{q − 1}.

On the right hand side, the determinant is Δ times

ζ − ζπB_{h} + (−1)^{h}π det (\begin{matrix}ζ^{q}B_{1} & ζ^{q^{2}}B_{2} & … & ζ^{q^{h−1}}B_{h−1} & 0 \\ 1 & B_{1}^{q} & ⋯ & B_{h−2}^{q} & B_{h−1}^{q} \\ 0 & 1 & ⋯ & B_{h−3}^{q^{2}} & B_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & B_{1}^{q^{h−1}} \end{matrix} ) ,

and by the identity in Eq. 4.2.4 this equals

ζ + (−1)^{h}π det (\begin{matrix}V_{1} & V_{2} & … & V_{h−1} & 0 \\ 1 & V_{1}^{q} & … & V_{h−2}^{q} & (ζ^{q} − ζ)V_{h−1}^{q} \\ 0 & 1 & … & V_{h−3}^{q^{2}} & (ζ^{q^{2}} − ζ)V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & (ζ^{q^{h−1}} − ζ)V_{1}^{q^{h−1}} \end{matrix} ) .

Equating determinants of both sides of Eq. (4.2.8) now yields

[π]_{LT}(W (ζ)) ≡

≡ ζΔ + (−1)^{h}πΔ det (\begin{matrix}V_{1} & V_{2} & … & V_{h−1} & 0 \\ 1 & V_{1}^{q} & … & V_{h−2}^{q} & (ζ^{q} − ζ)V_{h−1}^{q} \\ 0 & 1 & … & V_{h−3}^{q^{2}} & (ζ^{q^{2}} − ζ)V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & (ζ^{q^{h−1}} − ζ)V_{1}^{q^{h−1}} \end{matrix} ) (mod π^{δ})

The functions V_{1}, …, V_{h−1} vanish at the canonical point x; therefore so do the functions B_{1}, …, B_{h−1}. Applying the above congruence to x gives

[π]_{LT}(w(ζ)) ≡ ζΔ (mod π^{δ}).  (4.2.9)

We have W (ζ) = w(ζ) + (−1)^{h−1}ΔY (ζ), so that

[π]_{LT}(W (ζ)) = [π]_{LT}(w(ζ)) + (−1)^{h}πΔ(Y (ζ)^{q} − Y (ζ))

Therefore the congruence claimed in the proposition is valid modulo π^{ε}, where

ε = δ − 1 − \frac{1}{q − 1} ≥ q − 2 + \frac{q − 1}{q^{h} − 1} > 0.

The functions Y (ζ) on ℨ each generate a degree q algebra over the field of meromorphic functions on the polydisc 𝔛(π)^{1,x}. But the morphism ℨ → 𝔛(π)^{1,x}⊗E_{2} has degree q^{h}. We will now construct a linear combination of the Y (ζ) which generates the entire ring of integral analytic functions on ℨ as an algebra over O_{E_{2}}⟨V_{1}, …, V_{h−1}⟩. Let ζ, ζ^{q}, …, ζ^{q^{h}} be a basis for k_{h}/k, and let β ∈ k_{h} be such that

Tr_{k_{h}/k}(βζ^{q^{i}}) = \begin{cases}1, & i = 0, \\ 0 & i = 1, …, h − 1. \end{cases}  (4.2.10)

This implies that β, …, β^{q^{h−1}} is a basis for k_{h}/k as well. Let

Y = ∑_{i=0}^{h−1} β^{q^{i}}Y (ζ^{q^{i}}).  (4.2.11)

Then the stabilizer of Y in U_{𝔄}^{1} is exactly K_{2}.

