Component Groups of Purely Toric Quotients

William A. Stein  Brian Conrad

April 21, 2003

Abstract Suppose π : J → A is an optimal quotient of abelian varieties over a p-adic field, optimal in the sense that ker(π) is connected. Assume that J is equipped with a symmetric principal polarization θ (e.g., any Jacobian of a curve has such a polarization), that J has semistable reduction, and that A has purely toric reduction. In this paper, we express the group of connected components of the Néron model of A in terms of the monodromy pairing on the character group of the torus associated to J. We apply our results in the case when A is an optimal quotient of the modular Jacobian J_{0}(N). For each prime p that exactly divides N, we obtain an algorithm to compute the component group of A at p.

1 Introduction

Let A be an abelian variety over the rational numbers Q. Birch and Swinnerton-Dyer found a conjectural formula for the order of the Shafarevich-Tate group of A. The Tamagawa numbers c_{p} of A are among the quantities that appear in this formula. We now recall the definition of the Tamagawa numbers of an abelian variety (the definition of Néron model and component groups is given in Section 2).

Definition 1.1 (Tamagawa number). Let p be a prime, let A be the Néron model of A over the p-adic integers Z_{p}, and let Φ_{A,p} be the component group of A at p. Then the Tamagawa number c_{p} of A at p is the order of the subgroup Φ_{A,p}(F_{p}) of F_{p}-rational points in Φ_{A,p}(\overline{F}_{p}).

Remark 1.2. The Tamagawa number is defined in a different way in some other papers, but the definitions are equivalent.

When A has dimension one, A is called an elliptic curve, and A can be defined by a Weierstrass equation y^{2} = x^{3} + ax + b. Using that elliptic curves (and their related integral models) can be described by simple equations, Tate found an efficient algorithm to compute all of the Tamagawa numbers of A (see [18]). In the case when A is the Jacobian of a genus 2 curve, [7] discusses a method for computing the Tamagawa numbers of A. In this paper, we consider the situation in which A has purely toric reduction at p, with no constraint on the dimension of A. For such A we give an explicit description of the order of the group of connected components of the closed fiber of the Néron model of A. In the case when A = A_{f} is a quotient of J_{0}(N) attached to a newform f ∈ S_{2}(Γ_{0}(N)) and p || N, our method is completely explicit, and yields an algorithm to compute the Tamagawa number c_{p} of A (up to a bounded power of 2).

This paper is structured as follows. In Sections 2–6 we state and prove an explicit formula involving component groups of fairly general abelian varieties. Then in Section 7 we turn to quotients of modular Jacobians J_{0}(N). We give some tables and discussed the arithmetic of quotients of J_{0}(N) when N is prime. In Section 8 we prove a couple of facts about toric reduction that are used in the proof of Theorem 6.1.

Acknowledgement: This paper was inspired by lectures of R. Coleman and K. Ribet, and a letter from Ribet to Mestre (see [17]), which contains some of the results of the present paper in the case when A has dimension 1. The second author would like to thank A. Abbes, A. Agashe, D. Kohel, and D. Lorenzini, for helpful conversations. Both authors were partially supported by the NSF and the Clay Mathematics Institute during work on this paper.

2 The Main Results

In this section, we summarize the main contributions of this paper. First we recall the precise definition of the component group of an abelian variety, then we state our main theorem.

Let R be a discrete valuation ring with field of fractions K and maximal ideal 𝔪, and let k = R/𝔪 be the residue class field. Let A be an abelian variety over K.

Definition 2.1 (Néron model). A Néron model of A is a smooth commutative group scheme A over R such that A is its generic fiber and A satisfies the Néron mapping property: the restriction map

Hom_{R}(S, A) ⟶ Hom_{K}(S_{K}, A)

is bijective for all smooth schemes S over R.

The Néron mapping property implies that A is unique up to a unique isomorphism, so we will refer without hesitation to “the” Néron model of A. Néron models are separated and of finite type as opposed to just locally of finite type, even though their universal property is on the category of arbitrary smooth R-schemes. For more about Néron models see [2].

The closed fiber A_{k} of A is a group scheme over k, which need not be connected. Denote by A_{k}^{0} the connected component of A_{k} that contains the identity. We have an exact sequence

0 ⟶ A_{k}^{0} ⟶ A_{k} ⟶ Φ_{A} ⟶ 0,

where Φ_{A} is a finite étale group scheme over k. Equivalently, Φ_{A} is a commutative finite group equipped with a continuous action of Gal(\overline{k}/k).

Definition 2.2 (Component group). The component group of an abelian variety A over K is the group scheme Φ_{A} = A_{k}/A_{k}^{0}.

2.1 Statement of the Theorem

We now state our main result, supressing some of the definitions of the terms used until later (see Section 6 below for a more complete statement and the proof). Let K be as above, and suppose π : J → A is an optimal quotient. Assume that J is equipped with a symmetric principal polarization λ, in the sense of Definition 5.1. For example, the θ polarization of the Jacobian of a curve is a symmetric principal polarization. Also assume that J has semistable reduction, and that A has purely toric reduction.

We express the component group of A in terms of the monodromy pairing associated to J. Let m_{A} = \sqrt{deg(θ_{A})}, where θ_{A} : A^{∨} → A is induced by the canonical principal polarization of J arising from the θ-divisor. Let X_{J} be the character group of the toric part of the closed fiber of the Néron model of J. Let L be the saturation of the image of X_{A} in X_{J}. The monodromy pairing induces a map α : X_{J} → Hom(L, Z). Let Φ_{X} be the cokernel of α and m_{X} = [α(X_{J}) : α(L)] be the order of the finite group α(X_{J})/α(L). The main result of this paper is that

\frac{#Φ_{A}}{m_{A}} = \frac{#Φ_{X}}{m_{X}} ,

and this is recorded as Theorem 6.1 below.

Using the snake lemma, one sees that Φ_{X} is isomorphic to the image of the natural map Φ_{J} → Φ_{A}, and the above formula implies that the cokernel of the map Φ_{J} → Φ_{A} has order m_{A}/m_{X}. A non-obvious consequence of this is that m_{X} | m_{A}.

In the context of modular forms, if the optimal quotient J → A arises from a newform on Γ_{0}(N), then the quantities m_{A}, m_{X} and Φ_{X} can be explicitly computed, hence we can compute #Φ_{A}. Note that the authors have not computed the structure of Φ_{A} as a group.

3 Optimal Quotients

Let K be as in Section 2, let J be an abelian variety equipped with a symmetric principal polarization θ_{J} (see Definition 5.1). For example, J could be the Jacobian of a curve equipped with the canonical principal polarization arising from the θ-divisor.

Definition 3.1 (Optimal quotient). An optimal quotient of J is an abelian variety A and a smooth surjective morphism π : J → A whose kernel is connected (i.e., an abelian variety).

Remark 3.2. Any connected scheme of finite type over a field is geometrically connected if it contains a rational point (e.g., if it is a group scheme). See [8, IV_{2}, §4.5.13].

