TAMELY RAMIFIED HIDA THEORY

ASSAF GOLDBERGER AND EHUD DE SHALIT

Let N be a natural number, and p a prime not dividing N. Let X_{1}(N, p) denote the modular curve associated with the group Γ_{1}(N, p) = Γ_{1}(N) ∩ Γ_{0}(p), and X_{1}(Np^{ν}) the curve associated with Γ_{1}(Np^{ν}). Several authors (Hida, Mazur, Tilouine, Wiles...; see [H], [M-W1], [W]) studied the tower obtained from the transition maps between these curves, which correspond to the inclusions Γ_{1}(N, p) ⊃ Γ_{1}(Np^{ν}) ⊃ Γ_{1}(Np^{ν+1}). The Hecke ring acts as a ring of correspondences on this tower. In particular, the diamond operators induce an action of ℤ_{p}^{×}, and p-adic modules built from the tower can be studied as modules over the Iwasawa algebra ℤ_{p}[[ℤ_{p}^{×}]]. In this way the geometry of modular curves is connected with deformation theory of p-adic Galois representations.

Any module over ℤ_{p}[[ℤ_{p}^{×}]] breaks up as a direct sum according to the characters of 𝔽_{p}^{×} ⊂ ℤ_{p}^{×}. Thus the first step of the tower, the covering X_{1}(Np) → X_{1}(N, p), is exploited in a rather mild way, and is not the source of a true deformation. Around the mid 80’s it was noticed by Mazur and Tate that various results and conjectures in the p-adic theory of modular curves (or cyclotomic fields) had finite, “refined” versions, which could probably be linked - in the situation described here - to a deformation theory with group 𝔽_{p}^{×} [M-T]. In [dS1] and [dS2] one of us proposed to use the kernel of p − 1 on the Jacobian as a source of such “finite deformation theory”. This was applied there to a conjecture of Mazur and Tate [M-T] on the padic periods of elliptic curves with split multiplicative reduction, much in the style of Greenberg and Stevens [Gr-St], who used Hida theory to prove the conjecture of Mazur Tate and Teitelbaum [M-T-T], which predated the Mazur-Tate conjecture.

For non-trivial technical reasons, the articles [dS1] and [dS2] dealt with the case N = 1 (prime conductor). The purpose of this work is to return to the general set-up, and study the deformation theory which is obtained from the p − 1 torsion in the Jacobians of X_{1}(N, p) and X_{1}(Np). We call this “tame” Hida theory because the bad prime, where most of the action takes place, is still p. Clearly, what we do below is only a “weight 2” theory. The higher weights, which in Hida’s theory are subsumed by the weight 2 theory, have to await further developments.

The following is a brief summary. In Section 1 we collect preliminary results, and introduce notation and conventions to be used throughout. Section 2 is devoted to a detailed study of J_{1}(Np)[p − 1], as a deformation of its part fixed by the group of p-diamond operators, isomorphic to 𝔽_{p}^{×}. As we may treat the l-Sylow subgroup separately for each prime l dividing p − 1, we fix l (different from 2) and study J_{1}(Np)[r] where r is the highest power of l dividing p − 1. We address the structure over the group ring of 𝔽_{p}^{×}, Hecke structure, Galois action, and in particular the filtration for the action of the decomposition group at p. The results obtained in Section 2 are summarized in Theorem 2.8. Section 3 is devoted to questions of breaking the deformation into components, and in particular distinguishing the p-new components from the p-old ones or from other new components. Section 4 finally treats one such p-new component, and shows how the infinitesimal variation of the U_{p} operator in the deformation is related to the p-adic period matrix of the abelian variety. Theorem 4.6 may be considered the main result.

Date: August 31, 2000.

To illustrate the main result, let us consider an elliptic curve A over ℚ of conductor Np, with split multiplicative reduction at p. We normalize A within its ℚ-isogeny class by requiring that it appears as a factor of the Jacobian of the modular curve X_{1}(N, p). (This is a slight deviation from the more traditional point of view, as a factor of the Jacobian of X_{0}(Np).) Let q_{A} be the p-adic Tate period of A and write

q_{A} = up^{ν}

with u ∈ ℤ_{p}^{×} and ν ∈ ℤ.

Pick an odd prime l dividing p − 1, let r be the highest power of l dividing p − 1, and R = ℤ/rℤ. Let

u_{R} = u ⊗ 1 ∈ ℤ_{p}^{×} ⊗ R = 𝔽_{p}^{×} ⊗ R = μ_{r} and ν_{R} = ν ⊗ 1 ∈ R.

The ratio u_{R} : ν_{R} is the l-primary part of the refined L-invariant of A. Recall that the (p-adic) L-invariant of Mazur-Tate-Teitelbaum is the ratio log_{p}(u) : ν, which is blind to the p − 1 root of unity in q_{A}. The refined L-invariant is made precisely to capture that bit of information.

We make two technical assumptions. The first, that A is not l-Eisenstein (see [M] and Section 1.6 below for the precise meaning of this). The second, that the degree of the modular parametrization X_{1}(N, p) → A is prime to l. (This guarantees condition (A2) in Section 3.3.) Let us also assume, for simplicity, that l does not divide N.

Consider the Jacobians J_{0} of X_{1}(N, p) and J_{1} of X_{1}(Np). Because of our assumptions, A[r] ≅ R^{2} is a direct summand, as a Hecke and Galois module, of J_{0}[r]. On the other hand J_{1}[r] is also a Λ-module, where Λ = R[μ_{r}], and μ_{r} ⊂ 𝔽_{p}^{×} act via the p-diamond operators. We show that there is a unique direct summand (Hecke and Galois stable) eJ_{1}[r] ≅ Λ^{2} of J_{1}[r] which is a deformation of A[r] (Proposition 3.4).

We now examine the action of the Hecke operator U_{p}^{−2} on both pieces. On A[r] it acts trivially. On the deformation module eJ_{1}[r] it acts via multiplication by 1 + λ with λ in the augmentation ideal I of Λ. Identifying I/I^{2} with μ_{r}, we finally arrive at the desired relation

u_{R} = ν_{R}λ mod I^{2}.

Thus, up to a factor of −1/2, it may be said that the ratio u_{R} : ν_{R} is the derivative of the U_{p} operator in the deformation of A[r] which lives inside the Jacobian of X_{1}(Np).

1. Preliminaries and notation

1.1. Abstract diamond and Hecke operators. Let N ≥ 4 be an integer, and let p ≥ 7 be a prime which does not divide N. Let l be an odd prime dividing p − 1, and r = l^{n} the maximal power of l dividing p − 1. Write p − 1 = rr′, where (r′, r) = 1, so that 𝔽_{p}^{×} = μ_{r} × μ_{r′}. Let R = ℤ/rℤ and

(1.1)  \underline{Λ} = R[𝔽_{p}^{×}] = Λ ⊗ Λ′,

where Λ = R[μ_{r}] is local, and Λ′ = R[μ_{r′}] is étale over R. We write a typical element of \underline{Λ} as ∑ n_{a} ⟨a⟩ . Let I (resp. I′, resp. \underline{I}) be the augmentation ideal in Λ (resp. Λ′, resp. \underline{Λ}). We have \underline{I}=I ⊗ Λ′ + Λ ⊗ I′. Note that I is nilpotent, that the maximal ideal of Λ is (l) + I, and that ⟨−1⟩ − ⟨1⟩ belongs to I′.

Consider the commutative ring H generated over ℤ by the symbols T_{q} (q a prime, q ∤ Np), U_{q} (q|Np), ⟨t⟩_{N} for t ∈ (ℤ/Nℤ)^{×}, and ⟨a⟩_{p} , for a ∈ 𝔽_{p}^{×}. We call H the abstract Hecke ring, and the ⟨a⟩_{p} the p-diamond operators. All our Hecke rings will be quotients of H. If M is a module over H we denote by H(M) the image of H in End(M), and we call it the Hecke ring cut out by M. If M′ is a submodule of M, stable under H, both H(M′) and H(M/M′) are quotients of H(M) in a natural way.

We shall write M^{∗} for Hom(M, R), and let T ∈ H act on it via (T μ)(m) = μ(T m) (m ∈ M, μ ∈ M^{∗}).

If M is a module over a ring R, 𝔞 ⊂ R is an ideal, and a ∈ 𝔞, we denote by M[a] = Ker(a|_{M}) the kernel of multiplication by a, and by M[𝔞] = ⋂_{a∈𝔞} M[a].

In many cases the H-module M is killed by r. Then H(M) becomes a \underline{Λ}-algebra, in which 𝔽_{p}^{×} act by the p-diamond operators.

1.2. Modular curves and Jacobians. Let ℌ denote the upper half plane, and ℌ^{∗} = ℌ ∪ ℙ^{1}(ℚ) (with the usual topology). If Γ is a congruence group in SL_{2}(ℤ), we let X_{Γ} denote the corresponding complex modular curve, isomorphic as a Riemann surface to Γ\ℌ^{∗}. If Γ = Γ_{0} = Γ_{1}(N, p) = Γ_{1}(N) ∩ Γ_{0}(p) we shall denote by X_{0} = X_{1}(N, p) the well-known model of X_{Γ} over ℚ, whose non-cuspidal points parametrize elliptic curves with a point of order N and a cyclic subgroup of order p. If Γ = Γ_{1} = Γ_{1}(Np) we shall likewise denote by X_{1} the model of X_{Γ} over ℚ parametrizing elliptic curves with a point of order Np. The covering X_{1} → X_{0} is defined over ℚ, unramified, and is Galois with Galois group 𝔽_{p}^{×}. Note that our assumption N ≥ 4 insures that Γ_{0} is torsion-free, so acts faithfully on ℌ, and has no elliptic elements. We shall denote by Y_{Γ} the open modular curve, and by C_{Γ} = X_{Γ} − Y_{Γ} the (reduced, 0-dimensional) subscheme of the cusps. In the two examples mentioned above, Y_{0} and C_{0}, as well as Y_{1} and C_{1}, are defined over ℚ.

