Local Heights on Abelian Varieties

and Rigid Analytic Uniformization

Annette Werner

Received: October 1, 1998

Communicated by Peter Schneider

Abstract. We express classical and p-adic local height pairings on an abelian variety with split semistable reduction in terms of the corresponding pairings on the abelian part of the Raynaud extension (which has good reduction). Here we use an approach to height pairings via splittings of biextensions which is due to Mazur and Tate. We conclude with a formula comparing Schneider’s p-adic height pairing to the p-adic height pairing in the semistable ordinary reduction case defined by Mazur and Tate.

1991 Mathematics Subject Classification: 11G10, 11G25, 14G20

1 Introduction

In this paper we express classical and p-adic local height pairings on an abelian variety A_{K} with split semistable reduction in terms of the corresponding pairings on the abelian part B_{K} of the Raynaud extension. Since B_{K} is an abelian variety with good reduction, this result provides a rather explicit step from the class of local height pairings on all abelian varieties with good reduction to the class of local height pairings on arbitrary abelian varieties. As an application of this principle we show a formula comparing two local p-adic height pairings on A_{K}, namely the canonical Mazur-Tate pairing in the ordinary reduction case and Schneider’s norm-adapted pairing.

Besides these two p-adic height pairings, we study Néron’s classical real-valued pairing. We use an approach to height pairings developped in [Ma-Ta]. Let K be a non-archimedean local ground field. For any homomorphism ρ : K^{×} → Y to some abelian group Y, we can define a local height pairing on A_{K} with values in Y whenever we can continue ρ to a “bihomomorphic” map, a so-called ρ-splitting σ : P_{A_{K}×A′_{K}} (K) → Y on the K-rational points of the Poincaré biextension associated to A_{K} and its dual abelian variety A′_{K}. For our three types of height pairings the corresponding ρ-splittings can be uniquely characterized by certain properties. (We recall these facts in section 2.)

We assume that A_{K} has semistable reduction with split torus part, which can always be achieved after a finite base change. In section 3, we recall that A_{K} and A′_{K} are rigid analytic quotients of semiabelian varieties E_{K} respectively E′_{K} after certain lattices M_{K} and M′_{K}. Here the abelian quotients B_{K} respectively B′_{K} of E_{K} respectively E′_{K} have good reduction and are dual to each other. Let P_{B_{K}×B′_{K}} be the Poincaré biextension expressing the duality. We show that the biextension P_{A_{K}×A′_{K}}^{an}

is a quotient of the pullback of the biextension P_{B_{K}×B′_{K}}^{an} to E_{K}^{an} × E_{K}^{′an}.

Then, in section 4, we define (under a certain condition) for a given ρ-splitting σ on P_{B_{K}×B′_{K}} (K) a ρ-splitting τ on P_{A_{K}×A′_{K}} (K), and we describe the relation between the corresponding height pairings on B_{K} respectively on A_{K}.

In section 5 we show that if we start with the ρ-splitting σ corresponding to Néron’s local height pairing, the height pairing on A_{K} defined by our ρ-splitting τ is also Néron’s local height pairing. From this we can deduce a formula relating the Néron pairings on A_{K} and B_{K}. A similar formula was already proved in [Hi].

Then we investigate Schneider’s p-adic height pairing in section 6. First we show that if B_{K} has good ordinary reduction, our existence condition for τ is equivalent to the existence condition for Schneider’s height pairing on A_{K}, namely that the group of universal norms with respect to a certain ℤ_{p}-extension associated to ρ has finite index in A_{K}(K). Afterwards, we prove that if we use the ρ-splitting σ defining Schneider’s p-adic height pairing on B_{K} to construct τ, then the height pairing on A_{K} defined by τ is also Schneider’s p-adic pairing.

In the last section we compare the canonical Mazur-Tate ρ-splittings in the ordinary case on B_{K} and on A_{K}. Then we calculate the difference between Schneider’s p-adic height pairing and the p-adic Mazur-Tate pairing on A_{K}, using the fact that they coincide on abelian varieties with good ordinary reduction. Thereby we correct an error in the comparison formula for Tate curves given in [MTT], p.34.

This paper generalizes [We], where we showed formulas for Néron’s and Schneider’s height pairings on abelian varieties with split multiplicative reduction.

We adopt the following terminological conventions:

For a group scheme G over a base T we denote the unit section by e_{G/T}. When we refer to extensions or biextensions of T -group schemes, we work in the fppf-site over T. But note that we will often consider extensions and biextensions by 𝔾_{m}, so that many sequences will also be exact in the big Zariski site. For a biextension Q of T -group schemes X and X′ by G over T we refer to the element e_{Q/X}(e_{X/T}) ∈ Q(T) as the unit section of Q. By [SGA7, I], VII, 2.2, this is a symmetrical notion, i.e. e_{Q/X}(e_{X/T}) = e_{Q/X′}(e_{X′/T}).

We will often work with rigid analytic varieties over a complete non-archimedean field K, endowed with their rigid analytic Grothendieck topology. (See [BGR], 9.3, Def. 4.) There is a rigid analytic GAGA functor, associating to a K-scheme X locally of finite type a rigid analytic variety X^{an}, see [BGR], 9.3, Ex. 2. The analogies of Serre’s complex analytic GAGA theorems hold, see [Kö]. Extensions or biextensions of rigid analytic group varieties are always to be understood in the ”big Zariski site”, i.e. the category of rigid analytic varieties endowed with their Grothendieck topology.

Throughout this paper, K will be a non-archimedean field, locally compact with respect to a non-trivial absolute value, R will be its ring of integers, and k the residue class field. By v_{K} : K^{×} → ℤ we denote the valuation map, mapping a prime element to 1.

2 Height pairings

We fix an abelian variety A_{K} over K and a dual abelian variety (A′_{K}, P_{A_{K}×A′_{K}}), where P_{A_{K}×A′_{K}} is the Poincaré biextension expressing the duality. (See [SGA7, I], VII, 2.9). We write P = P_{A_{K}×A′_{K}}, when confusion seems unlikely. Note that P (K) is a biextension of A(K) × A′(K) by K^{×} in the category of sets. Let ρ : K^{×} → Y be a homomorphism to some abelian group Y. We call a map σ : P (K) → Y a ρ-splitting if it is compatible with the biextension structure on P (K), i.e. if the following conditions hold:

i) σ(αx) = ρ(α) + σ(x) for all α ∈ K^{×} and x ∈ P (K).

ii) For all a ∈ A_{K}(K) (respectively a′ ∈ A′_{K}(K)) the restriction of σ to P (K) ×_{(A_{K}(K)×A′_{K}(K))} {a} × A′_{K}(K) (respectively P (K) ×_{(A_{K}(K)×A′_{K}(K))} A_{K}(K) × {a′}) is a group homomorphism. (See [Ma-Ta], 1.4.)

Let Div^{0}A_{K} denote the group of divisors on A_{K} which are algebraically equivalent to zero, and let Z^{0}(A_{K}/K) denote the group of zero cycles on A_{K} with degree zero and K-rational support. By (Div^{0}A_{K} × Z^{0}(A_{K}/K))′ we denote the set of all pairs (D, z) with disjoint supports.

Whenever we have a homomorphism ρ : K^{×} → Y and a ρ-splitting σ : P (K) → Y, we can define a bilinear Mazur-Tate (height) pairing with values in Y:

( , )_{MT,σ} : (Div^{0}A_{K} × Z^{0}(A_{K}/K))′ ⟶ Y
(D, z)  ⟼ σ(s_{D}(z)),

where s_{D} is a rational section of P |_{A_{K}×{d}} → A_{K} with divisor D, and where d is the point in A′_{K}(K) corresponding to D. The rational section s_{D} is defined only up to a constant in K^{×} which vanishes when we continue s_{D} linearly to Z^{0}(A_{K}/K).

Let us denote by A respectively A′ the Néron models of A_{K} respectively A′_{K} over R, and by A^{0} respectively A^{′0} their identity components.

In this paper, we will deal with three situations in which good ρ-splittings can be singled out:

I) The Mazur-Tate splitting in the unramified case

Assume that ρ is unramified, i.e. that ρ vanishes on R^{×}, and that Y is uniquely divisible by m_{A}, the exponent of the group A_{k}(k)/A_{k}^{0}(k). There exists a biextension P_{A^{0}×A′} of A^{0} and A′ by 𝔾_{m,R} with generic fibre P, see [SGA7, I, exp. VIII], 7.1 b). The canonical ρ-splitting σ_{ρ} is defined as the unique ρ-splitting vanishing on P_{A^{0}×A′}(R) ⊂ P (K) ([Ma-Ta], 1.5.2). If ρ = log | |_{K} : K^{×} → ℝ, then the Mazur-Tate height pairing corresponding to the canonical ρ-splitting is just Néron’s local height pairing, see [Ma-Ta], 2.3.1.