Let π : J → A be an optimal quotient. Denote by J^{∨} and A^{∨} the abelian varieties dual to J and A, respectively. Upon composing the dual of π with θ_{J}^{∨} = θ_{J}, we obtain a map

A^{∨} ⟶^{π^{∨}} J^{∨} ⟶^{θ_{J}} J.

Proposition 3.3. The map θ_{J} ∘ π^{∨}: A^{∨} → J is a closed immersion.

Proof. Since θ_{J} is an isomorphism, we want to prove that π^{∨} is a closed immersion. It is a general fact that duals to surjections of abelian varieties with abelian variety kernel are closed immersions, but for lack of an adequate reference we recall the proof. Since a monomorphism between smooth finite type group schemes over a field is necessarily a closed immersion, it suffices to show that the commutative proper group scheme ker(π^{∨}) vanishes. Since a non-zero commutative proper group scheme G over a field F necessarily has a non-zero finite subgroup scheme G[n] for some n (since either (G_{/\overline{F}})_{red}^{0} is an abelian variety or else G is finite and non-zero), it suffices to show that ker(π^{∨})[n] vanishes for all positive integers n. In other words, it suffices to show that the induced map A^{∨}[n] → J^{∨}[n] is a closed immersion for all n.

Since Cartier duality interchanges faithfully flat maps and closed immersions, and the scheme-theoretic Weil pairing identifies the Cartier dual of the map induced by π^{∨} on ntorsion with π : J[n] → A[n], we just have to show that these latter maps are faithfully flat for all integers n. Using the short exact sequence

0 → ker(π) → J → A → 0

in the abelian category of fppf abelian sheaves over Spec(K), the snake lemma gives an exact sequence

0 → ker(π)[n] → J[n] → A[n] → 0

because n : ker(π) → ker(π) is a faithfully flat map (hence fppf surjective), as ker(π) is an abelian variety. This gives an isomorphism of group schemes

J[n]/ ker(π)[n] ≃ A[n]

compatible with the maps from J[n], whence J[n] → A[n] is faithfully flat.

Henceforth we will abuse notation and denote the injection A^{∨} → J by π^{∨}. We define θ_{A} to be the composite π ∘ π^{∨}, so the kernel of θ_{A} equals the scheme-theoretic intersection of A^{∨} and B = ker(π), as depicted in the following diagram:

\xymatrix{A^{∨} ∩ B \ar[r] \ar[d] & B \ar[d] \\ A^{∨} \ar@{^{(}->}[r]^{π^{∨}} \ar[dr]_{θ_{A}} & J \ar[d]^{π} \\  & A.}

Since θ_{A} is a polarization (due to how its definition uses the polarization θ_{J}) the degree of θ_{A} is a perfect square (see [16, §16, p. 150]).

Definition 3.4 (Degree). Define the degree of A as a quotient of J to be the integer

m_{A} = \sqrt{# ker(θ_{A})}.

4 The Closed Fiber of the Néron Model

In this section we recall some terminology associated with closed fibers of Néron models. Let K, R, and k be as in Section 2, and let Φ_{A} = A_{k}/A_{k}^{0} be the group scheme of connected components of the closed fiber A_{k}. By Chevalley’s structure theorem (see [3], or [4] for a modern account), if K is a perfect extension field of k (e.g., K = \overline{k}) then there is a unique short exact sequence

0 → C → A_{K}^{0} → B → 0

with C a smooth affine algebraic K-group and B an abelian variety. Moreover, there is a unique exact sequence

0 → T → C → U → 0

with T a torus and U unipotent.

Using the rigidity of tori, one can show that T is induced by a unique torus in A_{k}^{0}. In particular, the condition that B = U = 0 is equivalent to the condition that A_{k}^{0} be a torus, and the condition that U = 0 is equivalent to the condition that A_{k}^{0} be the extension of an abelian variety by a torus (i.e., be a semi-abelian variety). These conditions can be checked on a geometric closed fiber.

Definition 4.1. The abelian variety A is said to have purely toric reduction if A_{k}^{0} is torus, and to have semistable reduction if A_{k}^{0} is a semi-abelian variety (i.e., A0 ; \overline{k} has vanishing unipotent part).

4.1 The Monodromy Pairing on the Character Group

Definition 4.2 (Character group of torus). The character group

X_{A} = Hom\overline{k}(T\overline{k}, G_{m}\overline{k})

is a free abelian group of rank t contravariantly associated to A.

As discussed in [9], if A is semistable there is a monodromy pairing X_{A} × X_{A^{∨}} → Z and an exact sequence

0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A} → 0.

Also, the canonical isomorphism (A^{∨})^{∨} ≅ A induces an isomorphism

X_{A^{∨}} × X_{(A^{∨})^{∨}} ≅ X_{A} × X_{A^{∨}},

which identifies the monodromy pairing associated to A^{∨} with that associated to A. Example 4.3 (Tate curve). Suppose E = G_{m}/q^{Z} is a Tate curve over Q_{p}^{ur}. The monodromy pairing on X_{E} = q^{Z} is

⟨q, q⟩ = ord_{p}(q) = − ord_{p}(j).

Thus Φ_{E} is cyclic of order − ord_{p}(j).

Suppose J is an abelian variety equipped with a symmetric principal polarization. Since J is self dual via the given symmetric principal polarization, we can view the monodromy pairing on J as a pairing X_{J} × X_{J} → Z. Because the principal polarization on J is symmetric the resulting pairing X_{J} × X_{J} → Z is symmetric, so there is no ambiguity about left versus right definitions of X_{J} → Hom(X_{J}, Z). The above exact sequence then becomes

0 → X_{J} → Hom(X_{J}, Z) → Φ_{J} → 0.

5 The Degree of a Symmetric Isogeny

We next relate the degree of the isogeny A^{∨} → A defined at the end of Section 3 to the order of the cokernel of the induced map on the character groups of tori defined in Section 4.1. Let K be as in Section 2, and let A be an abelian variety over K.

Definition 5.1 (Symmetric isogeny). A symmetric isogeny φ : A^{∨} → A is an isogeny such that the map

φ^{∨} : A^{∨} → (A^{∨})^{∨} = A

is equal to φ.

If J and A are as in Section 3 then the principal polarization θ_{J} of J is symmetric, so the natural map A^{∨} → A is a symmetric isogeny.

Lemma 5.2. Suppose that A is a purely toric abelian variety over K and that φ : A^{∨} → A is a symmetric isogeny. Let φ_{a} : X_{A} → X_{A^{∨}} denote the induced map on character groups. Then

deg(φ) = # coker(φ_{a})^{2}.

Proof. By Corollary 8.7 applied to our isogeny φ (so what we are presently calling A^{∨} and A are respectively called A and B in the discussion surrounding Theorem 8.6), we deduce that

deg(φ) = # ker(φ) = # ker(φ_{t}) · # ker(φ_{t}^{∨})

where φ_{t} and φ_{t}^{∨} are the maps induced by φ and φ^{∨} on closed fiber tori.

Since the character group X_{A} is, by definition, Hom\overline{k}(T\overline{k}, G_{m}\overline{k}), where T is the toric part of the closed fiber of A, it follows that # ker(φ_{t}) = # coker(φ_{a}). Since φ = φ^{∨}, this proves the lemma.