We shall denote by J_{Γ} = P ic^{0}(X_{Γ}) the Jacobian variety of X_{Γ}, and by J_{Γ}^{#} the generalized Jacobian with respect to the modulus C_{Γ}. It classifies isomorphism classes of line bundles L of degree 0 on X_{Γ}, together with a trivialization of L|_{C_{Γ}}. Alternatively, it classifies divisors of degree 0 on X_{Γ} supported away from the cusps, modulo principal divisors of functions which are constant (≠ 0, ∞) along the cusps. Thus J_{Γ}^{#} is an extension of J_{Γ} by a (#C_{Γ} − 1)-dimensional torus:

(1.2)  0 → Hom(ℤ[C_{Γ}]_{0}, 𝔾_{m}) → J_{Γ}^{#} → J_{Γ} → 0,

where

(1.3)  ℤ[C_{Γ}]_{0} = Ker (deg : ℤ[C_{Γ}] → ℤ) .

A semi-abelian variety is, in general, not isogenous to the product of its toric part with an abelian variety. But in our case the Manin-Drinfel’d theorem, which asserts that divisors of degree 0 supported on C_{Γ} represent torsion points in J_{Γ}, precisely guarantees this. Indeed, applying the functor Hom(J_{Γ}, −) to the short exact sequence above we get an exact sequence

(1.4)  0 → Hom(J_{Γ}, J_{Γ}^{#}) → Hom(J_{Γ}, J_{Γ}) → Hom(ℤ[C_{Γ}]_{0}, J_{Γ}),

where we have used the principal polarization Ext^{1}(J_{Γ}, 𝔾_{m}) = J_{Γ}. The identity map id ∈ Hom(J_{Γ}, J_{Γ}) gets mapped here to the divisor class homomorphism δ ∈ Hom(ℤ[C_{Γ}]_{0}, J_{Γ}), which maps an element of ℤ[C_{Γ}]_{0} to its divisor class in J_{Γ}. By the Manin-Drinfel’d theorem δ is killed by some integer m, so m times the identity map of J_{Γ} lifts to a map J_{Γ} → J_{Γ}^{#}, proving that the extension (1.2) splits up to isogeny.

Following our example, we retain the symbols J_{0}, J_{0}^{#}, J_{1} and J_{1}^{#} for the models of the corresponding J_{Γ} and J_{Γ}^{#} over ℚ. We therefore have an exact sequence of group schemes over ℚ

(1.5)  0 → Hom(ℤ[C_{0}]_{0}, 𝔾_{m}) → J_{0}^{#} → J_{0} → 0

(and similarly for J_{1}^{#}), where as before

(1.6)  ℤ[C_{0}]_{0} = Ker (deg : ℤ[C_{0}] → ℤ)

and Hom(ℤ[C_{0}]_{0}, 𝔾_{m}) is the torus over ℚ whose character group is ℤ[C_{0}]_{0} (with the Galois action arising from the action on the cusps).

As a rule, we regard Jacobians as Picard schemes, so that correspondences on X_{Γ} induce endomorphisms of J_{Γ} by Picard (contravariant) functoriality. In particular we have maps J_{0} → J_{1} and J_{0}^{#} → J_{1}^{#} deduced from the covering (X_{1}, C_{1}) → (X_{0}, C_{0}), and these are defined over ℚ.

Let J_{0,old} be the p-old subvariety^{1} of J_{0}. It is the sum of the images of J_{1}(N) in J_{0}, under the maps induced from the two degeneracy maps X_{0} → X_{1}(N). Let

(1.7)  J_{0}^{new} = J_{0}/J_{0,old}

be the p-new quotient of J_{0}. There is a unique abelian subvariety J_{0,new} of J_{0} which is complementary to J_{0,old} in the sense that their intersection is finite, and their sum is the whole J_{0}. We call it the p-new subvariety of J_{0}, and J_{0}^{old} = J_{0}/J_{0,new} we call the p-old quotient. From now on we omit the prefix p- when referring to these abelian varieties. The new/old subvarieties/quotients are defined over ℚ, and the obvious maps J_{0,new} → J_{0}^{new} and J_{0,old} → J_{0}^{old} are isogenies.

1.3. The Hecke and diamond action on the modular curves and their Jacobians. We denote by T_{q} (q ∤ Np), U_{q} (q|Np), ⟨t⟩_{N} and ⟨a⟩_{p} (the latter being trivial in the case of X_{0}) the correspondences of X_{0} and X_{1} described, for example, in [M-W], Chapter 2, Section 5. They induce endomorphisms of the Jacobians (or generalized Jacobians), defined over ℚ, which we denote by the same letter. (Caution: T = T^{∗} is the endomorphism defined by Picard functoriality, while [M-W] choose to work with the endomorphisms defined by Albanese functoriality. One should take care of the U′_{q}s in particular. Not only are U_{q∗} and U_{q}^{∗} different - they do not commute). These endomorphisms all commute with each other, so we get an embedding

(1.8)  H(J_{0}) ↪ End(J_{0/ℚ}),

and similarly in other circumstances. Needless to say, the action of the p-diamond operators on J_{0} or J_{0}^{#} is trivial.

1As the referee pointed out, it is more common to use superscripts for the subvarieties, and subscripts for the quotient varieties. We hope that the reader will bear with our convention.

We shall have an occasion to use the following fact. Write N = N_{l}N′_{l} where N_{l} is the l-part of N and (l, N′_{l}) = 1. Let H_{l}(J_{0}) = H(J_{0}) ⊗ ℤ_{l}, a finite flat ℤ_{l}-algebra, so that

H′_{l}(J_{0}) = End(J_{0}) ⊗ ℤ_{l} ∩ H(J_{0}) ⊗ ℚ_{l}

is its saturation in the Endomorphism ring. The group H′_{l}(J_{0})/H_{l}(J_{0}) is a finite l-group. To each maximal ideal 𝔐 in the semi-local ring H_{l}(J_{0}) there is associated an idempotent e_{𝔐} in the ring, projecting onto its completion at 𝔐. If l divides N we call 𝔐 ordinary if U_{l} does not belong to 𝔐.

Proposition 1.1. ^{2}Let 𝔐 be a maximal ideal of H_{l}(J_{0}). If either (i) N_{l} = 1 or (ii) N_{l} = l and 𝔐 is ordinary, then

(1.9)  e_{𝔐}H_{l}(J_{0}) = e_{𝔐}H′_{l}(J_{0}).

Proof. We follow [M], proposition (9.5) on p.95. The elements of

H′(J_{0}) = End(J_{0}) ∩ H(J_{0}) ⊗ ℚ

act on the Néron model of J_{0} over R = ℤ_{l}[ζ_{N}], hence on its identity component, which is P ic_{𝔛/R}^{0} by Raynaud’s theorem ([BLR], 9.5/4), hence H′_{l}(J_{0}) acts on its Lie algebra, which is H^{1}(𝔛, O) ([BLR], 8.4/1). Here by 𝔛_{/R} we mean the proper and flat scheme which is the compactification of the scheme representing the moduli problem ([bal.Γ_{1}(N)]^{can}, Γ_{0}(p)) over R. Its special fiber is reduced, smooth if N_{l} = 1, and, if N_{l} = l, consists of 2 smooth complete curves intersecting transversally at the supersingular points. See [K-M], Chapter 3, for the terminology and for the precise structure of this moduli problem. Note that we chose to work with [bal.Γ_{1}(N)]^{can} over R and not with [Γ_{1}(N)] over ℤ_{l}. The moduli scheme for the second moduli problem has a non-reduced special fiber. (See the discussion of the fiber at p in section 2.5 below.) In particular the special fiber is a local complete intersection, hence Gorenstein and Cohen-Macaulay. We let Ω be the sheaf of regular differentials on 𝔛/R. See [M], section 3, p.67, and [De-Rap]. Since 𝔛/R is Gorenstein, this sheaf is invertible, and in fact is given by the recipe of [De-Rap], section 2.3. As in [M], section 3, one proves that H^{1}(𝔛, O) and H^{0}(𝔛, Ω) are free over R and dual to each other as R-modules. Their formation commutes with passing to the special fiber. For these facts it is important to know that the special fiber of 𝔛 is reduced, because the proof of lemma 3.3 in [M] uses the fact that the global functions on the special fiber are just the constants from the residue field of R.

Let 𝔐 be any maximal ideal of H_{l}(J_{0}), and let e_{𝔐} be the corresponding idempotent. We claim that e_{𝔐}H^{1}(𝔛, O) is free of rank 1 over e_{𝔐}H_{l}(J_{0}). By Nakayama’s lemma, it is enough to show that H^{1}(𝔛, O)/𝔐H^{1}(𝔛, O) is a 1-dimensional vector space over H_{l}(J_{0})/𝔐H_{l}(J_{0}). Dualizing, we deduce the desired result from the fact that H^{0}(𝔛, Ω)[𝔐] is one-dimensional (see [W], lemma 2.2). This “multiplicity one” result is a consequence of the q-expansion principle, and it is here that we need the assumption that 𝔐 is ordinary, if l|N.