II) Schneider’s p-adic height pairing

Here we take Y = ℚ_{p}. Let K be a finite extension of ℚ_{l}, and let ρ : K^{×} → ℚ_{p} be a non-trivial continuous homomorphism. Then ρ is continuous for the profinite topology on K^{×} and extends therefore uniquely to a homomorphism ρ^{∧} on the profinite completion K^{^{×}∧} of K^{×}. By local class field theory, K^{×∧} is topologically isomorphic to Gal(K^{ab}/K). Then ρ^{∧} determines a ℤ_{p}-extension K_{∞}/K with intermediate fields K_{ν} which are the uniquely determined cyclic extensions of degree p^{ν} of K such that ρ(N_{K_{ν}/K}K_{ν}^{×}) = p^{ν}ρ(K^{×}) ⊂ ℚ_{p} (see [Ma-Ta], 1.11.1). For any commutative group scheme G over K we denote by NG(K) ⊂ G(K) the group of universal norms with respect to K_{∞}/K. Furthermore, let P (K_{ν}, K) be the set of points in P (K_{ν}) which project to A_{K}(K_{ν}) × A′_{K}(K). We define NP (K) ⊂ P (K) as the intersection of all N_{K_{ν}/K}P (K_{ν}, K), where we use the group structure of P over A′_{K} to define norms.

If ρ is not unramified, assume that NA_{K}(K) has finite index in A_{K}(K). Then there exists a unique ρ-splitting σ_{ρ} : P (K) → ℚ_{p} vanishing on NP (K), see [Sch1], and [Ma-Ta], 1.11.5. If ρ is unramified (which e.g. is the case if the residue characteristic l is not equal to p), let σ_{ρ} the canonical ρ-splitting in case I).

We call ( , )_{MT,σ_{ρ}} Schneider’s local p-adic height pairing with respect to ρ. It was originally defined in [Sch1].

The following result characterizes the existence condition for Schneider’s height in the good reduction case.

Theorem 2.1 Assume that A_{K} has good reduction.

i)(Mazur) If A_{K} has good ordinary reduction, then for any non-trivial continuous homomorphism ρ the universal norm group NA_{K}(K) has finite index in A_{K}(K), i.e. Schneider’s local p-adic height pairing exists.

ii)(Schneider) Conversely, if ρ is not unramified, and if NA_{K}(K) has finite index in A_{K}(K), then A_{K} has good ordinary reduction.

Proof: See [Sch2], Theorem 2, and [Ma], 4.39. □ In section 6, we will investigate the existence condition for Schneider’s local height

in the case of semistable ordinary reduction.

III) The canonical Mazur-Tate splitting in the ordinary case

Let ρ : K^{×} → Y be a homomorphism. Assume that A has ordinary reduction, i.e. that the formal completion of A_{k} at the origin is isomorphic to a product of copies of 𝔾_{m}^{f} over the algebraic closure of k. This is equivalent to the fact that A_{k}^{0} is an extension of an ordinary abelian variety B_{k} by a torus T_{k}, see [Ma-Ta], 1.1. Note that in particular A has semistable reduction.

Now let T_{k} respectively T′_{k} be the maximal tori in A_{k} respectively A′_{k}, and denote by n_{A} respectively n_{A′} the exponents of A_{k}^{0}(k)/T_{k}(k) respectively A_{k}^{′0}(k)/T′_{k}(k). Assume that Y is uniquely divisible by m_{A}m_{A′}n_{A}n_{A′}. Moreover, denote by A^{t} and A^{′t} the formal completions of A and A′ along T_{k} and T′_{k}, and let P_{A^{0}×A′}^{t} be the formal completion of P_{A^{0}×A′} along the inverse image of T_{k} × T′_{k} in P_{A^{0}×A′}. Then P_{A^{0}×A′}^{t} is a formal biextension of A^{t} and A^{′t} by 𝔾_{mR}^{∧} (the formal completion of 𝔾_{m,R} along its special fibre). By [Ma-Ta], 5.11.1, P_{A^{0}×A′}^{t} admits a unique splitting σ_{0} : P_{A^{0}×A′}^{t} → 𝔾_{mR}^{∧}. Hence there exists a unique ρ-splitting σ̃ : P (K) → Y such that for all x ∈ P_{A^{0}×A′}^{t}(R) we have σ̃(x) = ρ ∘ σ_{0}(x). This defines a local p-adic height pairing ( , )_{MT,σ̃}.

If additionally ρ is unramified, then σ̃ coincides with the canonical splitting in case I).

On the other hand, if we take Y = ℚ_{p} and ρ is non-trivial and continuous but not unramified, case III) gives us a p-adic height pairing ( , )_{MT,σ̃} : (Div^{0}A_{K} × Z^{0}(A_{K}/K))′ → ℚ_{p}, if A has ordinary reduction. If A has good ordinary reduction, then ( , )_{MT,σ̃} coincides with Schneider’s p-adic height pairing from case II), see [Ma-Ta], 1.11.6. But in general both pairings may differ. We will compare these two p-adic height pairings in section 7.

Let us conclude this section with a general remark. If we start with a local field K and a homomorphism ρ : K^{×} → Y, we can extend ρ to any finite field extension L of K such that Y is uniquely divisible by [L : K] by the formula ρ_{L}(x) = [L :

K]^{−1}ρ(N_{L/K}(x)). Note that in all three cases discussed above the restriction of the canonical ρ_{L}-splitting to P (K) coincides with the canonical ρ-splitting. Hence we can investigate the corresponding local height pairings after finite base changes.

Finally, if we start with an abelian variety A_{F} over a global field F, we can define global height pairings by summing over all local ones, see [Ma-Ta], section 3.

3 Rigid analytic uniformization

We still fix A_{K}, A′_{K} and P_{A_{K}×A′_{K}}. We say that an abelian variety over K has split semistable reduction, if the special fibre of the identity component of its Néron model over R is an extension of an abelian variety by a split torus.

From now on we assume that A_{K} (and hence A′_{K}) has split semistable reduction. Note that by Grothendieck’s semistable reduction theorem, we are always in this situation after a finite base change. Even for an abelian variety A_{F} over a global field F we can find a finite extension E of F such that the Néron model of A_{F} ⊗ E has split semistable reduction at all finite places. Since our local height pairings are compatible with finite base changes, we can always place ourselves in the situation of the assumption if we want to deal with the local height pairings on A_{F} at the finite places of F.

Let us now recall some facts about the rigid analytic uniformization of A_{K} and A′_{K}. We can associate the following data to A_{K}, A′_{K} and P_{A_{K}×A′_{K}}:

i) Since A_{K} has split semistable reduction, there is an extension, the so-called Raynaud extension,

0 ⟶ T_{K} ⟶ E_{K} ⟶^{p} B_{K} ⟶ 0,

such that T_{K} is a split torus of dimension t over K, and B_{K} is an abelian variety over K with good reduction. Let M′ be the character group of T_{K}. Then M′ is a free ℤ-module of rank t. We denote the corresponding constant K-group scheme by M′_{K}. Fix once and for all a dual abelian variety (B′_{K}, P_{B_{K}×B′_{K}}) of B_{K}, where P_{B_{K}×B′_{K}} is the Poincaré biextension expressing the duality. We will always identify B′_{K} with \underline{Ext}^{1}(B_{K}, 𝔾_{m,K}) via b′ ↦ P_{B_{K}×B′_{K}} |_{B_{K}×\{b′\}} for functorial points b′ of B′_{K}. Then E_{K} corresponds to a homomorphism ϕ′ : M′ → B′_{K} (see e.g. [SGA7, I], VIII, 3.7.)

Besides, there is a rigid analytic homomorphism π : E_{K}^{an} → A_{K}^{an} inducing a short exact sequence

0 ⟶ M_{K}^{an} ⟶^{i} E_{K}^{an} ⟶^{π} A_{K}^{an} ⟶ 0,

where M_{K} is the constant group scheme corresponding to a free ℤ-module M of rank t. (See [Bo-Lü1], section 1, and [Ray].)

ii) We can construct a “dual” uniformization of A′_{K}: The embedding i : M_{K} → E_{K} induces a homomorphism ϕ : M_{K} →^{i} E_{K} →^{p} B_{K}, which gives us an extension E′_{K} (again by [SGA7, I], VIII, 3.7)

0 ⟶ T′_{K} ⟶ E′_{K} ⟶^{p′} B′_{K} ⟶ 0,

where T′_{K} is the split torus of dimension t over K with character group M. Besides, i induces a trivialization of the pullback P_{B_{K}×B′_{K}} |_{M_{K}×M′_{K}} of P_{B_{K}×B′_{K}} via the homorphism ϕ × ϕ′ : M_{K} × M′_{K} → B_{K} × B′_{K} in the following way: Fix m′ ∈ M′.

Then, tautologically, ϕ′(m′) ∈ B′_{K}(K) = Ext_{K}^{1}(B_{K}, G_{m}) corresponds to the extension P_{B_{K}×{ϕ′(m′)}}, and by the definition of ϕ′, this is the extension we get by pushout:

0 ⟶ T_{K} ⟶  E_{K}  ⟶ B_{K} ⟶ 0
m′↓  ^{h_{m′}} ↓  ↓
0 ⟶ 𝔾_{m,K} ⟶ P_{B_{K}×{ϕ′(m′)}} ⟶ B_{K} ⟶ 0

We define a bilinear map <, >: E_{K} × M′_{K} → P_{B_{K}×B′_{K}} by < e, m′ >:= h_{m′}(e).