6 Statement and Proof of the Main Theorem

Let K be as in Section 2, and let π : J → A be an optimal quotient. Assume that J is equipped with a symmetric principal polarization λ, that J has semistable reduction, and that A has purely toric reduction. Let X_{A}, X_{A^{∨}}, and X_{J} denote the character groups of the toric parts of the closed fibers of the abelian varieties A, A^{∨}, and J, respectively.

Let π : J → A be an optimal quotient, and let θ : A^{∨} → A denote the induced polarization. Let π_{∗}, π^{∗}, θ_{∗}, and θ^{∗} be the maps induced on character groups by the various functorialities, as indicated in the following two key diagrams:

\xymatrix{A^{∨} \ar@{^{(}->}[r]^{π^{∨}} \ar[dr]_{θ} & J \ar@{->>}[d]^{π} & X_{A} \ar@{^{(}->}[r]^{π^{∗}} \ar[dr]^{θ^{∗}} & X_{J} \ar@{->>}[d]^{π_{∗}} \\  & A &  & X_{A^{∨}}. \ar[ul]^{θ_{∗}}}

The surjectivity of π_{∗} is proved in Theorem 8.2. The injectivity of π^{∗} follows because

θ_{∗}π_{∗}π^{∗} = θ_{∗}θ^{∗} = deg(θ) ≠ 0,

and multiplication by a nonzero integer on a free abelian group is injective.

Let L be the saturation of π^{∗}X_{A} in X_{J}; thus π^{∗}X_{A} is a finite-index subgroup of L and the quotient X_{J}/L is torsion free. Let

α : X_{J} → Hom(π^{∗}X_{A}, Z)

be the map defined by the monodromy pairing restricted to X_{J} × π^{∗}X_{A}. For L of finite index in L, define the degree of L to be

m_{L} = [α(X_{J}) : α(L)],

and the component group of L to be

Φ_{L} = coker(X_{J} → Hom(L, Z)).

When L = L and A is fixed, for simplicity we write m_{X} = m_{L} and Φ_{X} = Φ_{L}.

Recall that Φ_{A} is the component group of A and m_{A} is the square root of the degree of the induced map A^{∨} → A.

Theorem 6.1. For any subgroup L of finite index in L, the following relation holds:

\frac{#Φ_{A}}{m_{A}} = \frac{#Φ_{L}}{m_{L}} .

6.1 Proof of the Main Theorem

The notation in this section is as in previous section.

Lemma 6.2. Let π_{∗} : X_{J} → X_{A^{∨}} and α : X_{J} → Hom(π^{∗}X_{A}, Z) be as in previous section. Then

ker(π_{∗}) = ker(α).

Proof. Suppose x ∈ ker(π_{∗}), and let y = π^{∗}z with z ∈ X_{A}. Then

⟨x, y⟩ = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩ = 0,

so x ∈ ker(α). Next let x ∈ ker(α). Then for all z ∈ X_{A},

0 = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩,

so π_{∗}x is in the kernel of the monodromy map

X_{A^{∨}} → Hom(X_{A}, Z).

Since X_{A^{∨}} and Hom(X_{A}, Z) are free of the same finite rank and the cokernel is torsion, the monodromy map is injective. Thus π_{∗}x = 0 and x ∈ ker(π_{∗}).

Let π^{∗} : X_{A} → X_{J} be as in previous section.

Lemma 6.3. The monodromy-pairing map X_{J} → Hom(X_{J}, Z) composed with restriction Hom(X_{J}, Z) → Hom(π^{∗}X_{A}, Z) gives rise to an exact sequence

X_{J} → Hom(π^{∗}X_{A}, Z) → Φ_{A} → 0.

Proof. Lemma 6.2 gives the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & X_{J}/ ker(α) \ar[d]^{≅} \ar[r] & Hom(π^{∗}X_{A}, Z) \ar[r] \ar[d]^{≅} & coker(α) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & X_{A^{∨}} \ar[r] & Hom(X_{A}, Z) \ar[r] & Φ_{A} \ar[r] & 0.}

By Lemma 6.2, the first vertical map is an isomorphism. The second is an isomorphism because it is induced by the isomorphism π^{∗} : X_{A} → π^{∗}X_{A}. It follows that coker(α) ≅ Φ_{A}, as claimed.

Recall that L denotes the saturation of π^{∗}X_{A} in X_{J}, and that L ⊂ L denotes a subgroup of finite index.

Lemma 6.4. The rational number \frac{#Φ_{L}}{m_{L}} is independent of the choice of L.

Proof. Suppose L′ is another finite index subgroup of L, and let n = [L : L′]. Here n is a rational number, the lattice index of L′ in L. Since α is injective when restricted to L, it follows that

m_{L′} = [α(X_{J}) : α(L′)] = [α(X_{J}) : α(L)] · [α(L) : α(L′)] = m_{L} · n.

Similarly, #Φ_{L′} = #Φ_{L} · n.

Recall that m_{A} = \sqrt{deg(θ)} and

Φ_{A} ≅ coker(X_{A^{∨}} → Hom(X_{A}, Z)),

where m_{A} is the degree of A and Φ_{A} is the component group of A.

Proof of Theorem 6.1. By Lemma 6.4 we may assume that L = π^{∗}X_{A}. With this choice of L, Lemma 6.3 asserts that Φ_{L} ≅ Φ_{A}. By Lemma 6.2, properties of the index, and Lemma 5.2 we have

m_{L} = [α(X_{J}) : α(L)]
= [π_{∗}(X_{J}) : π_{∗}(L)]
= [X_{A^{∨}} : π_{∗}(π^{∗}X_{A})]
= [X_{A^{∨}} : θ^{∗}X_{A}]
= # coker(θ^{∗})
= \sqrt{deg(θ)} = m_{A}.

Recall that Φ_{L} denotes the cokernel of the natural map X_{J} → Hom(L, Z) induced by composing the monodromy map X_{J} → Hom(X_{J}, Z) with the natural restriction map Hom(X_{J}, Z) → Hom(L, Z).

Proposition 6.5. The group Φ_{L} is canonically isomorphic to the image of the map from Φ_{J} to Φ_{A} induced by π : J → A. Thus

image(Φ_{J} → Φ_{A}) ≅ Φ_{L}.

Proof. Since π^{∗}X_{A} ⊂ L ⊂ X_{J}, an application of Lemma 6.3 gives the following commutative diagram with exact rows:

[figure: X_{J}; Hom(X_{J}, Z); Φ_{J}; 0; X_{J}; Hom(L, Z); Φ_{L}; 0; X_{J}; Hom(π^{∗}X_{A}, Z); Φ_{A}; 0.]

The map Hom(L, Z) → Hom(π^{∗}X_{A}, Z) is an isomorphism, so the map Φ_{L} → Φ_{A} is injective. Thus

image(Φ_{J} → Φ_{A}) ≅ image(Φ_{J} → Φ_{L}).