Write T′ = e_{𝔐}H′_{l}(J_{0}) and T = e_{𝔐}H_{l}(J_{0}). Then

T ⊂ T′ ↪ End_{T}(e_{𝔐}H^{1}(𝔛, O)) = T,

so T = T′, and the proposition is proved.

2We would like to thank K. Ribet and A. Agashe for help with the proof of this proposition.

1.4. The Atkin-Lehner “involutions”. There are important automorphisms of the curves X_{0} and X_{1}, denoted by w_{M} = w_{M,ζ}, for every M|Np such that (M, Np/M) = 1, which are defined over ℚ(ζ_{M}), and which are involutions up to diamond operators. Their definition depends on a choice of a root of unity. Fix an Np-th root of unity ζ = ζ_{Np} and let ζ_{M} = ζ_{Np}^{Np/M} for M|Np. We shall define w_{M,ζ} on X_{1}. The definition then descends to X_{0}. The curve X_{1} is a fine moduli space over ℚ for (isomorphism classes of) triples (E, P_{M}, P_{Np/M}) where E is an elliptic curve and P_{M} (resp. P_{Np/M}) is a point of exact order M (resp. Np/M) on E. We denote the isomorphism class of a triple (−) by [−]. Let

(1.10)  w_{M,ζ}([E, P_{M}, P_{Np/M}]) = [E′, P′_{M}, P′_{Np/M}],

where E′ = E/ ⟨P_{M}⟩, P′_{M} = P̃_{M} mod ⟨P_{M}⟩ , with P̃_{M} any point of exact order M, such that for the Weil e_{M}^{E}-pairing on E[M]

(1.11)  e_{M}^{E}(P_{M}, P̃_{M}) = ζ_{M},

and P′_{Np/M} = P_{Np/M} mod ⟨P_{M}⟩ . We denote by w_{M,ζ} also the corresponding auto-

morphisms of J_{0}, J_{0}^{#}, J_{1}and J_{1}^{#}, induced by Picard functoriality.

The following relations between the w-operators and the Hecke operators, and betwen the w-operators and themselves (as endomorphisms of the Jacobians, or generalized Jacobians) hold. We shall be interested in w_{N}, w_{p}, and w_{Np}. The corresponding relations in the ring of correspondences on the curve are obtained by reversing the order of composition. All are easy to check from the modular interpretation. See also [A-Li].

• (1) w_{Np} = ⟨N⟩_{p}^{−1} ∘ w_{N} ∘ w_{p} = ⟨p⟩_{N}^{−1} ∘ w_{p} ∘ w_{N}.

• (2) w_{Np}^{2} = 1, w_{p}^{2} = ⟨−p⟩_{N}, w_{N}^{2} = ⟨−N⟩_{p} .

• (3) ⟨a⟩_{p} ∘ w_{p} = w_{p} ∘ ⟨a⟩_{p}^{−1} , ⟨a⟩_{p} ∘ w_{N} = w_{N} ∘ ⟨a⟩_{p} .

• (4) T_{q} ∘ w_{p} = ⟨q⟩_{p} ∘ w_{p} ∘ T_{q}, T_{q} ∘ w_{N} = ⟨q⟩_{N} ∘ w_{N} ∘ T_{q} (for q ∤ Np),

• (5) U_{q} ∘ w_{p} = ⟨q⟩_{p} ∘ w_{p} ∘ U_{q} (for q|N), U_{p} ∘ w_{N} = ⟨p⟩_{N} ∘ w_{N} ∘ U_{p}. Note that there is no simple relation between w_{p} and U_{p}. The two do not

commute, not even up to diamond operators.

1.5. The action on the new part. Over ℂ we may identify S_{2}(Γ) with the cotangent space to J_{Γ}, and since we are in characteristic 0, the Hecke ring cut out by J_{Γ} is the same as the one cut out by the space of cuspforms. The decomposition

(1.12)  S_{2}(Γ_{0}) = S_{2}^{old}(Γ_{0}) ⊕ S_{2}^{new}(Γ_{0})

is stable under H. It follows that the same is true about the old/new subvariety/quotient of J_{0}, and that

(1.13)  H(J_{0,new}) = H(J_{0}^{new}) = H(S_{2}^{new}(Γ_{0})).

It is known, and easily checked from the modular interpretation, that U_{p} +w_{p} sends J_{0} into the old subvariety. It follows that on J_{0,new} we have the relation:

• (6) U_{p} = −w_{p}. It is also known that J_{0,new} is stable under all the Atkin-Lehner involutions. See

[A-Le], Theorem 3.

1.6. Eisenstein modules. The exact sequence

(1.14)  0 → Hom(ℤ[C_{1}]_{0}, 𝔾_{m}) → J_{1}^{#} → J_{1} → 0

induces a surjection H(J_{1}^{#}) → H(Hom(ℤ[C_{1}]_{0}, 𝔾_{m})). Let ℑ be its kernel, i.e. the annihilator of ℤ[C_{1}]_{0}. It is called the Eisenstein ideal. Any prime ideal of H(J_{1}^{#}) containing ℑ (i.e. in the support of H(J_{1}^{#})/ℑ) is called an Eisenstein prime, and any prime ideal containing (l, ℑ) is called an l-Eisenstein prime. If h is any quotient of H(J_{1}^{#}) we call h purely Eisenstein if its support (as an H(J_{1}^{#})-module) consists of Eisenstein primes, and non-Eisenstein if its support is disjoint from the Eisenstein primes. Finally if Z is a Hecke-stable subquotient of J_{1}^{#} we call Z purely Eisenstein or non-Eisenstein if H(Z) is. Thus Z is non-Eisenstein precisely when the image of ℑ under the map H(J_{1}^{#}) → H(Z) is the full H(Z).

A Hecke-stable subvariety A of J_{1} is called non l-Eisenstein if A[l^{n}] is non Eisenstein for all n, or, equivalently, if the support of H(A) contains no l-Eisenstein primes. Suppose that this is the case, and consider H(J_{1}^{#}[l^{n}]), which is an Artinian semi-local ring. Let e be the idempotent in this ring which is the projection onto the non-Eisenstein components. Then e kills Hom(ℤ[C_{1}]_{0}, μ_{l^{n}}) and therefore eJ_{1}^{#}[l^{n}] = eJ_{1}[l^{n}]. On the other hand eA[l^{n}] = A[l^{n}], and we may view A[l^{n}] as a submodule of J_{1}^{#}[l^{n}], even without having to find a map from A to J_{1}^{#}.

Suppose that A (assumed now non l-Eisenstein), has a Hecke-stable subvariety A^{⊥} complementary to it, in the sense that their intersection is finite, while their sum is the whole J_{1}. Assume, in addition, that this A^{⊥} satisfies

(1.15)  A[l^{n}] ∩ A^{⊥}[l^{n}] = (0).

Then for A′, the pre-image of A^{⊥} in J_{1}^{#}, we have the same thing : A[l^{n}]∩A′[l^{n}] = (0) (apply e to prove it!), and therefore

(1.16)  J_{1}^{#}[l^{n}] = A[l^{n}] ⊕ A′[l^{n}],

a direct sum which is of course Hecke and Galois-stable. These remarks will be used in chapter 3.

1.7. Néron models. Quite generally, if F is a finite extension of ℚ_{p} and A an abelian variety over F, we denote by N (A, O_{F}) the Néron model of A over O_{F} , by N (A, O_{F})^{0} its identity component, and by Φ(A, O_{F}) = N (A, O_{F})/N (A, O_{F})^{0} the group of connected components, which is a finite étale group over the residue field κ(F) of F. We shall need the following result.

Proposition 1.2. For any F, the group Φ(J_{0}, O_{F}) is new. More precisely, the exact sequence of abelian varieties over F

(1.17)  0 → J_{0,old} → J_{0} → J_{0}^{new} → 0

induces by functoriality maps between Néron models, hence maps between groups of connected components, and the map

(1.18)  Φ(J_{0}, O_{F}) → Φ(J_{0}^{new}, O_{F})

is an embedding.

Proof. Since ℤ_{p} is absolutely unramified and J_{0} has semi-stable reduction, Theorem 7.5/4(ii) of [BLR] gives the exactness of Néron models over ℤ_{p} (we denote here, for brevity J_{0} = N (J_{0}, ℤ_{p}) etc.)

(1.19)  0 → J_{0,old} → J_{0} → J_{0}^{new}

and therefore the exactness of

(1.20)  0 → J_{0,old} → (J_{0})^{0} → (J_{0}^{new})^{0} → 0.

Since for abelian varieties with semi-stable reduction the formation of the identity component of the Néron model commutes with finite flat base change ([BLR], 7.4/4), the sequence

(1.21)  0 → N (J_{0,old}, O_{F}) → N (J_{0}, O_{F})^{0} → N (J_{0}^{new}, O_{F})^{0} → 0

is exact. Consider the diagram

(1.22) \begin{aligned}0 & → & N (J_{0,old}, O_{F}) & → & N (J_{0}, O_{F})^{0} & → & N (J_{0}^{new}, O_{F})^{0} & → & 0 \\ & & || & & ↓ & & ↓ \\ 0 & → & N (J_{0,old}, O_{F}) & → & N (J_{0}, O_{F}) & → & N (J_{0}^{new}, O_{F}) \end{aligned} .