Then the restriction of <, > to M_{K} × M′_{K} →^{i×id} E_{K} × M′_{K} induces a trivialization of P_{B_{K}×B′_{K}} |_{M_{K}×M′_{K}}. On the other hand, this trivialization defines an embedding i′ : M′_{K} → E′_{K} such that p′ ∘ i′ = ϕ′, see [Bo-Lü1], 3.2. As above, by definition of ϕ we have a pushout diagram

0 ⟶ T′_{K} ⟶  E′_{K}  ⟶ B′_{K} ⟶ 0
m↓  ^{h_{m}}↓  ↓
0 ⟶ 𝔾_{m,K} ⟶ P_{{ϕ(m)}×B′_{K}} ⟶ B′_{K} ⟶ 0.

We get a bilinear map <, >: M_{K} × E′_{K} → P_{B_{K}×B′_{K}} defined by <m, e′>:= h_{m}(e′),

which coincides with the previous pairing on M_{K} × M′_{K} →^{id×i′} M_{K} × E′_{K}.

iii) The extension E′_{K} is a rigid analytic uniformization of A′_{K}: There is a rigid analytic homomorphism π′ : E_{K}^{′an} → A_{K}^{′an} such that the sequence

0 ⟶ M_{K}^{′an} ⟶^{i′} E_{K}^{′an} ⟶^{π′} A_{K}^{′an} ⟶ 0

is exact, and such that we have the following description of P_{A_{K}×A′_{K}}:

The 𝔾_{m,K}^{an}-torsor P_{A_{K}×A′_{K}}^{an} is the quotient of (p^{an} × p^{′an})^{∗}P_{B_{K}×B′_{K}}^{an} after the M × M′-linearization given by

u_{(m,m′)} : P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}} E_{K}^{an} × E_{K}^{′an} ⟶ P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}} E_{K}^{an} × E_{K}^{′an},

mapping a (functorial) point (ω, e, e′), such that ω in P_{B_{K}×B′_{K}}^{an} projects to p(e)×p′(e′)

in B_{K}^{an} × B_{K}^{′an}, to

(([<e, m′> • <m, m′>] ⊙ [<m, e′> • ω]), me, m′e′).

Here ⊙ is the group law on P_{B_{K}×B′_{K}}^{an} as a B_{K}-group, and • is the group law on P_{B_{K}×B′_{K}}^{an} as a B′_{K}-group. (See [Bo-Lü1], Theorem 6.8.)

Hence, in particular, we have a quotient morphism of analytic 𝔾_{m,K}^{an}-torsors:

θ : P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}} E_{K}^{an} × E_{K}^{′an} → P_{A_{K}×A′_{K}}^{an} .

Note that both sides also carry biextension structures. After multiplying θ by an element of 𝔾_{m,K}(K), we may assume that θ maps the unit section in P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}}

E_{K}^{an} × E_{K}^{′an}(K) to the unit section in P_{A_{K}×A′_{K}}^{an} (K).

Proposition 3.1 θ is a morphism of biextensions.

Proof: For better readability, we put P = P_{A_{K}×A′_{K}} and Q = (p × p′)^{∗}P_{B_{K}×B′_{K}}. We investigate the map

α : Q^{an} ×_{E_{K}^{′an}} Q^{an} ⟶ P^{an}

defined by α(x, y) = θ(xy)θ(x)^{−1}θ(y)^{−1}, where we multiply and take inverses with respect to the group structures over E_{K}^{′an} respectively A_{K}^{′an}. Since θ is a torsor homomorphism, the composition of α with the projection P^{an} → A_{K}^{an} × A_{K}^{′an} factorizes through the unit section of the A_{K}^{′an}-group A_{K}^{an}×A_{K}^{′an}, hence there is a A_{K}^{′an}-morphism α′ : Q^{an} ×_{E_{K}^{′an}} Q^{an} → 𝔾_{m,K}^{an} × A_{K}^{′an} which yields α when composed with the natural

embedding 𝔾_{m,K}^{an} ×A_{K}^{′an} → P^{an}. Besides, α (and hence α′) is equivariant with respect to the operation by 𝔾_{m,K}^{an} × 𝔾_{m,K}^{an} we get from the torsor structure of Q^{an}. Hence α′ is derived from a map

β : E_{K}^{an} × E_{K}^{an} × E_{K}^{′an} → 𝔾_{m,K}^{an}

by composition with the natural projection Q^{an} ×_{E_{K}^{′an}} Q^{an} → E_{K}^{an} ×E_{K}^{an} ×E_{K}^{′an}. Now

let ω_{1} and ω_{2} be (functorial) points in P_{B_{K}×B′_{K}}^{an} with the same projection to B_{K}^{′an},

and let e_{1}, e_{2} be points in E_{K}^{an} and e′ a point in E_{K}^{′an} such that x_{1} = (ω_{1}, e_{1}, e′) and x_{2} = (ω_{2}, e_{2}, e′) are in Q^{an}. Besides, fix m_{1}, m_{2} in M and m′ in M′. Then

u_{(m_{1},m′)}(x_{1}) • u_{(m_{2},m′)}(x_{2}) = u_{(m_{1}m_{2},m′)}(ω_{1} • ω_{2}, e_{1}e_{2}, e′),

where on the left hand side we use the symbol • also for the group law on Q^{an} as an E_{K}^{′an}-group. This implies α(u_{(m_{1},m′)}(x_{1}), u_{(m_{2},m′)}(x_{2})) = α(x_{1}, x_{2}). From that we can deduce that β is invariant under the action of M × M × M′ on E_{K}^{an} × E_{K}^{an} × E_{K}^{′an}, which implies that there is a morphism β_{1} : A_{K}^{an} × A_{K}^{an} × A_{K}^{′an} → 𝔾_{m,K}^{an} such that β = β_{1} ∘ (π × π × π′).

But since A_{K} and A′_{K} are projective, β_{1} must be constant. Since θ respects the unit sections, it follows that β_{1} is equal to 1 ∈ 𝔾_{m,K}^{an}(K), hence β = 1 and α factorizes through e_{P^{an}/A_{K}^{′an}}. This means that θ respects the group structures

over E_{K}^{′an} respectively A_{K}^{′an}. A parallel argument now shows that θ is also a group homomorphism with respect to the group structures over E_{K}^{an} respectively A_{K}^{an}. □

4 Definition of a local height pairing via the Raynaud extensions

For the rest of this paper, we fix an abelian variety A_{K} with split semistable reduction and its dual abelian variety (A′_{K}, P_{A_{K}×A′_{K}}). Let ρ : K^{×} → Y be a homomorphism to some commutative ring Y, and let σ : P_{B_{K}×B′_{K}} (K) → Y be a ρ-splitting on P_{B_{K}×B′_{K}}. We will show how to construct in certain cases from σ a ρ-splitting τ on P_{A_{K}×A′_{K}} (K).

We fix once and for all bases m_{1}, …, m_{t} for M and m′_{1}, …, m′_{t} for M′.

Definition 4.1 We call σ M-invertible, if the (t × t)-matrix (σ(<m_{i}, m′_{j}>)_{i,j}) with entries in Y is invertible over Y.

(This definition differs slightly from Definition 4.4 in [We].) For M-invertible σ we will now define a ρ-splitting τ^{∗} on (P_{B_{K}×B′_{K}} ×_{B_{K}×B′_{K}} E_{K} × E′_{K}) (K), which descends to a ρ-splitting τ on P (K).

Proposition 4.2 Assume that σ is M-invertible, and let Σ be the inverse matrix of (σ(<m_{i}, m′_{j}>)_{i,j}).

Define the ρ-splitting τ^{∗} : (P_{B_{K}×B′_{K}} ×_{B_{K}×B′_{K}} E_{K} × E′_{K})(K) → Y by

(ω, e, e′) ⟼ σ(ω) − (σ <e, m′_{1}>, …, σ <e, m′_{t}>) Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>)

for ω ∈ P_{B_{K}×B′_{K}} (K) and (e, e′) ∈ (E_{K} × E′_{K})(K) with the same projection to (B_{K} × B′_{K})(K).

Then there is a uniquely determined ρ-splitting τ : P (K) → Y such that τ^{∗} = τ∘θ.

Proof: First of all, note that τ^{∗} is indeed a ρ-splitting. Besides, we claim that for all m ∈ M, m′ ∈ M′, e ∈ E_{K}(K) and e′ ∈ E′_{K}(K) we have

i) τ^{∗}(<m, e′>, m, e′) = 0 and ii) τ^{∗}(<e, m′>, e, m′) = 0. We will only show i), since the argument for ii) is completely parallel. Note that

it suffices to prove i) for our basis m_{1}, …, m_{t}, since the left hand side is additive in m. By definition of Σ, we find that for all i = 1, …, t the vector (σ <m_{i}, m′_{1}>, …, σ <m_{i}, m′_{t}>)Σ is the i-th unit vector, hence

τ^{∗}(<m_{i}, e′>, m_{i}, e′)

= σ <m_{i}, e′> −(σ <m_{i}, m′_{1}>, …, σ <m_{i}, m′_{t}>) Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>)
= σ <m_{i}, e′> −σ <m_{i}, e′>= 0,

as desired. Now we can calculate

τ^{∗}(u_{(m,m′)}(ω, e, e′))

= τ^{∗}(<e, m′>, e, m′) + τ^{∗}(<m, m′>, m, m′) + τ^{∗}(<m, e′>, m, e′) + τ^{∗}(ω, e, e′)
= τ^{∗}(ω, e, e′).