The cokernel of Hom(X_{J}, Z) → Hom(L, Z) surjects onto the cokernel of Φ_{J} → Φ_{L}. Using the exact sequence

0 → L → X_{J} → X_{J}/L → 0,

we find that

coker(Hom(X_{J}, Z) → Hom(L, Z)) ⊂ Ext^{1}(X_{J}/L, Z).

Because L is saturated, the quotient X_{J}/L is torsion free, so the indicated Ext^{1} group vanishes. Thus the map Φ_{J} → Φ_{L} is surjective, from which the proposition follows.

Corollary 6.6. The cokernel of the map from Φ_{J} to Φ_{A} induced by π : J → A has order m_{A}/m_{L}. Thus

# coker(Φ_{J} → Φ_{A}) = \frac{m_{A}}{m_{L}} .

Proof. Combine Theorem 6.1 and Proposition 6.5.

7 Optimal Quotients of J_{0}(N)

In this section we specialize the general results of the rest of this paper to the concrete case in which J = J_{0}(N) is the Jacobian of a modular curve, and A = A_{f} is an optimal quotient of J attached to a modular forms. The paper [12] contains more computations like these.

7.1 Modular Curves and Semistable Reduction

Let X_{0}(N) be the modular curve associated to the subgroup Γ_{0}(N) of SL_{2}(Z) that consists of those matrices which are upper triangular modulo N. The algebraic curve X_{0}(N)_{C} can be constructed as a Riemann surface as the quotient

Γ_{0}(N)\ ({z : z ∈ C, Im(z) > 0} ∪ P^{1}(Q)) ,

and X_{0}(N) has a canonical structure of algebraic curve over Q.

It is well known that the p-new part of the Jacobian J_{0}(N) of X_{0}(N) has purely toric reduction at p when p || N. Let us briefly recall the reason, writing N = Mp. Using the description of closed fibers of modular curves [10, Ch. 13] and Raynaud’s result relating Néron models and Picard functors (as summarized in [2, Ch. 9]), the standard finite flat degeneracy maps X_{0}(Mp) → X_{0}(M) over Z_{(p)} induce a “pushfoward” map on Néron model connected components

Pic_{X_{0}(Mp)/Z_{(p)}}^{0} ⟶ Pic_{X_{0}(M)/Z_{(p)}}^{0} × Pic_{X_{0}(M)/Z_{(p)}}^{0}

which on the closed fiber is the map induced by pullback to the two components X_{0}(M)_{/F_{p}} in X_{0}(Mp)_{/F_{p}}. The kernel of this latter map is a torus [2, Ex. 9.2.8], yet this kernel is visibly isogenous to the semistable mod p fiber of the dual of J_{0}(Mp)^{new}, whence the purely toric conclusion.

7.2 Newforms and Optimal Quotients

The Hecke algebra

T = Z[… T_{n} …] ⊂ End(J_{0}(N))

is a commutative ring of endomorphisms of J_{0}(N) of Z-rank equal to the dimension of J_{0}(N). The character group X_{J,p} of J_{0}(N) at p is equipped with a functorial action of T. The Hecke algebra T also acts on the complex vector space S = S_{2}(Γ_{0}(N), C) of cusp forms.

Let f be a newform, and associate to f the ideal I_{f} of the Hecke algebra T of elements which annihilate f. Then O_{f} = T/I_{f} is an order in the ring of integers of the totally real number field K_{f} obtained by adjoining the Fourier coefficients of f to Q. The quotient

A_{f} = J_{0}(N)/I_{f}J_{0}(N)

is an optimal quotient of J_{0}(N) of dimension equal to [K_{f} : Q]. As discussed in the previous section, A_{f} is purely toric at p.

7.3 Tamagawa Numbers

Let Frob_{p} : X_{J} → X_{J} denote the map induced by the Frobenius automorphism. We have Frob_{p} = −W_{p}, where W_{p} is the map induced by the Atkin-Lehner involution on J_{0}(p). Let f be a newform, A = A_{f} the corresponding optimal quotient, and w_{p} the sign of the eigenvalue of W_{p} on f.

Proposition 7.1.

Φ_{A}(F_{p}) = \begin{cases}Φ_{A}(\overline{F}_{p}) & if w_{p} = −1, \\ Φ_{A}(\overline{F}_{p})[2] & if w_{p} = 1. \end{cases}

Proof. If w_{p} = −1, then Frob_{p} = 1 and the Gal(\overline{F}_{p}/F_{p})-action of Φ_{A}(\overline{F}_{p}) is trivial. In this case Φ(F_{p}) = Φ(\overline{F}_{p}). Next suppose w_{p} = 1. Recall that we have an exact sequence

0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A} → 0.

Since W_{p} acts as +1 on f, it also acts as +1 on each of the modules A, X_{A}, Hom(X_{A}, Z), and Φ_{A}. Thus Frob_{p} = −W_{p} acts as −1 on Φ_{A}. Since the subgroup of 2-torsion elements of a finite abelian group equals the subgroup of elements fixed under −1, it follows that Φ_{A}(F_{p}) = Φ_{A}(\overline{F}_{p})[2].

Warning: When extending this result to the whole of J_{0}(N), be careful. The action of Frob_{p} = T_{p} need not be by ±1, even though it must be by an involution of order 2. For example, the component group of J_{0}(65) at 5 is cyclic of order 42. The action of Frob_{5} is by multiplication by −13. Note that (−13)^{2} = 1 (mod 42). The fixed points of multiplication by −13 is the order 14 subgroup of Z/42Z.

7.4 Computing Component Groups

Using modular symbols, we can enumerate the optimal quotients A_{f} of J_{0}(N) (see, e.g., [1]) and compute the degree m_{A} (see [12, §3.1]). Suppose p is a prime that exactly divides N. As explained in [12], the method of graphs (see [14]) or the ideal theory of quaternion algebras (see [11]) can be used to compute X = X_{J_{0}(N),p} with its T-action and the monodromy pairing. We can then compute the following three modules:

1. the saturated submodule L = ⋂_{t∈I_{f}} ker(t) of X,

2. the character group degree m_{X} = m_{L}, and

3. Φ_{X} = Φ_{L}.

By Theorem 6.1 we obtain

#Φ_{A,p} = #Φ_{X} · \frac{m_{A}}{m_{X}} .

7.5 The Eisenstein Nature of Component Groups

The theorem below, which generalizes some of the results of [13] and [15], was conjectured by the second author after computing many component groups of quotients of J_{0}(p) using the results of this paper. M. Emerton read an early version of this paper and subsequently announced a proof of the theorem below (see [6]).

Theorem 7.2 (Emerton). Let p be a prime and let f_{1}, …, f_{n} be a set of representatives for the Galois-conjugacy classes of newforms in S_{2}(Γ_{0}(p)). Let A_{1}, …, A_{n} be the optimal quotients associated to f_{1}, …, f_{n}, respectively. Then for each i, i = 1, …, n, we have

#A_{i}(Q)_{tor} = #Φ_{A_{i}}(\overline{F}_{p}) = #Φ_{A_{i}}(F_{p}).

Furthermore,

#Φ_{J_{0}(p)}(\overline{F}_{p}) = ∏_{i=1}^{n} #Φ_{A_{i}}(\overline{F}_{p}).