The vertical arrows are injective, hence so is the first arrow in the bottom row. Since J_{0} has semi-stable reduction over F, the proof of theorem 7.5/4(ii) in [BLR] shows that the bottom row is exact too. (Note that the assumption (*) there about the absolute index of ramification of F was necessary only to get (i), but not to deduce (ii) from (i)). The lemma now follows from the snake lemma.

Corollary 1.3. On Φ(J_{0}, O_{F}) we have the relation U_{p} = −w_{p}.

The generalized Jacobians that we shall meet are all semi-abelian varieties (extensions of abelian varieties by tori). As such they have Néron (lft-)models too, which are constructed for example in [BLR], chapter 10. We shall denote the Néron lft-model of a semi-abelian variety A^{#} by the same notation N (A^{#}, O_{F}).

1.8. Two algebraic lemmas.

Lemma 1.4. Let W be an abelian group, and α, β ∈ End(W) two endomorphisms such that α + β = 1, and αβ is nilpotent. Then

(1.23)  W = W_{α} ⊕ W_{β}

where W_{α} is the subgroup of all the elements of W annihilated by some power of α, and similarly for W_{β}.

Proof. Elementary.

Lemma 1.5. Let Λ be a commutative ring, I a nilpotent ideal, and R = Λ/I. Let W be a Λ-module, and W̄ = W/IW. Let H ⊂ End_{Λ}(W) be a commutative subring. Suppose e_{0} ∈ H is such that ē_{0} ∈ End_{R}(W̄) is an idempotent. Then there exists a unique idempotent e ∈ H lifting ē_{0}, i.e. ē = ē_{0}.

Proof. For e, f ∈ End_{Λ}(W) write e ≡ f mod I^{m} if (e − f)W ⊂ I^{m}W. We shall define inductively a sequence e_{m} ∈ H with the property that

(1.24)  e_{m}^{2} ≡ e_{m} mod I^{m+1}, and e_{m+1} ≡ e_{m} mod I^{m+1}.

Simply let e_{m+1} = e_{m} + (e_{m}^{2} − e_{m})(1 − 2e_{m}). A straightforward computation yields

(1.25)  e_{m+1}^{2} − e_{m+1} = (e_{m}^{2} − e_{m})^{2}(4e_{m}^{2} − 4e_{m} − 3),

so the induction is carried on. For large enough m, e = e_{m} is an idempotent.

To prove the uniqueness, suppose that e and e′ are both idempotents lifting ē_{0}. Writing e′ = e + j, and assuming j ≡ 0 mod I^{m+1}, we get from

e + j = (e + j)^{2} = e + 2ej + j^{2}

that j(1 − 2e) = j^{2}, hence j = j^{2}(1 − 2e) ≡ 0 mod I^{m+2}, and so j = 0.

2. The deformation modules

In this section we introduce the main objects which we wish to study. They are finite groups built from the p − 1 torsion in the (generalized) Jacobians of X_{0} and X_{1}. It is clearly sufficient to treat their l-primary part for each l separately, and in doing so we shall exclude l = 2. Recall that r is the highest power of l dividing p − 1, and R = ℤ/rℤ.

2.1. The deformation. Write J_{1}^{#}[r, I′] for J_{1}^{#}[r][I′].

Proposition 2.1. Consider the map J_{0}^{#}[r] → J_{1}^{#}[r, I′], as a map of finite free R -modules. Then

(i) This map is injective.

(ii) It identifies J_{0}^{#}[r] with J_{1}^{#}[r, I′][I].

(iii) Consider the surjection

(2.1)  J_{1}^{#}[r, I′]^{∗} → J_{0}^{#}[r]^{∗}

where (−)^{∗} = Hom(−, R). Then J_{0}^{#}[r]^{∗} is identified with J_{1}^{#}[r, I′]^{∗} ⊗_{Λ} Λ/I.

(iv) As a Λ-module J_{1}^{#}[r, I′]^{∗} is isomorphic to a direct sum of a free Λ-module with one copy of Λ/I.

Proof. (i) We have canonical identifications J_{0}^{#}[r] = H_{ét,c}^{1}(Y_{0/ℚ̄}, μ_{r}) and J_{0}^{#}[r]^{∗} = H_{ét}^{1}(Y_{0/ℚ̄}, R). Artin’s comparison theorem allows us to identify these with the singular cohomology of Y_{0}(ℂ) (with or without compact supports, and with coefficients in the same groups). The latter can be computed as a group cohomology. We thus have

(2.2)  J_{0}^{#}[r]^{∗} = H_{ét}^{1}(Y_{0/ℚ̄}, R) = H_{sing}^{1}(Y_{0}(ℂ), R) = H^{1}(Γ_{0}, R),

where we have taken into account the fact that Γ_{0} acts faithfully and freely on ℌ. Similarly we have

(2.3)  J_{1}^{#}[r]^{∗} = H_{ét}^{1}(Y_{1/ℚ̄}, R) = H_{sing}^{1}(Y_{1}(ℂ), R) = H^{1}(Γ_{1}, R) = H^{1}(Γ_{0}, \underline{Λ}).

The last equality is a consequence of Shapiro’s lemma in group cohomology. Here Γ_{0} acts on \underline{Λ} via Γ_{0}/Γ_{1} ≃ 𝔽_{p}^{×}. To fix ideas we choose the isomorphism sending a matrix γ ∈ Γ_{0} to a_{γ} mod p (we could have chosen d_{γ} mod p too).

It can be checked that the diamond action on J_{1}^{#}[r]^{∗} is translated to the diamond action on the coefficients in H^{1}(Γ_{0}, \underline{Λ}), so

(2.4)  J_{1}^{#}[r, I′]^{∗} = H^{1}(Γ_{0}, Λ).

Next, the map J_{1}^{#}[r, I′]^{∗} → J_{0}^{#}[r]^{∗} is the map H^{1}(Γ_{0}, Λ) → H^{1}(Γ_{0}, R) coming from the augmentation map Λ → R. This map is surjective, since Γ_{0} is free.

Dualizing, we get (i). Assertions (ii) and (iii) are again dual to each other. Their proof can be found in [dS1], Proposition 1.3. (The assumption N = 1 is not used there.)

Assertion (iv) was proved in [dS1], Proposition 2.8, Step 1, under the assumption that l ≥ 5. Once again, the assumption N = 1 was not used there in the proof, and the proof carries over to the case l = 3, since Γ_{0} has no 3-torsion.

The injectivity in (i) is one place where it is advantageous to work with the generalized Jacobians. For the ordinary Jacobians, the map is not injective. Its kernel consists of the r-torsion in the Shimura subgroup.

Notation. Write

(2.5)  V = J_{0}[r]^{∗}, V^{#} = J_{0}^{#}[r]^{∗}
(2.6)  V = J_{1}[r, I′]^{∗}, V^{#} = J_{1}^{#}[r, I′]^{∗}.

These are modules over H, equipped with a commuting action of Gal(ℚ̄/ℚ).

2.2. The local filtration on V and V^{#}. The restriction of V or V^{#} to the decomposition group at p admits a 2-step filtration. It is defined geometrically, and its graded pieces are unramified. These facts are not new. At least in the case of V, and in the l-adic setting, they are used in [M-W] and in papers of Wiles from the same period. For completeness we review them.

Recall that X_{0} has a model 𝔛_{0} over ℤ_{p}, an open subset of which, 𝔜_{0}, is the fine moduli space for the moduli problem ([Γ_{1}(N)], [Γ_{0}(p)]). For every ℤ_{p}-algebra T, the points of 𝔜_{0}(T) are isomorphism classes [E, P_{N}, H] of triples consisting of an elliptic curve E over T, a point P_{N} ∈ E[N](T) “of exact order N” (see [K-M], p.99), and a cyclic finite flat subgroup scheme H ⊂ E[p] of rank p (loc.cit. p. 100). The cuspidal subscheme ℭ_{0} = 𝔛_{0} − 𝔜_{0} is finite étale over ℤ_{p} ([K-M], theorem 10.12.2).

The scheme 𝔛_{0} is flat and proper over ℤ_{p}, and regular as a 2-dimensional scheme ([K-M], theorem 6.6.2, p.167). Via the map “forget the Γ_{0}(p) structure”, it is finite and flat of rank p + 1 over X_{1}(N)_{/ℤ_{p}}.

The special fiber 𝔛_{0/𝔽_{p}} is reduced and consists of 2 irreducible smooth components X_{0}^{μ} and X_{0}^{et}, which are the curves 𝔐(Γ_{1}(N), (1, 0)) and 𝔐(Γ_{1}(N), (0, 1)) of [K-M], proposition 13.4.4, p.408. The map X_{0}^{μ} → X_{1}(N)_{/𝔽_{p}} is an isomorphism. The map X_{0}^{et} → X_{1}(N)_{/𝔽_{p}} is purely inseparable of degree p. Denoting by Y_{0}^{μ} and Y_{0}^{et} the intersections of X_{0}^{μ} and X_{0}^{et} with 𝔜_{0}, for any 𝔽_{p}-algebra T, points of Y_{0}^{μ}(T) are isomorphism classes of triples [E, P_{N}, KerF], where E is an elliptic curve over T, P_{N} ∈ E[N](T) is a point of exact order N, and F is the Frobenius homomorphism F : E → E^{(p)} over T. The curve Y_{0}^{et} classifies isomorphism classes of triples [E, P_{N}, H], where E and P_{N} are as before, and H is a cyclic finite flat subgroup scheme of rank p such that if π : E → E/H is the canonical isogeny, E ≃ (E/H)^{(p)}, and Ker(^{t}π) = KerF. Over a perfect base we have then H = Ker(V : E → E^{(p^{−1})}), where V is the Verschiebung homomorphism.