Hence there is a uniquely determined map τ : P_{A_{K}×A′_{K}} (K) → Y such that τ^{∗} = τ ∘ θ, and since θ is a homomorphism of biextensions by 3.1, τ is also a ρsplitting. □

Hence we can define a local height pairing on A_{K}

( , )_{MT,τ} : (Div^{0}A_{K} × Z^{0}(A_{K}/K))′ → Y

for any M-invertible ρ-splitting σ on P_{B_{K}×B′_{K}} (K).

In the next three sections, we will investigate the connection to the canonical height pairings in our three cases.

But first we will describe our local height pairing ( , )_{MT,τ} a bit more explicitely. Obviously, the description of P_{A_{K}×A′_{K}}^{an} as a quotient via θ implies that θ induces

an isomorphism P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}} E_{K}^{an} × E_{K}^{′an} ≃ (π × π′)^{∗}P_{A_{K}×A′_{K}}^{an}. Restricting this isomorphism we find for any a′ ∈ A′_{K}(K) and any preimage e′ ∈ E′_{K}(K) of a′ an isomorphism ν : π^{∗}P_{A_{K}×{a′}}^{an} → p^{an∗}P_{B_{K}×{p′(e′)}}^{an} which makes the following diagram commutative:

π^{∗}P_{A_{K}×{a′}}^{an} ⟶^{ν} p^{an∗}P_{B_{K}×{p′(e′)}}^{an}
↓  ↓^{θ(−,−,e′)}
P_{A_{K}×{a′}}^{an} ⟶^{=} P_{A_{K}×{a′}}^{an}.

Now consider a divisor D ∈ Div^{0}(A_{K}) with divisor class a′ ∈ A′_{K}(K). We choose a preimage e′ ∈ E′_{K}(K) of a′ and we denote by b′ the point p′(e′) ∈ B′_{K}(K). Let D^{∼} be a divisor in Div^{0}(B_{K}) whose class corresponds to b′. Then there is a meromorphic function h on E_{K}^{an} such that π^{∗}D^{an} = p^{an∗}D^{∼an} + div(h).

Let s_{D} and s_{D^{∼}} be rational sections corresponding to D respectively D^{∼} (both are uniquely defined up to a constant). s_{D} induces a meromorphic section s_{D}^{an} of P_{A_{K}×{a′}}^{an}, which we can pull back to a meromorphic section π^{∗}s_{D}^{an} of π^{∗}P_{A_{K}×{a′}}^{an} with divisor π^{∗}D^{an}. Via the isomorphism ν, this induces a meromorphic section ν(π^{∗}s_{D}^{an}) of p^{an∗}P_{B_{K}×{b′}}^{an} with the same divisor. Besides, the rational section s_{D^{∼}} gives a meromorphic section p^{an∗}s_{D^{∼}}^{an} of p^{an∗}P_{B_{K}×{b′}}^{an} with divisor p^{an∗}D^{∼an}.

Hence the meromorphic sections ν(π^{∗}s_{D}^{an}) and h · p^{an∗}s_{D^{∼}}^{an} of p^{an∗}P_{B_{K}×{b′}} differ by a function g ∈ Γ(E_{K}^{an}, O^{×}). We put h^{⋄} = hg. Then we have π^{∗}D^{an} = p^{an∗}D^{∼an} + div(h^{⋄}) and ν(π^{∗}s_{D}^{an}) = h^{⋄} · (p^{an∗}s_{D^{∼}}^{an}).

Now let a ∈ A_{K}(K) be a point not lying in the support of D. For any preimage e ∈ E_{K}(K) of a we can calculate

τ(s_{D}(a)) = τ(s_{D}(πe))
= τ(θ(−, −, e′) ∘ ν ∘ π^{∗}s_{D}^{an}(e))
= τ(θ(h^{⋄}(e) · s_{D^{∼}}(pe), e, e′))
= τ^{∗}(h^{⋄}(e) · s_{D^{∼}}(pe), e, e′)
= ρ(h^{⋄}(e)) + σ(s_{D^{∼}}(pe))

− (σ <e, m′_{1}>, …, σ <e, m′_{t}>) Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>).

This proves the following

Theorem 4.3 For (D, ∑_{i} n_{i}a_{i}) ∈ (Div^{0}A_{K} × Z^{0}(A_{K}/K))′ let a′ ∈ A′_{K}(K) be the point corresponding to D, and choose a preimage e′ ∈ E′_{K}(K) of a′. Then there exists a divisor D^{∼} ∈ Div^{0}(B_{K}) and a meromorphic function h^{⋄} on E_{K}^{an} such that π^{∗}D^{an} = p^{an∗}D^{∼an}+div(h^{⋄}) and such that for all rational sections s_{D} and s_{D^{∼}} corresponding to D respectively D^{∼} the meromorphic sections ν(π^{∗}s_{D}^{an}) and h^{⋄} · (p^{an∗}s_{D^{∼}}^{an}) of p^{an∗}P_{B_{K}×{p′(e′)}}^{an} differ by a constant.

For any choice of preimages e_{i} ∈ E_{K}(K) of the a_{i} we have the following formula for the canonical Mazur-Tate pairing associated to τ:

(D, ∑ n_{i}a_{i})_{MT,τ} = (D^{∼}, ∑ n_{i}p(e_{i}))_{MT,σ} + ∑ n_{i}ρ(h^{⋄}(e_{i}))

−(σ <∑ n_{i}e_{i}, m′_{1}>, …, σ <∑ n_{i}e_{i}, m′_{t}>) Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>).

5 Néron’s local height pairing

In this section we show that our ρ-splitting τ coincides with the canonical Mazur-Tate splitting in the unramified case if σ is the canonical Mazur-Tate splitting on B_{K}. Hence we can use τ to “calculate” Néron’s local height pairing on A_{K} in terms of Néron’s local height pairing on B_{K}.

We need some notation first. Put S = SpecR. We denote by A respectively A′ the Néron models of A_{K} respectively A′_{K}, and by B and B′ the Néron models of B_{K} and B′_{K}. Note that the split torus T_{K} and the semiabelian variety E_{K} have Néron models T and E over R by [BLR], 10.1, Proposition 7, and that the identity component T^{0} of T is isomorphic to 𝔾_{m,R}^{t} by [BLR], 10.1, Example 5. Similarly, let T′ and E′ be the Néron models of T′_{K} and E′_{K}.

By [SGA7, I], VIII, 7.1, we can (up to canonical isomorphism) uniquely extend P_{B_{K}×B′_{K}} to a biextension P_{B×B′} of B and B′ by 𝔾_{m,R}, and P_{A_{K}×A′_{K}} to a biextension P_{A^{0}×A′} of A^{0} and A′ by 𝔾_{m,R}. Now the sequences of identity components

0 ⟶ T^{0} ⟶ E^{0} ⟶ B ⟶ 0

and

0 ⟶ T^{′0} ⟶ E^{′0} ⟶ B′ ⟶ 0

are exact ([BLR], 10.1, proof of Proposition 7). Denote by D(T^{0}) and D(T^{′0}) the Cartier duals of T^{0} and T^{′0}. Then these sequences induce homomorphims ϕ : M_{S} ≃ D(T^{′0}) → B and ϕ′ : M′_{S} ≃ D(T^{0}) → B′, which extend our previous maps ϕ : M_{K} → B_{K} respectively ϕ′ : M′_{K} → B′_{K}. Here M_{S} and M′_{S} of course denote the constant S-group schemes corresponding to M and M′.

Hence we have pushout homomorphisms h_{m′} : E^{0} → P_{B×{ϕ′(m′)}} and h_{m} : E^{′0} → P_{{ϕ(m)}×B′} for m in M and m′ in M′. The pairings <, >_{S}: E^{0} × M′_{S} → P_{B×B′} and <, >_{S}: M_{S} × E^{′0} → P_{B×B′} defined by <e, m′ >_{S}= h_{m′}(e) and <m, e′ >_{S}= h_{m}(e′) extend the pairings from section 3, ii).

We will also use some results from formal and rigid geometry. Recall that there is a canonical functor associating to a formal S-scheme X, flat and locally of topologically finite type over S, its rigid analytic generic fibre X^{rig}, see [Bo-Lü2]. It is defined locally by associating to a formal affine scheme SpfA the affinoid variety Sp(A⊗_{R} K). Note that X(R) = Mor_{formal/R}(SpfR, X) = Mor_{rigid/K}(SpK, X^{rig}) = X^{rig}(K) by a standard argument: It suffices to check this for formal affine X = SpfA. Then one uses the fact that the supremum semi-norm is contractive ([BGR], Prop. 1, p. 238) to show that every K-homomorphism A ⊗_{R} K → K restricts to an R-homomorphism A → R.

Moreover, we use the theory of formal Néron models of rigid analytic groups as developped in [Bo-Sch]. A formal Néron model of a smooth rigid analytic K-variety Y is a smooth formal R-scheme Z such that its generic fibre Z^{rig} is an open rigid subspace of Y and such that for any smooth formal R-scheme Z′ all rigid K-morphisms Z^{′rig} → Y extend uniquely to formal R-morphisms Z′ → Z. For all S-schemes X, we denote by X^{∧} the completion along the special fibre. If X is separated and of finite type over S, there is a canonical open immersion X^{∧rig} → X_{K}^{an}. We will often use the fact that for a commutative smooth K-group scheme X_{K} of finite type, the formal completion X^{∧} of its ordinary Néron model X is a formal Néron model of X_{K}^{an}, see [Bo-Sch], Theorem 6.2.