Before Emerton proved the above assertion, the second author verified it using the algorithm of this paper for all p ≤ 757, and, up to a power of 2, for all p < 2000.

Remark 7.3. It is tempting to guess that, e.g., the natural map

Φ_{J_{0}(113)}(\overline{F}_{113}) → ∏_{i=1}^{4} Φ_{A_{i}}(\overline{F}_{113})

is an isomorphism, but this is incorrect. Two of the Φ_{A_{i}}(\overline{F}_{113}) have order 2, so the product ∏_{i=1}^{4} Φ_{A_{i}}(\overline{F}_{113}) is not a cyclic group. However, Mazur proved that the groups Φ_{J_{0}(p)}(\overline{F}_{p}) are cyclic for all primes p.

7.6 Examples

In this section we give some examples of the numbers involved in computing component groups of quotients of J_{0}(N). For more examples, see [12]. We use the notation for abelian varieties that is described in [1]. For example 65A is the “first” abelian variety quotient of J_{0}(65) attached to a newform.

7.6.1 Quotients of J_{0}(N)

Table 1 contains many of the quantities involved in the computation of component groups for each of the newform optimal quotients for N ∈ {65, 66, 68, 69}.

7.6.2 Quotients of J_{0}(p)^{−}

We computed the quantities m_{A}, m_{X}, and Φ_{X} for each abelian variety A_{f} associated to a newform of prime level p with p ≤ 631. Table 2 lists those A_{f} for which w_{p} = −1, along with the order of the corresponding component group. The first column, which is labeled “A” contains a description of A_{f}, the second column, labeled “d”, contains the dimension of A_{f}, and the third column, labeled “#Φ_{A}”, contains the order #Φ_{A_{f} ,p}(\overline{F}_{p}) of the component group.

Remark 7.4. Theorem 7.2 together with [13, Prop. II.17.10] imply that the component groups of the A_{f} for which w_{p} = +1 are trivial, so we do not list them. An optimal quotient A_{f} of J_{0}(p) with nonzero component group has nonzero rational torsion (by Theorem 7.2), so it factors through the Eisenstein quotient of J_{0}(p). Also w_{p} acts as −1 on the Eisenstein quotient of J_{0}(p), which is [13, Prop. II.17.10], and which is a deep result because of subtleties at the prime 2 (see the discussion in [13, III.1]).

Table 1: Component groups of quotients of J_{0}(N)

A  dim p w_{p} #Φ_{X} m_{X} m_{A} #Φ_{A}
65A 1 5 +  1  2  2  1
13 +  1  2  1
65B 2 5 +  3  2^{2}  2^{2}  3
13 −  3  2^{2}  3
65C 2 5 −  7  2^{2}  2^{2}  7
13 +  1  2^{2}  1
66A 1 2 +  1  2  2^{2}  2
3 −  3  2^{2}  3
11 +  1  2^{2}  1
66B 1 2 −  2  2  2^{2}  2^{2}
3 +  1  2^{2}  1
11 +  1  2^{2}  1
66C 1 2 −  1  2 2^{2} · 5 2 · 5
3 −  1  2^{2}  5
11 −  1 2^{2} · 5  1
68A 2 17 +  2  2 · 3 2 · 3  2
69A 1 3 −  2  2  2  2
23 +  1  2  1
69B 2 3 +  2  2 2 · 11 2 · 11
23 −  2 2 · 11  2

Table 2: Component groups of quotients of J_{0}(p)^{−}

A  d #Φ_{A}
11A 1  5
17A 1 2^{2}
19A 1  3
23A 2 11
29A 2  7
31A 2  5
37B 1  3
41A 3 2 · 5
43B 2  7
47A 4 23
53B 3 13
59A 5 29
61B 3  5
67A 1  1
67C 2 11
71A 3  5
71B 3  7
73A 1  2
73C 2  3
79B 5 13
83B 6 41
89B 1  2
89C 5 11
97B 4 2^{3}
101B 7 5^{2}
103B 6 17
107B 7 53
109A 1  1
109C 4 3^{2}
113A 1  2
113B 2  2
113D 3  7
127B 7 3 · 7
131B 10 5 · 13
137B 7 2 · 17
139A 1  1
139C 7 23
149B 9 37
151B 3  1
151C 6 5^{2}

A  d #Φ_{A}
157B 7  13
163C 7  3^{3}
167B 12 83
173B 10 43
179A 1  1
179C 11 89
181B 9 3 · 5
191B 14 5 · 19
193C 8  2^{4}
197C 10 7^{2}
199A 2  1
199C 10 3 · 11
211A 2  5
211D 9  7
223C 12 37
227B 2  1
227C 2  1
227E 10 113
229C 11 19
233A 1  2
233C 11 29
239B 17 7 · 17
241B 12 2^{2} · 5
251B 17 5^{3}
257B 14 2^{6}
263B 17 131
269C 16 67
271B 16 3^{2} · 5
277B 3  1
277D 9  23
281B 16 2 · 5 · 7
283B 14 47
293B 16 73
307A 1  1
307B 1  1
307C 1  1
307D 1  1
307E 2  3
307F 9  17
311B 22 5 · 31

A  d #Φ_{A}
313A 2  1
313C 12 2 · 13
317B 15 79
331D 16 5 · 11
337B 15 2^{2} · 7
347D 19 173
349B 17 29
353A 1  2
353B 3  2
353D 14 2 · 11
359D 24 179
367B 19 61
373C 17 31
379B 18 3^{2} · 7
383C 24 191
389A 1  1
389E 20 97
397B 2  1
397C 5  11
397D 10  3
401B 21 2^{2} · 5^{2}
409B 20 2 · 17
419B 26 11 · 19
421B 19 5 · 7
431B 1  1
431D 3  1
431F 24 5 · 43
433A 1  1
433B 3  1
433D 16 2^{2} · 3^{2}
439C 25 73
443C 1  1
443E 22 13 · 17
449B 23 2^{4} · 7
457C 20 2 · 19
461D 26 5 · 23
463B 22 7 · 11
467C 26 233
479B 32 239
487A 2  1

A  d #Φ_{A}
487B 2  3
487C 3  1
487D 16  3^{3}
491C 29 5 · 7^{2}
499C 23  83
503B 1  1
503C 1  1
503D 3  1
503F 26 251
509B 28 127
521B 29 2 · 5 · 13
523C 26 3 · 29
541B 24 3^{2} · 5
547C 25 7 · 13
557B 1  1
557D 26 139
563A 1  1
563E 31 281
569B 31 2 · 71
571A 1  1
571B 1  1
571C 2  1
571D 2  1
571F 4  1
571I 18 5 · 19
577A 2  3
577B 2  1
577C 3  1
577D 18  2^{4}
587C 31 293
593B 1  2
593C 2  1
593E 27 2 · 37
599C 37 13 · 23
601B 29 2 · 5^{2}
607D 31 101
613C 27 3 · 17
617B 28 2 · 7 · 11
619B 30 103
631B 32 3 · 5 · 7

8 Appendix: Some Facts Concerning Toric Reduction

Let R be a discrete valuation ring with fraction field K and residue field k. For any abelian variety A over K, with Néron model A over R, we denote by X_{A} the character group of the toric part of A_{k}^{0} (the connected component of the closed fiber of A). All group schemes below are understood to be commutative.