The two components of the special fiber intersect transversally at the set S ⊂ X_{1}(N)(𝔽̄_{p}) of the super-singular points. The supersingular point [E, P_{N}, KerF] ∈ X_{0}^{μ}(𝔽̄_{p}) is glued to [E, P_{N}, KerV] ∈ X_{0}^{et}(𝔽̄_{p}) because for a supersingular curve KerF = KerV ([K-M], theorem 13.4.7).

If L is a finite extension of ℚ_{p}, and x = [E, P_{N}, H] ∈ 𝔜_{0}(O_{L}) is a point with ordinary reduction, then x meets the special fiber in X_{0}^{μ} if and only if the reduction of H is connected, and it meets the special fiber in X_{0}^{et} if and only if the reduction of H is étale.

The cusps C_{0} reduce injectively in the special fiber (since the cuspidal scheme is étale) and break up as the union of C_{0}^{μ} and C_{0}^{et}, the cusps in X_{0}^{μ} and X_{0}^{et}.

The automorphism w_{p} extends to 𝔛_{0} and interchanges the two components of the special fiber. On the set S it has the same affect as the absolute Frobenius σ ∈ Gal(𝔽̄_{p}/𝔽_{p}). Whenever necessary we shall identify X_{0}^{et} with X_{1}(N) using w_{p}, followed by the isomorphism X_{0}^{μ} ≃ X_{1}(N) obtained from the map “forget level p”.

Let M = ℤ[S] be the free abelian group on the set S of super-singular points in X_{1}(N)(𝔽̄_{p}). Let M_{0} the kernel of the degree homomorphism from M to ℤ. Let Hom(M_{0}, 𝔾_{m}) denote the (non-split) torus over 𝔽_{p} whose character group is M_{0} (with the Galois action coming from its action on S).^{3}

The semi-abelian variety J_{0}^{#} has a Néron (lft-)model J_{0}^{#} over ℤ_{p} (see [BLR], 10.2/2). Similarly J_{0} has a Néron model J_{0} over ℤ_{p}. The following proposition describes the structure of the connected components of the special fibers of these Néron models.

Proposition 2.2. (i) Let T be the (lft-)Néron model of the torus Hom(ℤ[C_{0}]_{0}, 𝔾_{m}). Then the sequence

(2.7)  0 → T^{0} → J_{0}^{#0} → J_{0}^{0} → 0

is an exact sequence of group schemes over ℤ_{p}.

(ii) There are exact sequences of group schemes over 𝔽_{p}

(2.8)  \begin{aligned}0 & → & Hom(M, 𝔾_{m}) & → & J_{0/𝔽_{p}}^{#0} & → & J_{0}^{et#} × J_{0}^{μ#} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Hom(M_{0}, 𝔾_{m}) & → & J_{0/𝔽_{p}}^{0} & → & J_{0}^{et} × J_{0}^{μ} & → & 0 \end{aligned}

where J_{0}^{μ} (resp. J_{0}^{et}) is the Jacobian of X_{0}^{μ} (resp. X_{0}^{et}), and similarly J_{0}^{μ#} (resp. J_{0}^{et#}) is the generalized Jacobian with respect to the cusps C_{0}^{μ} (resp. C_{0}^{et}).

Proof. (i) Since ℭ_{0} is an étale scheme over ℤ_{p}, the torus Hom(ℤ[C_{0}]_{0}, 𝔾_{m}) is split over ℚ_{p}^{nr}. It follows that the connected component of the special fiber of its Néron (lft-)model is simply Hom(ℤ[C_{0}]_{0}, 𝔾_{m/𝔽_{p}}), and to prove the exactness of the sequence in (i) we may pass to an unramified extension, and assume therefore that our torus is split. In this case the claim follows from the proof of Proposition 7 in [BLR], Section 10.1.

(ii) According to Raynaud’s theorem, and its generalization to the generalized Jacobian with respect to C_{0} (which is valid since C_{0} extends to an étale subscheme over ℤ_{p}, see [dS3]), J_{0/𝔽_{p}}^{0} (resp. J_{0/𝔽_{p}}^{#0}) represents P ic_{𝔛_{0}/𝔽_{p}}^{0} (resp. the identity component of the relative Picard functor with trivialization along C_{0}). Any (trivialized) line bundle whose class belongs to P ic_{𝔛_{0}/𝔽_{p}}^{0} gives upon restriction to the two components points in the corresponding (generalized) Jacobians. The kernel of this map consists of line bundles which are trivial on each of the two components. Choosing trivializations and comparing them on S we get the homomorphism from M to 𝔾_{m}. If the trivializations are not given, they can be modified by a scalar factor on each component, and the resulting homomorphism is well-defined only on M_{0}. This establishes the exactness of the two rows in (ii), and the commutativity of the diagram is clear.

3A more canonical description of M_{0} is as the dual graph of 𝔛_{0/𝔽_{p}}. Thus, the action of w_{p} on M_{0} is via s ↦ −σ(s) (s ∈ S), although on S it acts like σ, because w_{p} interchanges the two components of the special fiber. One can check that this action is compatible with the inclusion of Hom(M_{0}, 𝔾_{m}) in the connected component of the special fiber of the Néron model of J_{0} over ℤ_{p} (see Proposition 2.2 below), and with the action of w_{p} induced on that connected component from the (Picard) action on J_{0}.

From the proposition we obtain the following exact sequences

(2.9) \begin{aligned}0 & → & Hom(M, μ_{r}) & → & J_{0}^{#0}(𝔽̄_{p})[r] & → & J_{0}^{et#}[r] × J_{0}^{μ#}[r] & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Hom(M_{0}, μ_{r}) & → & J_{0}^{0}(𝔽̄_{p})[r] & → & J_{0}^{et}[r] × J_{0}^{μ}[r] & → & 0 \end{aligned} .

Let us also remark that the two arrows in

(2.10)  J_{0}(ℚ_{p}^{nr})[r] ⟵ J_{0}(ℤ_{p}^{nr})[r] ⊃ J_{0}^{0}(ℤ_{p}^{nr})[r] → J_{0}^{0}(𝔽̄_{p})[r]

are isomorphisms (and similarly for the generalized Jacobians). For the one on the left this follows from the universal property of Néron models. For the one on the right note that J_{0}^{0}[r] is an étale group scheme, because r is relatively prime to p.

Definition 2.1. Let J_{0}[r]^{sub} (resp. J_{0}^{#}[r]^{sub}) denote the subgroup of J_{0}^{0}(ℤ_{p}^{nr})[r] (resp. J_{0}^{#0}(ℤ_{p}^{nr})[r]) consisting of the elements which map to 0 in J_{0}^{μ} (resp. in J_{0}^{μ#}). Let J_{0}[r]^{quot} = J_{0}[r]/J_{0}[r]^{sub} and similarly for the generalized Jacobians. Dualizing, we obtain exact sequences

(2.11)  \begin{aligned}0 & → & U & → & V & → & W & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & U^{#} & → & V^{#} & → & W^{#} & → & 0 \end{aligned}

where U = (J_{0}[r]^{quot})^{∗}, W = (J_{0}[r]^{sub})^{∗}, and similarly for the generalized Jacobians.

The vertical arrows are all injections. This is because J_{0}^{#}[r]^{sub} maps surjectively to J_{0}[r]^{sub}. The cokernels of the vertical maps fit into the exact sequence

(2.12)  0 → ℤ[C_{0}^{μ}]_{0} ⊗ μ_{r}^{−1} → ℤ[C_{0}]_{0} ⊗ μ_{r}^{−1} → ℤ[C_{0}^{et}] ⊗ μ_{r}^{−1} → 0.

The modules U, V, W, U^{#}, V^{#} and W^{#} are stable under the local Galois group at p. We shall now see that they are also stable under H. We shall check it for the operator U_{p}. Similar (and easier) computations apply to the other Hecke operators. Let x = [E, P_{N}, H] ∈ 𝔜_{0}(O_{L}) for some finite extension L of ℚ_{p}. Then in Div(𝔜_{0}(O_{L})) we have

(2.13)  U_{p}^{∗}(x) = ∑[E_{i}, P_{N,i}, H_{i}]

where (enlarging L if necessary) each x_{i} = (E_{i}, P_{N,i}, H_{i}) is defined over O_{L} and there is a cyclic isogeny λ_{i} of degree p defined over O_{L} carrying (E_{i}, P_{N,i}, H_{i}) to (E, P_{N}, H). In particular, the scheme theoretic intersection of H_{i} and Kerλ_{i} is 0, and H = λ_{i}(H_{i}) = λ_{i}(E_{i}[p]). If the reduction x̄ of x is ordinary, and x̄ belongs to X_{0}^{et}(𝔽̄_{p}), then each H_{i} must have an étale reduction, and so x̄_{i} belongs to X_{0}^{et}(𝔽̄_{p}). On the other hand if x̄ belongs to S, clearly every x̄_{i} belongs to S. It follows that if

δ = ∑ n_{i}x_{i} ∈ Div^{0}(𝔜_{0}(O_{L}))

is a divisor of degree 0 supported away from the cuspidal scheme, whose reduction is supported on X_{0}^{et}, then so is U_{p}^{∗}δ. Any point in J_{0}[r]^{sub} is the class of (the generic fiber of) a divisor of this form, and vice versa. Thus U_{p} = U_{p}^{∗} preserves J_{0}[r]^{sub}.