Lemma 5.1 Let Y be a commutative ring, and let v : K^{×} → Y be the homomorphism x ↦ v_{K}(x)1_{Y} given by the valuation map v_{K} : K^{×} → ℤ. We denote by σ_{v} the canonical v-splitting in the unramified case on P_{B_{K}×B′_{K}} (K). Then σ_{v} is M-invertible iff Y is uniquely divisible by m_{A}, the exponent of A_{k}(k)/A_{k}^{0}(k).

Proof: First of all, note that σ_{v} exists since B_{K} has good reduction (which implies m_{B} = 1). Our claim could be proven along the same lines as Lemma 4.9 in [We], but we prefer to give a different argument here.

Now M_{K} is a split lattice in E_{K} by [Bo-Lü1], Thm. 1.2, which means that the map M → ℝ^{t} given by m ↦ (σ_{v_{K}} <m, m′_{1}>, …, σ_{v_{K}} <m, m′_{t}>) is a bijection onto a lattice in ℝ^{t}. This implies that the pairing M × M′ → ℤ, mapping (m, m′) to σ_{v_{K}} (<m, m′>), induces an injection j : M → Hom(M′, ℤ). (Note that this pairing coincides with the monodromy pairing. This is claimed in [SGA7, I], IX, 14.2.5 and proven in [Co].) From the rigid analytic uniformization of A_{K} one can deduce that the component group ϕ_{A} = A_{k}/A_{k}^{0} is constant, and that there exists an exact sequence

0 → M →^{j} Hom(M′, ℤ) → ϕ_{A}(k) → 0.

(See e.g. [Bo-Xa], 5.2.) Hence the number of elements in ϕ_{A}(k) is equal to | det (σ_{v_{K}} (<m_{i}, m′_{j}>))|. Moreover, we have H^{1}(k_{ét}, A_{k}^{0}) = 0 by [La]. Hence we find that the natural inclusion A_{k}(k)/A_{k}^{0}(k) → ϕ_{A}(k) is actually an isomorphism.

This implies that det(σ_{v}(<m_{i}, m′_{j}>)) is a unit in Y iff m_{A} is a unit in Y. Hence our claim follows. □

Now we can compare our splitting τ to the canonical Mazur-Tate splitting:

Theorem 5.2 Let ρ : K^{×} → Y be an unramified homomorphism to the commutative ring Y, and let σ be the canonical ρ-splitting on P_{B_{K}×B′_{K}} (K) in the unramified case.

i) If σ is M-invertible, then Y is uniquely divisible by m_{A}. Conversely, if Y is uniquely divisible by m_{A}, and ρ(r) is a unit in Y for one (and hence for any) prime element r in R, then σ is M-invertible.

ii) Assume that σ is M-invertible. Then our ρ-splitting τ from 4.2 is the canonical ρ-splitting in the unramified case.

Proof: i) Since ρ is unramified, we find ρ(x) = v_{K}(x)ρ(r), where r is a prime element in K^{×}. Let σ_{v} denote as above the canonical v-splitting of P_{B_{K}×B′_{K}} (K). Then for all z ∈ P_{B_{K}×B′_{K}} (K) we have σ(z) = σ_{v}(z)ρ(r). Hence det(σ(<m_{i}, m′_{j}>)_{i,j}) = ρ(r)^{t} det(σ_{v}(<m_{i}, m′_{j}>)_{i,j}), which, together with 5.1, implies our claim.

ii) In order to show that τ coincides with the canonical Mazur-Tate-splitting, we have to show that τ vanishes on P_{A^{0}×A′}(R) ⊂ P_{A_{K}×A′_{K}} (K). We fix a point a′ ∈ A′_{K}(K) = A′(R) and a preimage e′ of a′ in E′_{K}(K) = E′(R). Let b′ ∈ B′_{K}(K) = B′(R) be the projection of e′. Since P_{A_{K}×{a′}} is semiabelian, it has a Néron model Q over S, which is an extension of A by G_{S}, the Néron model of 𝔾_{m,K}. Its formal completion Q^{∧} is a formal Néron model of P_{A_{K}×{a′}}^{an}.

Let us write ϑ for the map θ(−, −, e′) : P_{B_{K}×{b′}}^{an} ×_{B_{K}^{an}} E_{K}^{an} → P_{A_{K}×{a′}}^{an}. By the universal property of formal Néron models, the homomorphism of rigid analytic K-groups

(P_{B×{b′}}^{∧} ×_{B^{∧}} E^{0∧})^{rig} ↪ P_{B_{K}×{b′}}^{an} ×_{B_{K}^{an}} E_{K}^{an} ⟶^{ϑ} P_{A_{K}×{a′}}^{an}

is induced from a unique formal morphism

P_{B×{b′}}^{∧} ×_{B^{∧}} E^{0∧} ⟶^{f} Q^{∧},

which means that it coincides with

(P_{B×{b′}}^{∧} ×_{B^{∧}} E^{0∧})^{rig} ⟶^{f^{rig}} (Q^{∧})^{rig} ↪ P_{A_{K}×{a′}}^{an}.

Moreover, f is a homomorphism of formal S-group schemes. Now P_{B×{b′}}^{∧} ×_{B^{∧}} E^{0∧} is connected (use e.g. [EGA IV] 4.5.7), hence its image via f is contained in Q^{0∧}, the identity component of Q^{∧}. Hence we get the following commutative diagram:

(P_{B×{b′}}^{∧} ×_{B^{∧}} E^{0∧})^{rig} ⟶ P_{B_{K}×{b′}}^{an} ×_{B_{K}^{an}} E_{K}^{an}
↓  ↓^{ϑ}
(Q^{0∧})^{rig}  ⟶  P_{A_{K}×{a′}}^{an}

which induces on K-rational points the commutative diagram

(P_{B×{b′}} ×_{B} E^{0})(R) ⟶^{⊂} (P_{B_{K}×{b′}}^{an} ×_{B_{K}^{an}} E_{K}^{an})(K)
↓  ↓^{ϑ}
Q^{0}(R)  ⟶^{⊂}  P_{A_{K}×{a′}}^{an}(K)

According to [SGA7, I], VIII, 7.1, taking the generic fibre induces a fully faithful functor from the category of extensions of A^{0} by 𝔾_{m,R} to the category of extensions of A_{K} by 𝔾_{m,K}. Hence there is an isomorphism Q^{0} ⟶^{∼} P_{A^{0}×{a′}} inducing the identity on the generic fibre. This implies that θ maps (P_{B×{b′}} ×_{B} E^{0})(R) to P_{A^{0}×{a′}}(R).

Now take a point x in P_{A^{0}×{a′}}(R) projecting to a ∈ A^{0}(R). The homomorphism E^{0∧} → A^{0∧} induced by π : E_{K}^{an} → A_{K}^{an} is an isomorphism (see [Bo-Xa], Thm. 2.3), hence we find a point y ∈ (P_{B×{b′}}×_{B} E^{0})(R) projecting to a ∈ A^{0}(R). Since ϑ(y) lies in P_{A^{0}×{a′}}(R), it follows that x = αϑ(y) = ϑ(αy) = θ(αy, e′) for some α ∈ 𝔾_{m,R}(R). So τ(x) = τ^{∗}(z) for some z ∈ (P_{B×B′} ×_{B×B′} (E^{0} × E′))(R). Since σ vanishes on P_{B×B′}(R), and since <e, m′>=<e, m′>_{R} ∈ P_{B×B′}(R) for all e ∈ E^{0}(R) and m′ ∈ M′, it follows from the definition of τ^{∗}, that τ(x) = τ^{∗}(z) = 0. Hence τ coincides with the canonical Mazur-Tate splitting in the unramified case. □

From this theorem we immediately get the following

Corollary 5.3 If ρ = log | |_{K} : K^{×} → ℝ, and σ : P_{B_{K}×B′_{K}} (K) → ℝ is the canonical ρ-splitting, then σ is M-invertible. Define τ as in 4.2. Then ( , )_{MT,τ} coincides with Néron’s local height pairing on A_{K}.

According to this corollary, Theorem 4.3 expresses Néron’s local height pairing on A_{K} with the one on B_{K}. There is a similar result by Hindry who even relates the Néron functions for arbitrary divisors on A_{K} to certain Néron functions on B_{K}. Let us denote by ( , )_{N,A_{K}} the local Néron pairing on A_{K}. For D ∈ Div^{0}(A_{K}) completely antisymmetric, i.e. of the shape D = (−1)^{∗}D′ − D′ for some divisor D′ on A_{K}, Hindry’s result is the following: There exists a completely antisymmetric divisor D^{∼} ∈ Div^{0}(B_{K}) and a meromorphic function h with h(e^{−1}) = h(e)^{−1} on E_{K}^{an} such that π^{∗}D^{an} = p^{an∗}D^{∼an} + div(h) and such that for ∑_{i} n_{i}a_{i} ∈ Z^{0}(A_{K}/K) disjoint from the support of D and all preimages e_{i} of a_{i} in E_{K}(K) the following formula holds

(D, ∑_{i} n_{i}a_{i})_{N,A_{K}} = ∑_{i} n_{i} log |h(e_{i})|_{K} + (D^{∼}, ∑_{i} n_{i}p(e_{i}))_{N,B_{K}} + ∑_{i} n_{i}J(e_{i}),

where J : E_{K}(K) → ℝ is a linear function determined by its values on M which are given by

J(m) = log |\frac{h(e)}{h(me)}|_{K} + (D^{∼}, p(e) − p(me))_{N,B_{K}}

for arbitrary e. (See [Hi], Lemme 3.4 and Théorème D, but note that our height pairing differs from his by a sign, since we started with ρ = log | |_{K}.)