Proposition 4.3. There exists ε > 0 for which the congruence

Y^{q^{h}} − Y ≡ (\begin{matrix}V_{1}^{q^{h}} − V_{1} & V_{2}^{q^{h}} − V_{2} & ⋯ & V_{h−1}^{q^{h}} − V_{h−1} & 0 \\ 1 & V_{1}^{q} & ⋯ & V_{h−2}^{q} & V_{h−1}^{q} \\ 0 & 1 & ⋯ & V_{h−3}^{q^{2}} & V_{h−2}^{q^{2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & V_{1}^{q^{h−1}} \end{matrix} ) (4.2.12)

holds modulo π^{ε} in the ring of integral analytic functions on ℨ.

Proof. We have

Y^{q^{h}} − Y = ∑_{j=0}^{h−1} β^{q^{j}} (Y (ζ^{q^{j}})^{q^{h}} − Y (ζ^{q^{j}}))

= ∑_{j=0}^{h−1} β^{q^{j}} ∑_{i=0}^{h−1}(Y (ζ^{q^{j}})^{q} − Y (ζ^{q^{j}}))^{q^{i}}

≡ ∑_{i=0}^{h−1} ∑_{j=0}^{h−1} β^{q^{j}} det (\begin{matrix}V_{1}^{q^{i}} & V_{2}^{q^{i}} & … & V_{h−1}^{q^{i}} & 0 \\ 1 & V_{1}^{q^{i+1}} & … & V_{h−2}^{q^{i+1}} & (ζ^{q^{i+j+1}} − ζ^{q^{i+j}})V_{h−1}^{q^{i+1}} \\ 0 & 1 & … & V_{h−3}^{q^{i+2}} & (ζ^{q^{i+j+2}} − ζ^{q^{i+j}})V_{h−2}^{q^{i+2}} \\ ⋮ & & ... & & ⋮ \\ 0 & 0 & ⋯ & 1 & (ζ^{q^{i+j+h−1}} − ζ^{q^{i+j}})V_{1}^{q^{h−1}} \end{matrix} )

modulo π^{ε}, by Prop. 4.2. We now apply the orthogonality relations in Eq. (4.2.10). The term with i = 0 is (−1)^{h−1}B_{h}, and the term with 1 ≤ i ≤ h−1 is

det (\begin{matrix}V_{1}^{q^{i}} & V_{2}^{q^{i}} & ⋯ & V_{h−i}^{q^{i}} & ⋯ & V_{h−1}^{q^{i}} & 0 \\ 1 & V_{1}^{q^{i+1}} & ⋯ & V_{h−i−1}^{q^{i+1}} & ⋯ & V_{h−2}^{q^{i+1}} & 0 \\ 0 & 1 & ⋯ & V_{h−i−2}^{q^{i+2}} & ⋯ & V_{h−3}^{q^{i+2}} & 0 \\ ⋮ & & & & & & ⋮ \\ 0 & 0 & ⋯ & V_{1}^{q^{h−1}} & ⋯ & V_{i}^{q^{h−1}} & 0 \\ 0 & 0 & ⋯ & 1 & ⋯ & V_{i−1}^{q^{h}} & V_{i}^{q^{h}} \\ 0 & 0 & ⋯ & 0 & ⋯ & V_{i−2}^{q^{h+1}} & 0 \\ ⋮ & & & ⋮ & & & ⋮ \\ 0 & 0 & ⋯ & 0 & ⋯ & 1 & 0 \end{matrix}) = (−1)^{h−1}V_{i}^{q^{h}}B_{h−i}^{q^{i}},

so that

Y^{q^{h}} − Y ≡ (−1)^{h−1}(B_{h} + V_{1}^{q^{h}}B_{h−1} + V_{2}^{q^{h}}B_{h−2} + ⋯ + V_{h−1}^{q^{h}}B_{1}) (mod π^{ε}).

This last expression agrees with the determinant in the proposition, as can be seen by expanding along the first row.