(Note that if we used the Albanese functoriality we should have defined J_{0}[r]^{sub} using J_{0}^{μ} instead of J_{0}^{et}.) The proof for the generalized Jacobian is identical. Compare the arguments here with [M-W], chapter 2, section 9, in particular Propositions 2 and 3 there.

2.3. The local filtration on V and V^{#}. At the Γ_{1} level the geometry is a little more complicated, and one has to go to ℚ_{p}(ζ_{p}), or at least to its subfield K of degree r over ℚ_{p}, to get a good understanding of the picture. It turns out that there is a similar filtration there, which is again stable under the local Galois group and Hecke. It is even stable under the full Gal(ℚ̄_{p}/ℚ_{p}), but the graded pieces are not unramified anymore. One of them inherits a “geometric inertia action” of Gal(K^{nr}/ℚ_{p}^{nr}) via the μ_{r}-diamond operators.

It is well-known that J_{1} acquires semi-stable reduction over ℚ_{p}(ζ_{p}). However, we want to be able to work over a field of absolute index of ramification strictly less than p − 1. This is possible because J_{1}[r, I′] is in fact the r-torsion of an abelian variety which acquires semi-stable reduction over the field K mentioned above. We now explain this point.

Let X_{1} → X_{2} → X_{0} be the intermediate cover (defined over ℚ) with Gal(X_{2}/X_{0}) cyclic of order r. Let J_{2} be its Jacobian variety. In

(2.14)  J_{0} → J_{2} → J_{1}

the kernel of J_{2} → J_{1} is a μ-type subgroup, of order (p − 1)/r. Since r and (p − 1)/r are relatively prime, J_{2}[r] ⊂ J_{1}[r], and in fact we claim that

(2.15)  J_{2}[r] = J_{1}[r, I′],

so V = J_{2}[r]^{∗}. Indeed, quite generally, if X′ → X is an unramified cyclic covering of curves over an algebraically closed field k of characteristic 0, or at least relatively prime to the order of Gal(X′/X) = Γ, and J and J′ are their Jacobians, we have an exact sequence

(2.16)  0 → Hom(Γ, k^{×}) → J(k) → J′(k)^{Γ} → 0.

This is proved using Galois cohomology of cyclic groups.

Lemma 2.3. The abelian variety J_{2} admits semi-stable reduction over the unique subfield K of ℚ_{p}(ζ_{p}) of degree r over ℚ_{p}.

Proof. Let q be any prime not dividing p(p − 1)/r (for example, we could take q = l). Then by the discussion above we have for the rational Tate modules

(2.17)  V_{q}J_{2} = V_{q}J_{1}[I′].

Let L = ℚ_{p}(ζ_{p}) and let I_{L} be the inertia subgroup of Gal(L̄/L). Then since J_{1} acquires semistable reduction over L, V = V_{q}J_{1} has a subspace W = V^{I_{L}} such that I_{L} = Gal(L̄/L^{nr}) acts trivially on W and V/W. Since L is normal over ℚ_{p}, W is stable under the full inertia group at p, and it can be computed that the action of Gal(L̄/ℚ_{p}^{nr}) on W and on V/W factors through the diamond operators, i.e. through the canonical isomorphism of Gal(L^{nr}/ℚ_{p}^{nr}) ≃ Gal(L/ℚ_{p}) ≃ 𝔽_{p}^{×} with the group of diamond operators. It follows that Gal(K̄/K^{nr}) acts trivially on the subspace W [I′] ⊂ V [I′] = V_{q}J_{2} and on the quotient V_{q}J_{2}/W [I′], hence by the criterion for semi-stable reduction ([BLR], 7.4/6), J_{2} acquires semi-stable reduction over K.

Alternatively, the above lemma follows from the semi-stable, regular model of X_{2} over O_{K} which we describe ahead, and Raynaud’s theorem.

Recall that for any finite extension F of ℚ_{p} and any abelian variety A over F we denote by N (A, O_{F}) the Néron model of A over O_{F} . Thus, for example, J_{0} = N (J_{0}, ℤ_{p}). Like in the case of X_{0}, the curve X_{2/K} has a regular, flat and proper model 𝔛_{2/O_{K}}, the special fiber of which is reduced and consists of two smooth irreducible components X_{2/𝔽_{p}}^{ét} and X_{2/𝔽_{p}}^{μ} (Igusa curves) which intersect transversally at the supersingular points S. In the language of [K-M], 𝔛_{2} contains an open set 𝔜_{2} which is the fine moduli space for the moduli problem ([Γ_{1}(N)], ([bal.Γ_{1}(p)]/μ_{r′} × μ_{r′})^{can}) (see [K-M], theorem 10.12.2, p.326 and the remark following it). The scheme of cusps ℭ_{2} = 𝔛_{2} − 𝔜_{2} is finite étale over O_{K}.

The map X_{2} → X_{0}, which is Galois unramified of degree r, extends over O_{K} to a map 𝔛_{2} → 𝔛_{0/O_{K}} , which on the special fiber induces maps X_{2}^{ét} → X_{0}^{ét} and X_{2}^{μ} → X_{0}^{μ}. These two maps are totally ramified over the super-singular points, but are étale over the ordinary locus. They are interchanged by the “involution” w_{p}.

Let J_{2}^{ét} and J_{2}^{μ} be the Jacobians of the curves X_{2/𝔽_{p}}^{ét} and X_{2/𝔽_{p}}^{μ}. Then by Raynaud’s theorem we have for any field F containing K a diagram of group schemes over the residue field κ_{F}

(2.18)  \begin{aligned}0 & → & Hom(M_{0}, 𝔾_{m}) & → & N (J_{0}, O_{F})_{/κ_{F}}^{0} & → & J_{0}^{ét} × J_{0}^{μ} & → & 0 \\ & & || & & ⋂ & & ⋂ \\ 0 & → & Hom(M_{0}, 𝔾_{m}) & → & N (J_{2}, O_{F})_{/κ_{F}}^{0} & → & J_{2}^{ét} × J_{2}^{μ} & → & 0 \end{aligned}

with exact rows. The injectivity of the middle vertical arrow is deduced from the injectivity of the right vertical arrow, which is a consequence of the fact that the coverings X_{2}^{ét} → X_{0}^{ét} and X_{2}^{μ} → X_{0}^{μ} inducing the maps between the Jacobians are totally ramified over S.

A similar analysis applies to the generalized Jacobians (always with respect to the cusps). Thanks to the fact that ℭ_{2} is finite étale over O_{K}, the proposition analogous to 2.2 is valid. Note that there are precisely r cusps in C_{2} above any given cusp of C_{0}. Thus we have exact sequences

(2.19) \begin{aligned}0 & → & Hom(M, μ_{r}) & → & J_{2}^{#0}(𝔽̄_{p})[r] & → & J_{2}^{et#}[r] × J_{2}^{μ#}[r] & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Hom(M_{0}, μ_{r}) & → & J_{2}^{0}(𝔽̄_{p})[r] & → & J_{2}^{et}[r] × J_{2}^{μ}[r] & → & 0 \end{aligned}

where we have denoted by J_{2} = N (J_{2}, O_{K}) etc. We can now define the filtration on V and V^{#}.

Definition 2.2. Let J_{2}[r]^{sub} (resp. J_{2}^{#}[r]^{sub}) denote the subgroup of J_{2}^{0}(O_{K}^{nr})[r] (resp. J_{2}^{#0}(O_{K}^{nr})[r]) consisting of the elements which map to 0 in J_{2}^{μ} (resp. in J_{2}^{μ#}). Let J_{2}[r]^{quot} = J_{2}[r]/J_{2}[r]^{sub} and similarly for the generalized Jacobians. Dualizing, we obtain exact sequences

(2.20)  \begin{aligned}0 & → & U & → & V & → & W & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & U^{#} & → & V^{#} & → & W^{#} & → & 0 \end{aligned}

where U = (J_{2}[r]^{quot})^{∗}, W = (J_{2}[r]^{sub})^{∗}, U^{#} = (J_{2}^{#}[r]^{quot})^{∗}, W^{#} = (J_{2}^{#}[r]^{sub})^{∗}.

As in the case of J_{0}, the vertical maps are injections, and the six modules are stable under the local Galois group of K, Gal(K̄/K), and under the Hecke algebra H. As we shall see later, they are even stable under Gal(ℚ̄_{p}/ℚ_{p}).

2.4. Structure of the filtered deformation. It will be important to analyze the filtered deformation V^{#} → V^{#} in the way it was done, without the filtration, in Proposition 2.1. We shall show that this surjective map induces surjective maps on the sub/quotient modules, and that the structure as Λ-modules is as “clean” as possible, given Proposition 2.1(iv). These tasks turn out to be somewhat delicate, and we shall break them to several steps. We shall then recollect all the information we have gathered in one theorem.

Assume that F is a field containing K, and that J_{0}[r] and J_{2}[r] are all F -rational. If Σ denotes the Shimura subgroup of J_{0},then Σ[r] = Ker(J_{0} → J_{2}). Consider the diagram

(2.21) \begin{aligned}0 & → & 0 & → & N (J_{0}, O_{F})^{0}(O_{F})[r] & → & N (J_{2}, O_{F})^{0}(O_{F})[r] \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Σ [r] & → & J_{0}[r] & → & J_{2}[r] \end{aligned}  .

with exact rows.