Our result in 4.3 can be used to deduce an expression for Hindry’s linear term J(e) for general e ∈ E_{K}(K).

6 Schneider’s local p-adic height pairing

Let K be a finite extension of ℚ_{l}, and let ρ : K^{×} → ℚ_{p} be a non-trivial continuous homomorphism with corresponding ℤ_{p}-extension K_{∞}/K. Since we already dealt with the unramified case in section 5, we will in this section assume that ρ is not unramified. Recall that this implies that l = p. Since R^{×} ⊂ K^{×} is mapped to a non-trivial compact subgroup of ℚ_{p}, there is an integer s so that ρ(K^{×}) = p^{s}ℤ_{p}. The goal of this section is to prove the following two theorems:

Theorem 6.1 Assume that ρ is not unramified, and that B_{K} has ordinary reduction. Let σ_{ρ} be the canonical Schneider ρ-splitting on P_{B_{K}×B′_{K}} (K). Then the universal norm group NA_{K}(K) with respect to K_{∞}/K has finite index in A_{K}(K) iff σ_{ρ} is M-invertible, i.e. iff the matrix (σ_{ρ} <m_{i}, m′_{j}>_{i,j}) is invertible over ℚ_{p}.

With other words, this theorem says that in the semistable ordinary reduction case Schneider’s local p-adic height exists iff our ρ-splitting τ from 4.2 exists.

Theorem 6.2 Assume that B_{K} has ordinary reduction, and let σ be the canonical Schneider ρ-splitting on P_{B_{K}×B′_{K}} (K). If σ is M-invertible, our ρ-splitting τ from 4.2 is equal to Schneider’s ρ-splitting σ_{ρ} on P_{A_{K}×A′_{K}} (K).

Hence ( , )_{MT,τ} coincides with Schneider’s p-adic height pairing on A_{K}.

Let us prove two lemmas first. We will use the notation from the beginning of section 5.

Lemma 6.3 The map

μ : E^{0} → P_{B×{ϕ′(m′_{1})}} ×_{B} … ×_{B} P_{B×{ϕ′(m′_{t})}} =: ×_{B}P_{B×{ϕ′(m′_{j})}}
x ↦ (<x, m′_{1}>_{S}, …, <x, m′_{t}>_{S})

is an isomorphism.

Proof: Look at the following commutative diagram in the category of abelian sheaves on the big flat site over S:

0 ⟶ T^{0} ⟶  E^{0}  ⟶ B ⟶ 0
(m′_{1},...,m′_{t})↓  ^{μ}↓  ∥  |
0 ⟶ 𝔾_{m,S}^{t} ⟶ ×_{B}P_{B×{ϕ′(m′_{j})}} ⟶ B ⟶ 0.

Both horizontal sequences are exact. This is clear for the upper one. For the lower one, it follows from the exactness of the sequences

0 ⟶ 𝔾_{m,S} ⟶ P_{B×{ϕ′(m′_{j})}} ⟶ B ⟶ 0

for all j. Hence, since (m′_{1}, … m′_{t}) : T^{0} → 𝔾_{m,S}^{t} is an isomorphism, μ is also an isomorphism. □

Lemma 6.4 Assume that B_{K} has ordinary reduction. Then taking universal norms with respect to K_{∞}/K in the short exact sequence 0 → T_{K} → E_{K} → B_{K} → 0 yields a short exact sequence

0 ⟶ NT_{K}(K) ⟶ NE_{K}(K) ⟶ NB_{K}(K) ⟶ 0.

Proof: Recall Theorem 2.1,i) and imitate the proof of Lemma 3 in section 2 of [Sch1], substituting 𝔾_{m,K} by T_{K}. □

Now we are ready to prove Theorem 6.1:

Proof of Theorem 6.1: We write N_{ν} for the norm map N_{K_{ν}/K} where K_{ν}/K is the intermediate layer of degree p^{ν} in the ℤ_{p}-extension K_{∞} belonging to ρ.

(1) π induces an isomorphism

E_{K}(K)/ ∩_{ν} (MN_{ν}E_{K}(K_{ν})) ⟶^{∼} A_{K}(K)/NA_{K}(K).

(2) Since B_{K} has good ordinary reduction, the quotient B_{K}(K)/NB_{K}(K) is finite by 2.1. Let d be the number of elements of this group. Since B′_{K} is isogeneous to B_{K}, the quotient B′_{K}(K)/NB′_{K}(K) is also finite, and we denote its cardinality by d′.

By construction, Schneider’s canonical ρ-splitting σ maps a point x ∈ P_{B_{K}×B′_{K}} (K) to σ(x) = d^{−1}ρ(α), where α is an element in K^{×} such that x^{d} ∈ αNP (K). Hence the image of σ is contained in d^{−1}ρ(K^{×}) = d^{−1}p^{s}ℤ_{p}. Besides, if we assume that x ∈ N_{ν}P_{B_{K}×{b′}}(K_{ν}) for some b′ ∈ B′_{K}(K) and some index ν, we find that σ(x) = d^{−1}ρ(α) for some α ∈ K^{×} ∩ N_{ν}P_{B_{K}×{b′}}(K_{ν}). The proof of Lemma 3 in section 2 of [Sch1], applied to X = P_{B_{K}×{b′}}, shows that α^{d′} is contained in N_{ν}(K_{ν}^{×}). Hence σ(x) lies in d^{−1}d^{′−1}ρ(N_{ν}K_{ν}^{×}) = d^{−1}d^{′−1}p^{ν+s}ℤ_{p}.

Since σ induces homomorphisms P_{B_{K}×{ϕ′(m′_{j})}}(K) → d^{−1}p^{s}ℤ_{p} for each j, we can define a homomorphism

ω : E_{K}(K) →^{μ_{K}} ×_{B_{K}} P_{B_{K}×\{ϕ′(m′_{j})\}}(K) → ∏_{j=1}^{t} P_{B_{K}×\{ϕ′(m′_{j})\}}(K) \underrightarrow{∏ σ} (d^{−1}p^{s}ℤ_{p})^{t},

where μ_{K} is the generic fibre of the isomorphism μ from Lemma 6.3. Let us first show that the cokernel of ω is annihilated by d. Namely, consider (α_{1}, …, α_{t}) ∈ p^{s}ℤ_{p}^{t} with α_{j} = ρ(x_{j}) for x_{j} ∈ K^{×}. Then (x_{1}, …, x_{t}) is an element of 𝔾_{m}(K)^{t} which embeds naturally into ×_{B}P_{B×{ϕ′(m′_{j})}}(K). Hence there exists some z ∈ E_{K}(K) such that μ_{K}(z) = (x_{1}, … x_{t}). Then ω(z) = (ρ(x_{1}), …, ρ(x_{t})) = (α_{1}, …, α_{t}). This proves our claim.

Let us now show that ω(∩_{ν}(MN_{ν}E_{K}(K_{ν}))) is contained in the p-adic closure \overline{ω(M)} of ω(M). Take an element z in ∩_{ν}(MN_{ν}E_{K}(K_{ν})). Then ω(z) lies in ω(M) + ω(N_{ν}E_{K}(K_{ν})) for all ν. Now ω(N_{ν}E_{K}(K_{ν})) is contained in ∏_{j} σ(N_{ν}P_{B×{ϕ′(m′_{j})}}(K_{ν})), which is contained in ∏_{j}(d^{−1}d^{′−1}p^{ν+s}ℤ_{p}), as we showed above. This implies that ω(z) lies indeed in \overline{ω(M)}. Hence ω induces a homomorphism

\overline{ω} : E_{K}(K)/ ∩_{ν} (MN_{ν}E_{K}(K_{ν})) ⟶ (d^{−1}p^{s}ℤ_{p})^{t}/\overline{ω(M)}.

Since the cokernel of ω is annihilated by d, the same holds for the cokernel of \overline{ω}. Let us now study the kernels of ω and \overline{ω}. For all z ∈ E_{K}(K), z^{d} projects to an element b in NB_{K}(K) ⊂ B_{K}(K). According to Lemma 6.4, b has a preimage z′ in NE_{K}(K). Hence z^{d} = z′α for some α ∈ T_{K}(K). Since σ vanishes on NP (K), we find ω(z′) = 0. Hence if we assume that ω(z) = 0, it follows that ω(α) = (ρ(m′_{1}α), …, ρ(m′_{t}α)) = 0. Therefore all m′_{j}α lie in the kernel of ρ, which is equal to N𝔾_{m}(K), so that α ∈ NT_{K}(K). Hence z^{d} is contained in NE_{K}(K), which implies that Kerω/NE_{K}(K) is annihilated by d.