Our aim in this appendix is to prove a couple of facts (Theorem 8.2 and Theorem 8.6) which are no doubt well-known to experts but for which published proofs do not appear to be readily available. We begin with a simple and basic lemma.

Lemma 8.1. Let f : G → G′ be a map between multiplicative (resp. étale) finite flat group schemes over R. The map f is a closed immersion (resp. faithfully flat) if and only if the generic fiber map f_{K} is a closed immersion (resp. faithfully flat).

Proof. Cartier duality interchanges étaleness and multiplicativeness, as well as closed immersions and faithfully flat maps (as the latter two properties may be checked on the closed fiber, for which one is reduced to the standard case of finite commutative group schemes over a field). Thus, it suffices to consider the étale case. By faithfully flat base change to a strict henselization of R, we are reduced to the case where our finite étale group schemes are constant. Since faithful flatness is equivalent to surjectivity (for maps between étale schemes over a base), the lemma is now physically clear.

Now we turn to the first of the two main results we want to prove. Let π : J → A be an optimal quotient of abelian varieties over K (i.e., we assume that ker π is an abelian variety over K), and assume that J has semistable reduction over R (so A does too). We do not yet make any hypotheses of purely toric reduction. The dual abelian varieties A^{∨} and J^{∨} again have semistable reduction, as they are isogenous to A and J respectively.

Theorem 8.2. With notation as above, the map X_{J^{∨}} → X_{A^{∨}} induced by π is surjective.

Proof. The underlying idea comes down to two facts: Lemma 8.1 and the fact that we can lift tori on the level of ℓ-divisible groups for any prime ℓ. More precisely, we argue as follows. By Proposition 3.3, the map π^{∨} : A^{∨} → J^{∨} is a closed immersion of abelian varieties. We will use this to prove that the induced map π_{t}^{∨} on closed fiber tori of Néron models is a closed immersion. Since the “character group” functor sets up an anti-equivalence of categories between tori over a field F and finite free Z-modules with continuous action of Gal(F_{s}/F), identifying closed immersions of tori with surjections of character groups and surjections of tori with “saturated injections” of character groups (i.e., injections with torsion-free cokernel), the closed immersion property for π_{t}^{∨} on the closed fiber tori will yield the desired surjection of character groups.

In general the “Néron model” functor doesn’t behave well for closed immersions. That is, just because π^{∨} is a closed immersion, it does not follow purely formally that π^{∨} induces a closed immersion on Néron models. Nevertheless, we claim quite generally that if B → B′ is a closed immersion of abelian varieties over K with semistable Néron models, then the induced map T → T′ on closed fiber tori is a closed immersion. For this it is sufficient to prove that the induced map on ℓ-divisible groups T [ℓ^{∞}] → T′[ℓ^{∞}] is a closed immersion for all primes ℓ (i.e., all maps T [ℓ^{n}] → T′[ℓ^{n}] are closed immersions). Indeed, suppose we verify this closed immersion property on torsion, and let H be the kernel of T → T′, so H[ℓ^{n}] = 0 for all primes ℓ and positive integers n. The torus (H_{/\overline{k}}^{0})_{red} must vanish (as it has no non-trivial torsion) and hence H is finite. If N is the order of H, then H = H[N] = 0. The map T → T′ is then a monomorphism between algebraic groups over a field and hence is a closed immersion, as desired.

In order to verify that the ℓ-divisible group maps T [ℓ^{∞}] → T′[ℓ^{∞}] are closed immersions for all ℓ, we can make the faithfully flat base change to the henselization of R (which commutes with formation of Néron models) to reduce to the case where R is henselian. Now we recall the following basic result of Grothendieck:

Lemma 8.3. Let R be a henselian local ring, G a quasi-finite separated R-scheme of finite presentation. There is a unique decomposition

G = G_{f} ∐ G′

into disjoint clopen pieces with G_{f} finite over R (called the “finite part”of G) and G′ having empty closed fiber. The formation of G_{f} is functorial in G and is compatible with henselian local base change and formation of fiber products over R.

If moreover G is a group scheme over R, then G_{f} is a clopen subgroup scheme and there exists a unique multiplicative closed R-subgroup scheme G_{μ} inside of G whose closed fiber is the multiplicative part of the closed fiber of G (G_{μ} is called the “multiplicative part” of G). The formation of G_{μ} is functorial in G.

Proof. For the first part, see [8, IV_{4}, 18.5.11(c)] (aside from the functorial properties, which are obvious). The second part, concerning group schemes, is a mechanical consequence of the first part (including the functoriality of the finite part). For example, the existence of G_{μ} follows from considering the connected-étale sequence of the Cartier dual of G_{f} over the henselian local base R, and the uniqueness and functoriality follows from the functoriality of G ⇝ G_{f} and the functoriality of the connected-étale sequence.

Remark 8.4. Assuming R in Lemma 8.3 is a discrete valuation ring (with fraction field K and residue field k), let us make some observations concerning the behavior of Lemma 8.3 with respect to primary components, as this will be useful later. Let’s suppose that N and M are relatively prime integers with NM divisible by the order of G_{K}, and hence killing G. Thus, by functoriality we have G = G[N] ×_{R} G[M] where G[N] and G[M] are quasi-finite separated R-group schemes. We claim that G[N] and G[M] are also flat over R, whence it follows that the formation of G_{f} and G_{μ} is compatible with passage to “primary components”.

In other words, if ℓ is a prime and ℓ^{n} is divisible by the ℓ-part of the order of G_{K}, then we claim that G[ℓ^{n}] is R-flat. From the clopen decomposition G = G_{f} ∐ G′, it is easy to see that G[ℓ^{n}] = G_{f}[ℓ^{n}] ∐ X_{n} for some finite K-scheme X_{n}, so for the issue of R-flatness we can replace G with G_{f}. We are thereby reduced to the finite flat case, so we can use the proof of [10, 1.7.2].

The significance of Lemma 8.3 for our purposes is the following standard consequence.

Corollary 8.5. Let A be an abelian variety over the fraction field K of a henselian discrete valuation ring R with residue field k. Let A be the Néron model of A, and assume that A has semistable reduction. For every prime ℓ, there exists a unique multiplicative ℓ-divisible group Γ_{ℓ} inside of A whose closed fiber is the ℓ-divisible group of the torus T of A_{k}^{0}. The formation of Γ_{ℓ} is functorial in A.