4.3 Conclusion of the proof.

We now complete the proof of Thm. 1.1. Let x be an unramified canonical point on the Lubin-Tate tower. Since the unramified canonical points in 𝔛(1) lie in the same orbit under O_{B}^{×} = Aut Σ, we may assume that x lies above the point with u_{1} = ⋯ = u_{h−1} = 0 in 𝔛(1). Recall that 𝔛(π)^{1,x} ⊂ 𝔛(π) ⊗ E_{1} is the affinoid defined by the conditions v(u_{i}) ≥ 1 for i = 1, …, h − 1 and v(X_{r}^{(1)} − x_{r}) ≥ v(x_{r}^{q}) for r = 1, …, h; we showed in Prop. 4.1 that 𝔛(π)^{1,x} is a polydisc over E_{1}. The quotient 𝔛(K_{x,2}) → 𝔛(π) is Galois with group H = U_{𝔄}^{1}/K_{x,2} ≈ F_{q^{h}}. After passing to E_{2} coefficients, the affinoid ℨ was defined as the inverse image of 𝔛(π)^{1,x} in this quotient. Therefore ℨ → 𝔛(π)^{1,x} ⊗ E_{2} is an étale cover of affinoids with group H. Consider the integral coordinate Y on ℨ produced by Prop. 4.3: the calculation in §5.1 below shows that the action of a nonzero element of H translates Y by a nonzero element of F_{q^{h}}. Thus the reduction of the cover ℨ → 𝔛(π)^{1,x} ⊗ E_{2} is an étale cover of affine hypersurfaces over \overline{k}, also with group H. For a tuple V = (V_{1}, …, V_{h−1}), let d(V) denote the determinant appearing on the right hand side of Eq. (4.2.12). Let \overline{ℨ}′ denote the hypersurface over \overline{k} with equation Y^{q^{h}} − Y = d(V); then \overline{ℨ}′ → A^{h−1} is an Artin-Schreier cover of affine hypersurfaces with group H. Prop. 4.3 shows that \overline{ℨ} → A^{h−1} factors through an H-equivariant morphism \overline{ℨ} → \overline{ℨ}′. Since \overline{ℨ} and \overline{ℨ}′ are both étale covers of A^{h−1} with group H, we find that \overline{ℨ} → \overline{ℨ}′ is an isomorphism. Finally, \overline{ℨ}′ is isomorphic to the hypersurface described in Thm. 1.1 via Y = (−1)^{h−1}V_{h}. This concludes the proof of Thm. 1.1.

5 Group actions on a hypersurface

We close with a discussion of various group actions on the affinoid ℨ, with an eye towards linking Thm. 1.1 with the local Langlands correspondence and the Jacquet-Langlands correspondence. What follows is meant to indicate further directions of research; no proofs will be given. A large open subgroup of GL_{h}(F) × B^{×} × W_{F} acts on the Lubin-Tate tower 𝔛(π^{n}), cf. the introduction to [HT01]. To investigate the question of whether the cohomology of the affinoid ℨ realizes the appropriate correspondences among the three factor groups, it will be useful to compute the stabilizer of ℨ in each group, along with the action of the stabilizer on the reduction \overline{ℨ}. We do precisely this for the groups GL_{h}(F) and W_{F}. The hypersurface \overline{ℨ}, when considered as an abstract variety over \overline{k}, admits a nontrivial action by a large subquotient of B^{×}, but we cannot prove this action arises from the actual action of B^{×} on the Lubin-Tate tower. Let X be the F_{q^{h}}-rational model for \overline{ℨ}/\overline{F}_{q} from Conj. 1.6. That is, X ⊂ A_{F_{q^{h}}}^{h} is the hypersurface with equation

det (\begin{matrix}V_{1}^{q^{h}} − V_{1} & V_{2}^{q^{h}} − V_{2} & V_{3}^{q^{h}} − V_{3} & ⋯ & V_{h−1}^{q^{h}} − V_{h−1} & V_{h}^{q^{h}} − V_{h} \\ 1 & V_{1}^{q} & V_{2}^{q} & ⋯ & V_{h−2}^{q} & V_{h−1}^{q} \\ 0 & 1 & V_{1}^{q^{2}} & ⋯ & V_{h−3}^{q^{2}} & V_{h−2}^{q^{2}} \\ ⋮ & & & ... & & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & V_{1}^{q^{h−1}} \end{matrix}) = 0.