Let now z be an element in E_{K}(K) such that ω(z) lies in \overline{ω(M)}. Hence for all ν we find some m_{ν} ∈ M such that ω(z) − ω(m_{ν}) ∈ p^{ν}((p^{s}ℤ_{p})^{t}) = (ρ(N_{ν}K_{ν}^{×}))^{t}. So we find (α_{1}, …, α_{t}) ∈ (N_{ν}K_{ν}^{×})^{t} such that ω(z) − ω(m_{ν}) = (ρ(α_{1}), …, ρ(α_{t})), which is equal to ω(t_{ν}) for the element t_{ν} ∈ N_{ν}T_{K}(K_{ν}) satisfying m′_{j}(t_{ν}) = α_{j} for all j. Therefore zm_{ν}^{−1}t_{ν}^{−1} is contained in the kernel of ω, which implies that z^{d} lies in MN_{ν}E_{K}(K_{ν}) for all ν. Hence we find that the kernel of \overline{ω} is annihilated by d.

(3) We deduce from (1) and (2) that if B_{K} has ordinary reduction, then A_{K}(K)/ NA_{K}(K) is finite if and only if d^{−1}ρ(K^{×})^{t}/\overline{ω(M)} is finite. (Note that the torsion parts of both groups are finite. For the first group, this follows from Mattuck’s Theorem, and the second group is a finitely generated ℤ_{p}-module.) So it remains to show that d^{−1}ρ(K^{×})^{t}/\overline{ω(M)} is finite iff M is σ-invertible.

Note that ω(M) is generated by (σ <m_{i}, m′_{1}>, …, σ <m_{i}, m′_{t}>) for i = 1, …, t. If d^{−1}ρ(K^{×})^{t}/\overline{ω(M)} is finite, then \overline{ω(M)} contains a ℚ_{p}-basis of ℚ_{p}^{t}. Hence the same holds for ω(M), which implies that M is σ-invertible. Conversely, assume that M is σ-invertible. Then ω(M), and hence \overline{ω(M)} contains a ℚ_{p}-basis of ℚ_{p}^{t}. Thus d^{−1}ρ(K^{×})^{t}/\overline{ω(M)} is a finitely generated torsion ℤ_{p}-module, hence finite. □

Actually, the proof of 6.1 shows a more general result. By Mattuck’s Theorem (or the existence of a logarithm), A_{K}(K)/NA_{K}(K) contains a subgroup U of finite index, which is isomorphic to a free ℤ_{p}-module. We define rk_{ℤ_{p}}A_{K}(K)/NA_{K}(K) to be the rank of this module. The properties of our map \overline{ω} show that if B_{K} has ordinary reduction, then

rk_{ℤ_{p}}A_{K}(K)/NA_{K}(K) = rk_{ℤ_{p}}(d^{−1}p^{s}ℤ_{p})^{t}/\overline{ω(M)} = t − rk((σ <m_{i}, m′_{j}>)_{i,j}).

Question 6.5 Is there a formula for the ℤ_{p}-rank of A_{K}(K)/NA_{K}(K) in terms of data given by the rigid analytic uniformization if B_{K} has arbitrary (good) reduction?

Certainly, in such a formula the ℤ_{p}-rank of B_{K}(K)/NB_{K}(K) should appear. Note that Schneider’s result [Sch2], Theorem 2 gives a formula for rk_{ℤ_{p}}B_{K}(K)/NB_{K}(K), which does not depend on the choice of a ramified ℤ_{p}-extension.

Proof of Theorem 6.2: Since Schneider’s ρ-splitting on P_{A_{K}×A′_{K}} (K) is uniquely determined by the fact that it vanishes on NP_{A_{K}×A′_{K}} (K), it suffices to show that our ρ-splitting τ vanishes on this universal norm group. Let a′ be a point in A′_{K}(K), and let x be an element of NP_{A_{K}×{a′}}(K). Fix a preimage e′ of a′ in E′_{K}(K), and put b′ = p′(e′). Note that for any intermediate extension K_{ν}/K the map

P_{B_{K}×{b′}}(K_{ν}) ×_{B_{K}(K_{ν})×{b′}} (E_{K}(K_{ν}) × {e′}) ⟶^{θ} P_{A_{K}×{a′}}(K_{ν})

is surjective. We abbreviate again N_{ν} = N_{K_{ν}/K}. For any ν, there exists some x_{ν} ∈ P_{A_{K}×{a′}}(K_{ν}) such that x = N_{ν}x_{ν}. Let (ω_{ν}, e_{ν}, e′) ∈ P_{B_{K}×{b′}}(K_{ν}) ×_{B_{K}(K_{ν})×{b′}} (E_{K}(K_{ν})×{e′}) be a preimage of x_{ν}. Then N_{ν}(ω_{ν}, e_{ν}, e′) ∈ P_{B_{K}×{b′}}(K)×_{B_{K}(K)×{b′}} (E_{K}(K) × {e′}) projects to x under θ. Hence

τ(x) = τ^{∗}(N_{ν}(ω_{ν}, e_{ν}, e′)) = σ(N_{ν}ω_{ν}) − (σ <N_{ν}e_{ν}, m′_{1}>, …, σ <N_{ν}e_{ν}, m′_{t}>)Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>).

We denote again by d respectively d′ the number of elements of B_{K}(K)/NB_{K}(K) respectively B′_{K}(K)/NB′_{K}(K). Now recall from the proof of 6.1 that σ maps N_{ν}P_{B_{K}×{b̃}}(K_{ν}) to d^{−1}d^{′−1}p^{ν+s}ℤ_{p} for all b̃ ∈ B′_{K}(K). If we denote by u ≤ 0 an integer such that the vector Σ ^{t}(σ <m_{1}, e′>, …, σ <m_{t}, e′>) (which does not depend on ν) is contained in (p^{u}ℤ_{p})^{t}, we find that τ(x) is contained in d^{−1}d^{′−1}p^{ν+s+u}ℤ_{p} for all ν. Hence τ(x) = 0. □

7 The canonical Mazur-Tate height in the ordinary reduction case

Let us put S_{n} = SpecR/M^{n+1}, where M is the maximal ideal in R, and let us indicate base changes over S = SpecR with S_{n} by subscripts n. We continue to assume that A_{K} has split semistable reduction and that B_{K} has ordinary reduction, and we use the notation of section 5. In particular, we write Z^{∧} for the completion of a S-scheme along the special fibre.

The rigid analytic uniformization map π : E_{K}^{an} → A_{K}^{an} induces a homomorphism of formal Néron models E^{∧} → A^{∧}, which is an isomorphism on the identity components by [Bo-Xa], 2.3. This induces compatible isomorphisms E_{n}^{0} ⟶^{∼} A_{n}^{0} for all n. In particular, the abelian part of A_{k}^{0} is isomorphic to B_{k}. Hence A_{K} has semistable ordinary reduction and n_{A} = n_{B}. The same reasoning applies to A′_{K}.

Let ρ : K^{×} → Y be a homomorphism to an abelian group Y which is uniquely divisible by m_{A}m_{A′}n_{A}n_{A′}. Then the canonical Mazur-Tate splittings in the ordinary case σ̃_{A} : P_{A_{K}×A′_{K}} (K) → Y respectively σ̃_{B} : P_{B_{K}×B′_{K}} (K) → Y exist.

Let ν : P_{A^{0}×A^{′0}} → A^{0} × A^{′0} and μ : P_{B×B′} → B × B′ be the natural projections. We will ususally write X^{Z} for the formal completion of a scheme X along a closed subscheme Z, with some exceptions: We denote by A^{0t}, A^{′0t} respectively P_{A^{0}×A^{′0}}^{t} the completions along T_{k}, T′_{k} respectively ν^{−1}(T_{k} × T′_{k}), and by B^{e}, B^{′e}, respectively P_{B×B′}^{e} the completion along the unit sections of the special fibre respectively along the preimage of the unit section of B_{k} × B′_{k} under μ. Similar conventions hold for (B_{n})^{e}, (B′_{n})^{e} and (P_{B×B′})_{n}^{e}.

The isomorphisms E_{n}^{0} ⟶^{∼} A_{n}^{0} provide all A_{n}^{0} with the structure of an extension of B_{n} by T_{n}^{0} in a compatible way. T_{n}^{0} is (up to canonical isomorphism) the uniquely determined torus lifting T_{k}^{0} and T_{n}^{0} ↪ A_{n}^{0} is the unique lift of T_{k}^{0} ↪ A_{k}^{0}. Let p_{n} : A_{n}^{0} → B_{n} be the projection map. We can deduce from [SGA7, I], IX, 7.5 that there exists a compatible system of isomorphisms

(p_{n} × p′_{n})^{∗}(P_{B×B′})_{n} ⟶^{∼} (P_{A^{0}×A^{′0}})_{n}.