Proof. Fix ℓ. By the semistability hypothesis, the multiplication maps ℓ^{n} : A → A are quasi-finite flat, so A[ℓ^{n}] is a quasi-finite flat separated R-group scheme. Let A[ℓ^{n}]_{μ} denote its multiplicative part (as in Lemma 8.3), so the multiplicative T [ℓ^{n}] ↪ A[ℓ^{n}]_{k} lies inside of (A[ℓ^{n}]_{μ})_{k}. The “closed fiber” functor is an equivalence of categories between finite flat multiplicative group schemes over R and k (since Cartier duality reduces this to the étale case, and the “closed fiber” functor is an equivalence of categories between finite étale R-schemes and finite étale k-schemes [8, IV_{4}, 18.5.12]). Thus, there exists a unique multiplicative closed R-subgroup scheme Γ^{(n)} ↪ A[ℓ^{n}]_{μ} whose closed fiber is T [ℓ^{n}].

Moreover, using the equivalence of categories just mentioned, A[ℓ^{n}]_{μ} lies inside of A[ℓ^{n+1}]_{μ} and Γ^{(n)} lies inside of Γ^{(n+1)}. The resulting system Γ_{ℓ} = {Γ^{(n)}} over R forms an ℓ-divisible group on the closed fiber and hence is an ℓ-divisible group over R. This settles the desired existence, as well as the desired uniqueness. The functoriality of Γ_{ℓ} in A follows from the functoriality of toric parts on the closed fiber of Néron models.

Returning to the proof of Theorem 8.2, recall that we were studying the map of toric parts j_{t} : T → T′ induced by a closed immersion j : B ↪ B′ of semistable abelian varieties over K, with R henselian. We wanted the map

j_{t}[ℓ^{∞}] : T [ℓ^{∞}] → T′[ℓ^{∞}]

to be a closed immersion for all primes ℓ (as we have seen that this forces T → T′ to be a closed immersion, which is what we really want to show). Fix ℓ. By Corollary 8.5 there exist unique multiplicative ℓ-divisible groups Γ and Γ′ over R in the respective Néron models B and B′ such that Γ and Γ′ respectively lift the ℓ-divisible groups of the tori of the closed fibers. Hence, it suffices to show that the R-map γ : Γ → Γ′ induced by the Néron functoriality map N(j) is a closed immersion. The generic fiber map γ_{K} is a closed immersion since it “sits inside” the generic fiber ℓ-divisible groups of B and B′, the map between which is a closed immersion since j : B → B′ is a closed immersion. Now we use Lemma 8.1 (applied at all finite torsion levels) to conclude that γ is a closed immersion. This completes the proof of Theorem 8.2.

We now turn to a result which requires a stronger hypothesis on the closed fiber. Note that we retain the hypothesis that R is henselian (this hypothesis arose in the proof of Theorem 8.2, even though it wasn’t needed for the statement). Let A and B be abelian varieties over K with purely toric reduction (i.e., their Néron models have closed fiber connected components which are tori). Let φ : A → B be an isogeny, and let φ_{t} : T_{A} → T_{B} be the induced map on the closed fiber toric parts (i.e., connected components) of the Néron models. We denote by φ_{t}^{∨} : T_{B^{∨}} → T_{A^{∨}} the analogous map induced by the dual isogeny φ^{∨}. Since the map φ_{t} is an isogeny (by functoriality), the kernel ker(φ_{t}) is a finite multiplicative k-group scheme.

For any finite multiplicative k-group scheme G, we let G̃ denote the (unique) multiplicative finite flat R-group scheme with closed fiber G. For example, { \atop ker(φ_{t})} is a multiplicative R-group scheme which lies inside of

ker(N(φ))_{μ}

(where N(φ) is the map induced by Néron functoriality). Thus, we have a natural closed immersion

{ \atop (ker φ_{t})_{K}} ↪ ker φ

and likewise we have a natural quotient map

ker(φ^{∨})^{∨} → { \atop ker(φ_{t}^{∨})}_{K}^{∨}

dual to the natural closed immersion using the isogeny φ^{∨}.

By the duality theory for abelian varieties (particularly the adjointness of φ and φ^{∨} with respect to the scheme-theoretic Weil pairing over K), there is a canonical perfect duality K-group scheme duality between ker(φ) and ker(φ^{∨}) over K, whence there is a natural quotient map of K-group schemes

ker(φ) ≃ ker(φ^{∨})^{∨} → { \atop ker(φ_{t}^{∨})}_{K}^{∨}.

Theorem 8.6. The diagram of K-group schemes

0 → { \atop ker(φ_{t})}_{K} → ker(φ) → { \atop ker(φ_{t}^{∨})}_{K}^{∨} → 0

is exact.

The content of the proof is the Grothendieck Orthogonality Theorem. Moreover, Theorem 8.6 is implicit in Grothendieck’s construction of the monodromy pairing for semiabelian varieties.

Proof. The exact sequence of the theorem says that the finite flat K-group schemes p

ker(φ)/{ \atop ker(φ_{t})}_{K} and { \atop ker(φ_{t}^{∨})}_{K}

are canonically Cartier dual to each other compatibly with the perfect duality between ker(φ) and ker(φ^{∨}). More precisely, let A and B denote the Néron models of A and B, respectively, let

G = ker(A → B), G^{∨} = ker(B^{∨} → A^{∨}),

so G and G^{∨} are both quasi-finite flat separated R-group schemes whose generic fibers are the ker(φ) and ker(φ^{∨}) in the theorem (the R-flatness of G and G^{∨} arises from the semiabelian condition, since any quasi-finite morphism between semi-abelian schemes is necessarily flat, as can be checked on geometric fibers). Being quasi-finite flat and separated, the R-group schemes G and G^{∨} have canonical respective “finite parts” G_{f} and G_{f}^{∨} and “multiplicative parts” G_{μ} and G_{μ}^{∨} (as in Lemma 8.3). Beware that we do not claim G_{f}^{∨} (resp. G_{μ}^{∨}) is the Cartier dual to G_{f} (resp. G_{μ}); usually such duality does not hold.

Since G_{μ} and G_{μ}^{∨} are finite flat R-group schemes, the quotients G/G_{μ} and G^{∨}/G_{μ}^{∨} make sense as quasi-finite flat separated R-group schemes. The theorem almost says that there is a canonical duality between (G/G_{μ})_{K} = G_{K}/(G_{μ})_{K} and (G_{μ}^{∨})_{K}, induced by the duality between G_{K} = ker(φ) and G_{K}^{∨} = ker(φ^{∨}), except for the mild problem that G_{μ} might be

larger than { \atop ker(φ_{t})} (i.e., possibly (G_{μ})_{k} is not entirely inside of A_{k}^{0}) and likewise G_{μ}^{∨} might

be larger than { \atop ker(φ_{t}^{∨})}.

We will work on ℓ-primary components for each prime ℓ individually. In order to permit this, we use Remark 8.4. We will first treat the more subtle case when ℓ is the residue characteristic, and then we’ll handle the case when it isn’t. The advantage of working with the case in which ℓ is the residue characteristic is that multiplicative finite k-group schemes are automatically connected. Thus, in this case (G_{μ})_{ℓ} = { \atop ker(φ_{t})}_{ℓ} and (G_{μ}^{∨})_{ℓ} = { \atop ker(φ_{t}^{∨})}_{ℓ}.