Let

(2.22)  Ψ = (J_{0}[r]/Σ[r] ∩ N (J_{2}, O_{F})^{0}(O_{F})[r]) /N (J_{0}, O_{F})^{0}(O_{F})[r].

Then Ψ embeds in N (J_{2}, O_{F})_{/κ_{F}}^{0} /N (J_{0}, O_{F})_{/κ_{F}}^{0} (note that r torsion in the Néron model maps injectively under the specialization to the special fiber because p does not divide r). On the other hand Ψ is a submodule of J_{0}[r]/ (Σ[r] + N (J_{0}, O_{F})^{0}(O_{F})[r]) , hence

(2.23)  Ψ ⊂ Φ(J_{0}, O_{F})/Σ[r]

and in particular we have the identity U_{p} = −w_{p} on Ψ.

Let H be the subgroup of Ψ which maps to J_{2}^{ét}/J_{0}^{ét} × {0}. Note that H is stable under U_{p} because J_{2}^{ét}/J_{0}^{ét} × {0} is stable under U_{p}, but H can not be stable under w_{p} because it interchanges J_{2}^{ét}/J_{0}^{ét} with J_{2}^{μ}/J_{0}^{μ}. However, we have noticed that U_{p} = −w_{p} on H. This proves that H = 0.

Proposition 2.4. Inside J_{1}[r], we have

(2.24)  Im J_{0}[r] ∩ J_{1}[r, I′]^{sub} = Im J_{0}[r]^{sub}.

Proof. As noted before J_{1}[r, I′] = J_{2}[r]. Let F be any field containing K so that J_{2}[r] and J_{0}[r] are rational over F. Let x_{F} ∈ J_{2}[r]^{sub} be the image of y_{F} ∈ J_{0}[r]. Extend x_{F} and y_{F} to sections x and y of the Néron models over O_{F} . Then x ∈ N (J_{2}, O_{F})^{0}(O_{F}) maps to J_{2}^{ét} × {0} so by the last remark before the proposition, the image of y in Ψ lies in H = 0, hence we can change y by an element of Σ[r] to bring it into the connected component. We may therefore assume that y ∈ N (J_{0}, O_{F})^{0}(O_{F}). It is now clear that if the intersection of x with the special fiber projects to J_{2}^{ét}, then so does the intersection of y with the special fiber.

Proposition 2.5. In the following diagram

(2.25)  \begin{aligned}0 & → & J_{0}^{#}[r]^{sub} & → & J_{0}^{#}[r] & → & J_{0}^{#}[r]^{quot} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & J_{1}^{#}[r, I′]^{sub} & → & J_{1}^{#}[r, I′] & → & J_{1}^{#}[r, I′]^{quot} & → & 0 \end{aligned}

all the vertical arrows are injective.

Proof. (Compare [dS1], lemma 2.7, p.85-86). We only have to prove that

J_{0}^{#}[r] ∩ J_{1}^{#}[r, I′]^{sub} ⊂ J_{0}^{#}[r]^{sub}.

Let x ∈ J_{0}^{#}[r] ∩ J_{1}^{#}[r, I′]^{sub}. By the previous proposition, and after modifying x by an element of J_{0}^{#}[r]^{sub}, we may assume that x lies in the kernel of the map to J_{1}[r, I′]^{sub}, i.e.

(2.26)  x ∈ Hom(ℤ[C_{2}^{ét}], μ_{r}).

Here we made use of the diagram (over 𝔽_{p}) (2.27)

0 → Hom(ℤ[C_{2}]_{0}, μ_{r})  → J_{2}^{#}[r]^{0}  → J_{2}[r]^{0}  → 0
↓  ↓  ↓
0 → Hom(ℤ[C_{2}^{ét}]_{0} × ℤ[C_{2}^{μ}]_{0}, μ_{r}) → J_{2}^{ét#}[r] × J_{2}^{μ#}[r] → J_{2}^{ét}[r] × J_{2}^{μ}[r] → 0

and the exact sequence

(2.28) 0 → Hom(ℤ[C_{2}^{ét}], μ_{r}) → Hom(ℤ[C_{2}]_{0}, μ_{r}) → Hom(ℤ[C_{2}^{μ}]_{0}, μ_{r}) → 0.

Consider now the diagram (2.29)

0  → 0  → Σ[r]  −− −−
↓  ↓  ↓
0  → Hom(ℤ[C_{0}]_{0}, μ_{r}) → J_{0}^{#}[r] → J_{0}[r]  → 0
↓  ↓  ↓
0  → Hom(ℤ[C_{2}]_{0}, μ_{r}) → J_{2}^{#}[r] → J_{2}[r]  → 0
↓  ↓  ↓
−− → Hom(M_{0}, μ_{r})  → X  → J_{2}[r]/ Im J_{0}[r] → 0

(obtained from the snake lemma) with M_{0} = Ker(ℤ[C_{2}]_{0} → ℤ[C_{0}]_{0}). The image of Σ[r] in Hom(M_{0}, μ_{r}) can be computed explicitly (use the action of the diamond operators Δ_{r} on Σ[r] = Hom(Δ_{r}, μ_{r})). But in any case it is invariant under w_{p}, because Σ is invariant under it. In particular, it intersects trivially Hom(M^{ét}, μ_{r}) ⊂ Hom(M_{0}, μ_{r}), where

(2.30)  M^{ét} = Ker(ℤ[C_{2}^{ét}] → ℤ[C_{0}^{ét}]),

because w_{p} interchanges the ét and the μ cusps. Similarly we have another diagram (2.31)

\begin{aligned}& & 0 & → & 0 & → & 0 & −− & −− \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Hom(ℤ[C_{0}^{ét}], μ_{r}) & → & J_{0}^{#}[r]^{sub} & → & J_{0}[r]^{sub} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Hom(ℤ[C_{2}^{ét}], μ_{r}) & → & J_{2}^{#}[r]^{sub} & → & J_{2}[r]^{sub} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ −− & → & Hom(M^{ét}, μ_{r}) & → & Y & → & J_{2}[r]^{sub}/J_{0}[r]^{sub} & → & 0 \end{aligned} .

Now x ∈ Hom(ℤ[C_{2}^{ét}], μ_{r}) ∩ J_{0}^{#}[r]. Chasing the diagrams we see that its image in Hom(M_{0}, μ_{r}) lies in

(2.32)  Hom(M^{ét}, μ_{r}) ∩ Σ[r] = {0}

and therefore x ∈ Hom(ℤ[C_{0}^{ét}], μ_{r}) ⊂ J_{0}^{#}[r]^{sub}.

2.5. The action of the full decomposition group. We wish to show that J_{1}^{#}[r, I′]^{sub} and J_{1}[r, I′]^{sub} are stable under Gal(ℚ̄_{p}/ℚ_{p}), and to compute its action. It will follow that the six modules in Definition 2.2 are acted upon by the full decomposition group, and not only by Gal(K̄/K). For that purpose consider the model 𝔛′_{2/ℤ_{p}} of X_{2} which is associated with the moduli problem ([Γ_{1}(N)], [Γ_{1}(p)]/μ_{r′}). See [K-M], theorem 5.5.1 for the representability. 𝔛′_{2} is proper and flat over ℤ_{p}, regular as a 2-dimensional scheme, but unlike the model 𝔛_{2} over the ring O_{K}, its special fiber is non-reduced. In fact, by [K-M], theorem 13.5.4 on p.416, 𝔛′_{2/𝔽_{p}} is the union of two irreducible components X_{2}^{′μ} and X_{2}^{′et}, intersecting transversally at the set S of supersingular points. The curve X_{2}^{′μ} is non-reduced, and its multiplicity as an abstract curve is r. When we reduce it, (X_{2}^{′μ})^{red} becomes equal to X_{1}(N). The curve X_{2}^{′et} is reduced, and it is an Igusa curve. In fact it is isomorphic to X_{2}^{et}.

The Néron model N (J_{2}, ℤ_{p}) can be computed by Raynaud’s theorem from this model. As 𝔛′_{2} is not semi-stable, the connected component of the special fiber of N (J_{2}, ℤ_{p}) has a unipotent part. In fact N (J_{2}, ℤ_{p})_{/𝔽_{p}}^{0} will be an extension of J_{2}^{′et} × J_{2}^{′μ} by an affine algebraic group, which is itself the extension of the torus Hom(M_{0}, 𝔾_{m}) by a unipotent group. This unipotent part will contribute nothing to the r-torsion. From the map

(2.33)  N (J_{2}, ℤ_{p}) ×_{ℤ_{p}} O_{K} → N (J_{2}, O_{K}) = J_{2}

(obtained by the universal property of Néron models) we shall get a map

N (J_{2}, ℤ_{p})_{/𝔽_{p}}^{0} → J_{2/𝔽_{p}}^{0},

and the pre-image of J_{2}^{′et}[r] × {0} = J_{2}^{et}[r] × {0} under the map

(2.34)  N (J_{2}, ℤ_{p})^{0}(𝔽̄_{p})[r] → J_{2}^{′et}[r] × J_{2}^{′μ}[r]

will map isomorphically onto J_{2}[r]^{sub}, both being extensions of J_{2}^{et}[r] by Hom(M_{0}, μ_{r}). Since this pre-image is a subgroup of N (J_{2}, ℤ_{p})^{0}(ℤ_{p}^{nr}) which is stable under the full decomposition group, this proves our claim, and at the same time shows that J_{1}[r, I′]^{sub} is unramified for the full decomposition group. The same holds for the generalized Jacobian.