We have a natural commutative diagram of formal biextensions

(P_{B×B′})_{n}^{e} ×_{B_{n}^{e}×_{Sn} B_{n}^{′e}} (A^{0T_{n}^{0}} ×_{S_{n}} A^{′0T_{n}^{′0}}) ⟶ (P_{A^{0}×A^{′0}})_{n}^{ν_{n}^{−1}(T_{n}^{0}×T_{n}^{′0})}
↓  ↓
(P_{B×B′})_{n} ×_{B_{n}×_{Sn} B′_{n}} (A_{n}^{0} ×_{S_{n}} A_{n}^{′0}) ⟶  (P_{A^{0}×A^{′0}})_{n}

Passing to the limit, we find a commutative diagram of formal biextensions

(P_{B×B′})^{e} ×_{B^{e}×_{S^{∧}} B^{′e}} (A^{0t} ×_{S^{∧}} A^{′0t}) ⟶ (P_{A^{0}×A^{′0}})^{t}
↓  ↓
(P_{B×B′})^{∧} ×_{B^{∧}×_{S^{∧}} B^{′∧}} (A^{0∧} ×_{S^{∧}} A^{′0∧}) ⟶_{ξ}^{∼} (P_{A^{0}×A^{′0}})^{∧}

Hence the (uniquely determined) splitting (P_{B×B′})^{e} → 𝔾_{m,R}^{∧} induces the uniquely determined splitting of the biextension (P_{A^{0}×A^{′0}})^{t}. This implies that the relation between σ̃_{A} and σ̃_{B} is the following:

Lemma 7.1 For x ∈ P_{A_{K}×A′_{K}} (K) we denote by x^{(m_{A},m_{A′})} the point we get by applying to x the m_{A}-th power map with respect to the group structure over A′_{K} and the m_{A′}- th power map with respect to the group structure over A_{K}. Let α ∈ K^{×} and y ∈ P_{A^{0}×A^{′0}}(R) be such that x^{(m_{A},m_{A′})} = αy. Let ω be the projection to P_{B×B′}(R) of ξ^{−1}(y). Then

σ̃_{A}(x) = \frac{1}{m_{A}m_{A′}} (ρ(α) + σ̃_{B}(ω)).

Proof: Both sides are ρ-splittings which are equal to ρ ∘ σ_{0} on (P_{A^{0}×A^{′0}})^{t}(R), where σ_{0} : P_{A^{0}×A^{′0}}^{t} → 𝔾_{m,R}^{∧} is the unique splitting. □

Note that ξ induces a homomorphism of biextensions

ξ^{rig} : (P_{B×B′})^{∧rig} ×_{B^{∧rig}×B^{′∧rig}} (E^{0∧rig} × E^{′0∧rig}) → (P_{A^{0}×A^{′0}})^{∧rig} ↪ P_{A_{K}×A′_{K}}^{an} .

Hence ξ^{rig} differs from the restriction of θ to (P_{B×B′})^{∧rig} ×_{B^{∧rig}×B^{′∧rig}} (E^{0∧rig} × E^{′0∧rig}) ↪ P_{B_{K}×B′_{K}}^{an} ×_{B_{K}^{an}×B_{K}^{′an}} E_{K}^{an} × E_{K}^{′an} by a bilinear map E^{0∧rig} × E^{′0∧rig} → 𝔾_{m,K}^{an}, which must be equal to one. We find that ξ^{rig} is equal to the restriction of θ. Hence for y ∈ P_{A^{0}×A^{′0}}(R) the point ξ^{−1}(y) is just the unique preimage of y under θ which lies in P_{B×B′}(R) ×_{B(R)×B′(R)} (E^{0}(R) × E^{′0}(R)) (after identifying E^{0}(R) respectively E^{′0}(R) with A^{0}(R) respectively A^{′0}(R)).

If σ̃_{B} is M-invertible, and we use it to construct our ρ-splitting τ, then we can calculate the difference between σ̃_{A} and τ using 7.1. We apply this to compare Schneider’s p-adic height pairing to the one defined by Mazur and Tate in the semistable ordinary reduction case.

Theorem 7.2 Let ρ : K^{×} → ℚ_{p} be a non-trivial, continuous homomorphism, and assume that B_{K} has ordinary reduction and that NA_{K}(K) has finite index in A_{K}(K). Let σ_{ρ,A} respectively σ_{ρ,B} denote the canonical Schneider ρ-splittings on A_{K} respectively B_{K}. For x ∈ P_{A_{K}×A′_{K}} (K) projecting to (a, a′) ∈ A_{K}(K) × A′_{K}(K) let e and e′ be the uniquely determined preimages of a^{m_{A}} respectively a^{′m_{A′}} in E^{0}(R) respectively E^{′0}(R). Then

σ̃_{A}(x) = σ_{ρ,A}(x)+

\frac{1}{m_{A}m_{A′}} (σ_{ρ,B} <e, m′_{1}>, …, σ_{ρ,B} <e, m′_{t}>) Σ ^{t}(σ_{ρ,B} <m_{1}, e′>, …, σ_{ρ,B} <m_{t}, e′>).

Proof: Recall that σ̃_{B} is equal to σ_{ρ,B} since B_{K} has good ordinary reduction, and that σ_{ρ,A} is equal to τ by 6.2. Then our claim follows from 7.1 and the definition of τ. □

Note that in [MTT], p.34, a comparison formula between these two p-adic height pairings is stated for Tate curves. Let us apply 7.2 to a Tate curve A_{K} over K with Tate parameter q ∈ K^{×}, i.e. E_{K} = 𝔾_{m,K} and M =< q >⊂ K^{×}. We identify E′_{K} with 𝔾_{m,K} via the character q. Then Theorem 7.2 boils down to

Corollary 7.3 Assume that ρ(q) ≠ 0. For x ∈ P_{A_{K}×A′_{K}} (K) projecting to (a, a′) in A_{K}(K) × A′_{K}(K) let (e, e′) be the uniquely defined preimage of (a^{m_{A}}, a^{′m_{A′}}) in R^{×} × R^{×} ⊂ E_{K}(K) × E′_{K}(K). Then

σ̃_{A}(x) = σ_{ρ,A}(x) + \frac{1}{m_{A}m_{A′}} (\frac{ρ(e)ρ(e′)}{ρ(q)}) .

Hence our formula differs from the one in [MTT] by the factors ord_{v}(q_{v}) appearing in the denominators of their correction term. The author consulted the authors of [MTT] about this discrepancy who agreed that the result in [MTT] needs a correction.

Note that a similar formula (without factors ord_{v}(q_{v})) describes the relation between Schneider’s height and Nekovar’s canonical height on Tate curves, see [Ne], 7.14. It seems very probable that Nekovar’s height coincides with the canonical Mazur-Tate height for abelian varieties with semistable ordinary reduction, cf. [Ne], 8.2.

References

[BGR] Bosch, S., Güntzer, U., Remmert, R. : Non-Archimedean analysis, Springer 1984.

[Bo-Lü1] Bosch, S., Lütkebohmert, W.: Degenerating abelian varieties, Topology 30 (1991) 653-698.

[Bo-Lü2] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry I, Math. Ann. 295 (1993) 291-317.

[BLR] Bosch, S., Lütkebohmert, W., Raynaud, A.: Néron models (Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 21) Springer 1990.

[Bo-Sch] Bosch, S., Schlöter, K.: Néron models in the setting of formal and rigid geometry, Math. Ann. 301 (1995) 339-362.

[Bo-Xa] Bosch, S., Xarles, X.: Component groups of Néron models via rigid uniformization, Math. Ann. 306 (1996) 459-486.

[Co] Coleman, R.: The monodromy pairing, Preprint 1996.

[EGA IV] Grothendieck, A.: Éléments de géométrie algébrique IV, Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966), 32 (1967).

[Hi] Hindry, M.: Sur les hauteurs locales de Néron sur les variétés abéliennes, Prépublications Mathématiques de l’U.R.A. 212 “Théories Géométriques” no. 51, Université Paris 7 1993.

[Kö] Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schriftenreihe Math. Inst. Univ. Münster, 2. Serie, Heft 7.

[La] Lang, S.: Algebraic groups over finite fields, Amer. J. Math. 78 (1956) 555-563.

[Ma] Mazur, B.: Rational points of Abelian varieties with values in towers of number fields, Invent. Math. 18 (1972) 183-266.

[Ma-Ta] Mazur, B., Tate, J.: Canonical height pairings via biextensions. In: Arithmetic and Geometry, Papers dedicated to I.R. Shafarevich, Progress in Mathematics 35, Birkhäuser 1983 195-235.

[MTT] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986) 1-48.

[Ne] Nekovar, J.: On p-adic height pairings. In: Séminaire de Théorie des Nombres Paris 1990-91, Progress in Mathematics 108, Birkhäuser 1993 127-202.

[Ray] Raynaud, M.: Variétés abéliennes et géométrie rigide, Actes, Congrès intern. math. 1970, Tome 1 473-477.

[Sch1] Schneider, P.: P -adic height pairings I, Invent. Math. 69 (1982) 401-409.

[Sch2] Schneider, P.: Arithmetic of formal groups and applications I: Universal norm subgroups, Invent. Math. 87 (1987) 587-602.

[SGA7, I] Grothendieck, A. et al.: Séminaire de géométrie algébrique du Bois-Marie 1967/69, Groupes de monodromie en géométrie algébrique I, Lecture Notes in Mathematics 288, Springer 1972.

[We] Werner, A.: Local heights on Abelian varieties with split multiplicative reduction, Comp. Math. 107 (1997) 291 - 320.

Annette Werner

Mathematisches Institut Universität Münster Einsteinstr. 62 D-48149 Münster werner@math.uni-muenster.de 320

Documenta Mathematica 3 (1998)