Since

0 → G_{K} → A → B → 0

is an exact sequence of abelian sheaves on the fppf site over Spec(K), by the usual snake lemma argument (and the fact that the ℓ-part (G_{K})_{ℓ} of G_{K} is killed by a big power of ℓ) we obtain an exact sequence

0 → (G_{K})_{ℓ} → A[ℓ^{∞}] → B[ℓ^{∞}] → 0.

Arguing as in Corollary 8.5, there is an exact sequence over Spec(R)

0 → (G_{μ})_{ℓ} → A[ℓ^{∞}]_{t} → B[ℓ^{∞}]_{t} → 0

which lifts the exact sequence involving ℓ-divisible groups of tori on the closed fiber (as (G_{μ})_{ℓ} must be in the relative connected component of A). Passing to the generic fiber over K gives us a commutative diagram with exact rows and closed immersions along columns

\xymatrix{0 \ar[r] & ((G_{μ})_{K})_{ℓ} \ar[r] \ar[d] & A[ℓ^{∞}]_{t} \ar[r] \ar[d] & B[ℓ^{∞}]_{t} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & (G_{K})_{ℓ} \ar[r] & A[ℓ^{∞}] \ar[r] & B[ℓ^{∞}] \ar[r] & 0}

where A[ℓ^{∞}]_{t} denotes the K-fiber of the ℓ-divisible group {A[ℓ^{n}]_{t}}, and likewise for B[ℓ^{∞}]_{t}.

Using the snake lemma in the category of fppf abelian sheaves over Spec(K), we get a short exact sequence of cokernels

0 → ((G/G_{μ})_{K})_{ℓ} → A[ℓ^{∞}]/(toric) → B[ℓ^{∞}]/(toric) → 0

where all maps are the natural ones, and right two terms are ℓ-divisible groups over K.

The Grothendieck Orthogonality Theorem (see [9, Exp. IX, Prop 5.6]) asserts that the perfect scheme-theoretic Weil pairing between A[ℓ^{n}] and A^{∨}[ℓ^{n}] makes A[ℓ^{n}]_{t} and A^{∨}[ℓ^{n}]_{f} exact annhilators, where A[ℓ^{n}]_{f} denotes the K-fiber of the finite part of the A^{0}[ℓ^{n}] and A[ℓ^{n}]_{t} denotes the K-fiber of the unique R-subgroup scheme in A^{0}[ℓ^{n}] lifting the ℓ^{n}-torsion on the closed fiber torus. By the purely toric condition applied to A^{∨}, we see A^{∨}[ℓ^{n}]_{f} = A^{∨}[ℓ^{n}]_{t}. Thus, the orthogonality theorem says that A[ℓ^{n}]/A[ℓ^{n}]_{t} and A^{∨}[ℓ^{n}]_{t} are in perfect duality via the scheme-theoretic Weil pairing over K.

Passing to the limit, we get a canonical isomorphism of ℓ-divisible groups

A[ℓ^{∞}]/(toric) = (A^{∨}[ℓ^{∞}]_{t})^{∨}.

But φ and φ^{∨} are adjoint with respect to Weil pairing, so we conclude that the diagram

[figure: A[ℓ^{∞}]/(toric); φ; (A^{∨}[ℓ^{∞}]_{t})^{∨}; (φ_{t}^{∨})^{∨}; B[ℓ^{∞}]/(toric); (B^{∨}[ℓ^{∞}]_{t})^{∨}]

commutes. Thus, we get an isomorphism between the kernels of these vertical isogenies. The kernel of the left column is ((G/G_{μ})_{K})_{ℓ}, as we saw above. Meanwhile, the kernel of the right is (by duality theory of ℓ-divisible groups) exactly the dual of ker(φ_{t}^{∨}) = (G_{μ}^{∨})_{K}. This gives the desired perfect duality between (G/G_{μ})_{K} and (G_{μ}^{∨})_{K} on ℓ-primary parts for ℓ equal to the residue characteristic.

Now we consider the case when ℓ is not equal to the residue characteristic. There is no loss of generality in passing to the case of a strictly henselian base R. Thus, the closed fiber tori have constant ℓ-divisible groups. Also, we can work with Z_{ℓ}-modules of geometric points (over K) via Tate’s construction. The “toric” part of the ℓ-adic Tate module T_{ℓ}(A) is exactly the (saturated) maximal submodule with trivial Galois action, since a compatible system of ℓ-power torsion points in A(K) = A(R) must lie entirely inside of A^{0}(R) (thanks to the finiteness of the component group) and we can identify A^{0}(R)[ℓ^{n}] with the (constant) ℓ^{n}-torsion on the split torus A_{k}^{0} over the separably closed k.

Using inverse limits, we see that T_{ℓ}(A) → T_{ℓ}(B) is injective with cokernel (G_{K})_{ℓ} (= geometric points of ℓ-part), and this cokernel is exactly ker(φ)_{ℓ}. Likewise, the cokernel of the map

T_{ℓ}(A)_{t} → T_{ℓ}(B)_{t}

on “toric” parts (i.e., ℓ-adic Tate module generic fibers of the lifts of the ℓ-divisible groups of closed fiber tori) is ({ \atop ker(φ_{t})}_{K})_{ℓ}.

Thus, we get a commutative diagram with horizontal exact sequences

\xymatrix{0 \ar[r] & T_{ℓ}(A)_{t} \ar[r] \ar[d] & T_{ℓ}(B)_{t} \ar[r] \ar[d] & ({ \atop ker(φ_{t})_{K}})_{ℓ} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & T_{ℓ}(A) \ar[r] & T_{ℓ}(B) \ar[r] & ker(φ)_{ℓ} \ar[r] & 0}

with columns given by the natural maps. These vertical maps are all injective, so by the snake lemma we get a short exact sequence of cokernels. We can now use the exact same Weil pairing arguments with the Grothendieck orthogonality theorem (now in the easier “ℓ ≠ p” form of the orthogonality theorem [9, Exp IX, 2.4]), essentially just as we argued in the previous case. One distinction is that the use of inverse limit Tate modules rather than direct limits causes some maps to switch direction.

More specifically we have a commutative square with horizontal isomporphisms (thanks to the orthogonality theorem)

[figure: T_{ℓ}(A)/T_{ℓ}(A)_{t}; φ; T_{ℓ}(A^{∨})_{t}^{∨}; T_{ℓ}(φ_{t}^{∨})^{∨}; T_{ℓ}(B)/T_{ℓ}(B)_{t}; T_{ℓ}(B^{∨})_{t}^{∨}]

This induces an isomorphism

((ker φ)/{ \atop ker(φ_{t})}_{K})_{ℓ} ≃ (({ \atop ker φ_{t}^{∨}})_{K})_{ℓ}^{∨}

between the vertical cokernels, and by construction this isomorphism is compatible with Weil pairings, whence the desired perfect pairing has been shown.

Let R be an arbitrary dvr (not necessarily henselian), let φ : A → B be an isogeny, and let φ_{t} : T_{A} → T_{B} be the induced map on the closed fiber toric parts, as above.

Corollary 8.7. The order of ker(φ) is the product of the orders of ker φ_{t} and ker φ_{t}^{∨}.

Proof. Pass to the henselization of R and use Theorem 8.6.

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