The actions on \overline{ℨ} we consider in this paragraph all descend to actions on the F_{q^{h}}-rational model X.

5.1 The action of GL_{h}(F)

The affioid ℨ is stabilized by the group U_{𝔄}^{1} = 1 + πM_{h}(O_{F}), and the action of U_{𝔄}^{1} on ℨ factors through the quotient H = U_{𝔄}^{1}/K_{2}. The action of H on the reduction \overline{ℨ} can be made completely explicit. We identify H with k_{h} = F_{q^{h}} via the isomorphism 1 + γπ ↦ γ, γ ∈ k_{h}. From Eq. (4.2.1) and the construction of Y in Eqs. (4.2.3) and (4.2.11) we see that the action of an element γ ∈ H on \overline{ℨ} preserves the variables V_{1}, …, V_{h−1} and has the following effect on V_{h}:

V_{h} ↦ V_{h} + ∑_{j=1}^{h−1} β^{q^{j}} Tr_{k_{h}/k}(ζ^{q^{j}}γ) = V_{h} + γ.  (5.1.1)

Of course, this action descends to an action of H on X by F_{q^{h}}-rational automorphisms. We offer some brief remarks relating the characters of the group H to the theory of Bushnell-Kutzko types for GL_{h}(F), wherein supercuspidal representations are constructed by induction from compact-mod-center subgroups. In fact, in our particular situation, the construction goes back to Howe [How77]. Suppose ψ is a character of H ≈ F_{q^{h}} which does not factor through Tr_{F_{q^{h}} /F_{q^{d}}} for any proper divisor d of h. This character pulls back to a character of U_{𝔄}^{1} = 1 + πM_{h}(O_{F}), which we also call ψ. Recall that we have fixed an embedding of E into M_{h}(F). Choose a character θ of E^{×} for which θ|_{1+𝔭_{E}} = ψ|_{1+𝔭_{E}}. Then θ is an admissible character in the sense that there is no proper subextension E′ ⊂ E of E/F for which θ factors through the norm map E^{×} → (E′)^{×}. The character θ has conductor 𝔭_{E}^{2}. Let η be the unique character of J = E^{×}U_{𝔄}^{1} for which η|_{E^{×}} = θ and η|_{U_{𝔄}^{1}} = ψ. Then π(θ) = Ind_{J}^{GL_{h}(F)} η is a supercuspidal representation of GL_{h}(F) of level π^{2}: This is a special case of the construction used to prove Theorem 2 of [How77]. Therefore the question of whether the cohomology of ℨ realizes the Bushnell-Kutzko types for GL_{h}(F) is a matter of determining which characters of H appear in the cohomology of X; this is discussed in Conj. 5.1 below.

5.2 The action of inertia

The action of the inertia subgroup I_{F} ⊂ W_{F} on \overline{ℨ} can be made explicit as well. Let I_{2} = Gal(E_{2}/E^{nr}); we identify I_{2} with (O_{E}/π^{2}O_{E})^{×} via the reciprocity map of local class field theory. Thus if α ∈ O_{E}^{×}, and x ∈ 𝔛(π^{2}) is an unramified canonical point corresponding to a basis x_{1}, …, x_{h} of ℱ_{0}[π^{2}], then α(x) corresponds to the basis αx_{1}, …, αx_{h}. Since the definition of the affinoid ℨ only depends on the image of x in 𝔛(π), the stabilizer of ℨ in I_{2} is the group (1 + πO_{E})/(1 + π^{2}O_{E}). The action of an element 1 + γπ ∈ 1 + πO_{E} on \overline{ℨ} is exactly as in Eq. (5.1.1).

5.3 The action of B^{×}

More subtle is the action of O_{B}^{×} = Aut Σ. The algebra O_{B} is generated over O_{F} by O_{E} and Φ, where Φ^{h} = π and Φα = α^{q}Φ, α ∈ O_{E}. For n ≥ 1, let U_{B}^{n} = 1 + Φ^{n}O_{B}. Let C^{B} be the orthogonal complement of O_{E} in O_{B}, so that

C^{B} = O_{E}Φ ⊕ ⋯ ⊕ O_{E}Φ^{h−1},

and define a subgroup K_{2}^{B} of O_{B}^{×} by

K_{2}^{B} = 1 + 𝔭_{E}^{2} + 𝔭_{E}C,

so that K_{2}^{B} lies properly between U_{B}^{1} and U_{B}^{2}. Let H^{B} = U_{B}^{1}/K_{2}^{B}. Let R be the noncommutative ring F_{q^{h}}[τ]/(τ^{h+1}) whose multiplication is given by the rule ατ = τα^{q}, α ∈ F_{q^{h}}. Then H^{B} is isomorphic to 1 + τR. As we observed in Rmk. 1.2, R^{×} acts on X. It seems likely that the stabilizer of ℨ in O_{B}^{×} is U_{B}^{1}, and that the action of U_{B}^{1} on \overline{ℨ} factors through the this action of H^{B} ≅ 1+τR. The action of such a large group of F_{q^{h}}-rational automorphisms has consequences for the cohomology of X which allow us to reinterpret Conj. 1.6. First, let us provide a short description of the representation theory of the nilpotent group H^{B}. The subgroup Z = 1 + τ^{h}R is the center of 1 + τR ≅ H^{B}. Let ψ be a character of Z ≈ F_{q^{h}} which does not factor through Tr_{F_{q^{h}} /F_{q^{d}}} for any proper divisor d of h. There is a unique representation V_{ψ} of H^{B} lying over ψ, of dimension q^{h(h−1)/2}. Let H = H_{c}^{∗}(X ⊗ \overline{F}_{q}, \overline{Q}_{ℓ}), considered as a virtual module for the action of Gal(\overline{F}_{q}/F_{q^{h}}) × H^{B}. Conj. 1.6 now takes the following alternate form:

Conjecture 5.1. Let H_{ψ} = Hom_{H^{B}} (V_{ψ}, H), considered as a virtual module for the action of Gal(\overline{F}_{q}/F_{q^{h}}). Then dim H_{ψ} = (−1)^{h−1}, and the eigenvalue of Frob_{q^{h}} on H_{ψ} is q^{h(h−1)/2}.

The formalism of Bushnell-Kutzko types for GL_{h}(F) in [BK93] has been extended to the context of its anisotropic form B^{×} by Broussous [Bro95]. Granting Conj. 5.1, it will not be difficult to detect the types for B^{×} in the middle cohomology of ℨ. The types for B^{×} appearing in H_{c}^{h−1}(\overline{ℨ}, \overline{Q}_{ℓ}) should correspond exactly to those types for GL_{h}(F) which appear there; indeed this space should realize the correspondence between types. There has already been much work towards an “explicit Jacquet-Langlands correspondence”, whereby the admissible square-integrable duals of GL_{h}(F) and of B^{×} are linked via the explicit parameterizations of each dual via types, see [Hen93], [BH00], [BH05c]. However there are still outstanding cases where the explicit Jacquet-Langlands correspondence is not established, including (in some instances) the supercuspidals π(θ) of §5.1. For these there may be some advantage to the cohomological point of view, given that the Jacquet-Langlands correspondence is already known to be realized in the cohomology of the Lubin-Tate tower, cf. [HT01], [Str08a].

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Jared Weinstein Dept. of Mathematics University of California Los Angeles Box 951555 Los Angeles, CA 90095-1555 USA jaredsweinstein@gmail.com 1008

Documenta Mathematica 15 (2010)