To compute the action of σ, the arithmetic Frobenius at p, let us start with a non-cuspidal point x = [E, P_{N}, P_{p}] ∈ X_{1}^{et}(𝔽̄_{p}) where P_{N} is a point of exact order N, and P_{p} a point of exact order p which generates Ker(V : E → σ^{−1}E) as a Cartier divisor. See the discussion in Section 2.2 about the moduli problem represented by X_{0}^{et}. The difference between X_{1}^{et} and X_{2}^{et} is that in the latter P_{p} is only taken modulo the action of μ_{r′}. To simplify the notation, we shall do the computations at the level of X_{1}. Now

(2.35)  U_{p}^{∗}(x) = ∑_{i=1}^{p}[E/⟨P_{p}⟩, ⟨p⟩_{N}^{−1}P_{N} mod⟨P_{p}⟩, Q_{p,i} mod⟨P_{p}⟩]

where Q_{p,i} ∈ E(𝔽̄_{p}) is a solution of pQ_{p,i} = P_{p}. There are p^{2} solutions of this equation (taken with multiplicities) and modulo ⟨P_{p}⟩ there are p. If x was an ordinary point then ⟨P_{p}⟩ is a reduced subgroup, and all the Q_{p,i} are equal modulo ⟨P_{p}⟩. If x was in S then ⟨P_{p}⟩ = KerF = KerV and of course all the Q_{p,i} are 0. Thus in any case the sum on the right hand side consists of p equal terms. Applying σ to any of them yields

(2.36)  [E/E[p], ⟨p⟩_{N}^{−1}P_{N} mod E[p], Q_{p,i} mod E[p]] = [E, P_{N}, P_{p}] = x.

Since a point of J_{1}[r, I′]^{sub} or J_{1}^{#}[r, I′]^{sub} is represented by a divisor of degree 0 with support on x′s as above, and since the U_{p} action is by Picard functoriality we arrive at the formula

(2.37)  U_{p}(x) = pσ^{−1}(x).

Proposition 2.6. (Compare [dS1], Proposition 3.2) The modules U^{#} and W^{#} are stable under Gal(ℚ̄_{p}/ℚ_{p}). The action of the decomposition group on W^{#} is through the unramified character ϕ sending the arithmetic Frobenius σ to ϕ(σ) = U_{p} (and therefore U_{p} acts invertibly on W^{#}).

Proof. This follows from the above discussion, the observation that p ≡ 1 mod r, and the conventions about Galois and Hecke actions after dualizing.

The next proposition shows that the action of the decomposition group on U^{#} is ramified.

Proposition 2.7. Let ω : Gal(ℚ̄_{p}/ℚ_{p}) → (ℤ/Nℤ)^{×} × 𝔽_{p}^{×} ⊂ H be the character which associates to τ the diamond operator

(2.38)  ω(τ) = ⟨χ(τ)⟩_{Np}

where χ(τ) is determined by τ(ζ) = ζ^{χ(τ)} for ζ ∈ μ_{Np}(ℚ̄_{p}). Then Gal(ℚ̄_{p}/ℚ_{p}) acts on U^{#} via the character ωϕ^{−1}.

Proof. A proof can be modeled on the proof of proposition 3.7 in [dS1], paying attention to the Γ_{1}(N) level structure, which was missing there. Alternatively (for U at least), we can use the twisted Weil pairing of [M-W2], p.243, defined by

(2.39)  [x, y] = ⟨x, w_{Np}y⟩

for x, y ∈ J_{1}[r, I′]. (Here w_{Np} = w_{Np}^{∗} = w_{Np∗} because w_{Np}^{2} = 1.) This twisted pairing has two advantages over the original one: (a) The transpose of U_{p}^{∗} (or, for that matter, any T ∈ H) under it, is U_{p}^{∗} itself (resp. T itself), and not U_{p∗}, which may not be in H. (b) J_{1}[r, I′]^{sub} is maximal isotropic for it, and therefore set in perfect duality with J_{1}[r, I′]^{quot}. Indeed, by Grothendieck’s “Théorème d’orthogonalité” ([SGA7], exposé IX, Thm. 2.4) and by the fact that w_{Np} preserves J_{1}^{0}[r, I′], the restriction of [·, ·] to J_{1}^{0}[r, I′] factors through the projection to J_{2}^{et}[r] × J_{2}^{μ}[r], but since w_{Np} interchanges the two components of the special fiber, if both x and y project to J_{2}^{et}[r] (i.e. belong to J_{1}[r, I′]^{sub}) then [x, y] = 0.

We can now compute for x ∈ J_{1}[r, I′]^{sub} and y ∈ J_{1}[r, I′]^{quot} and any τ in the local Galois group

[x, y] = τ[x, y] = ⟨τx, τw_{Np}y⟩ = ⟨τx, w_{Np}⟨χ(τ)⟩_{Np}τy⟩ =
(2.40)  = [ϕ^{−1}(τ)x, ω(τ)τy] = [x, ϕ^{−1}ω(τ)τy]

so by non-degenracy, τy = ω^{−1}ϕ(τ)y. Dualizing we see that τ acts on U via ωϕ^{−1}(τ). Observe that implicit in the definition of w_{Np} there is a choice of an Np-th root of unity ζ, hence τ does not commute with w_{Np}, but rather satisfies τ ∘ w_{Np} = w_{Np} ∘ ⟨χ(τ)⟩_{Np}^{∗} ∘ τ.

2.6. Structure over Λ. Recall the identifications

(2.41)  \begin{aligned}J_{1}^{#}[r, I′]^{∗} & = & H^{1}(Γ_{1}, R)_{I′} & = & H^{1}(Γ_{0}, Λ) \\ ↓ & & ↓ Cor & & ↓ can. \\ J_{0}^{#}[r]^{∗} & = & H^{1}(Γ_{0}, R) & = & H^{1}(Γ_{0}, R) \end{aligned}

from section 2.1, where Cor is the corestriction map, and can the canonical map induced by the projection Λ → R. Let γ_{0}, γ_{1}, …, γ_{2m} be a set of generators of Γ_{0} as a free group, such that γ_{i} ∈ Γ_{1} for i ≥ 1. (The rank of Γ_{0} is odd. In fact

(2.42)  2m + 1 = 2genus(X_{0}) + #C_{0} − 1

and #C_{0} is even.) Then γ_{0} projects to a generator of the cyclic group Γ_{0}/Γ_{1} ≃ 𝔽_{p}^{×}. Using the γ_{i} we write

(2.43)  H^{1}(Γ_{0}, R) = Hom(Γ_{0}, R) ≃ R^{2m+1}, h ↦ (h(γ_{0}), …, h(γ_{2m})).

Similarly, since Γ_{0} is free,

(2.44)  Z^{1}(Γ_{0}, Λ) ≃ Λ^{2m+1}, z ↦ (z(γ_{0}), …, z(γ_{2m})),

and under this isomorphism B^{1}(Γ_{0}, Λ) ≃ I ×(0)×···×(0). This gives a realization of the isomorphism encountered in (2.1)(iv), which depends on the choice of the γ_{i}. Let v_{0}, …, v_{2m} denote the elements of J_{1}^{#}[r, I′]^{∗} = H^{1}(Γ_{0}, Λ) ≃ (Λ/I)⊕Λ⊕···⊕Λ which correspond under the isomorphism to (1, 0, …, 0), …, (0, …, 1). Let u_{i} denote the images of v_{i} in J_{0}^{#}[r]^{∗} = H^{1}(Γ_{0}, R) ≃ R^{2m+1}.

Note that while the line Rv_{0} depends on the choice of the γ_{i}, the line Ru_{0} does not, because under any change of the γ_{i}, v_{0} goes over to ∑ λ_{i}v_{i} with λ_{0} ∈ R^{×} and λ_{i} ∈ Λ[I] ⊂ I for i ≥ 1. This “canonical line” Ru_{0} ⊂ J_{0}^{#}[r]^{∗} = H^{1}(Γ_{0}, R) simply consists of homomorphism from Γ_{0} to R factoring through Γ_{0}/Γ_{1}. Now giving u ∈ J_{0}^{#}[r]^{∗} is the same as giving a cyclic covering X → X_{0} defined over ℚ̄, unramified outside the cusps, of degree dividing r, together with a distinguished generator of Gal(X/X_{0}). Such a u comes from J_{0}[r]^{∗} ⊂ J_{0}^{#}[r]^{∗} if and only if it is also unramified at the cusps. Consider in particular the covering X_{2} → X_{0} which is of the above type. It corresponds to the line Ru_{0}, and since it is unramified also at the cusps, we conclude that Ru_{0} ⊂ J_{0}[r]^{∗}.

We claim that Ru_{0} is mapped injectively into W = (J_{0}[r]^{sub})^{∗}, hence also into W^{#}. A-fortiori, Rv_{0} will map injectively into W^{#}. This was proved in the case of prime conductor ([dS1], p.87), by an argument on Mumford curves. Here we need a different argument. What we have to show is that u_{0}(J_{0}[r]^{sub}) is of order r. By the non-degeneracy of the Weil pairing on J_{0}[r] there is a unique ũ_{0} ∈ J_{0}[r] such that

(2.45)  u_{0}(x) = ⟨ũ_{0}, x⟩

for every x ∈ J_{0}[r], where ⟨·, ·⟩ denotes the Weil e_{r}-pairing, followed by a fixed isomorphism μ_{r} ≃ R. Unwinding the definitions we see the following. The homomorphism u_{0} corresponds to the cyclic covering X_{2} → X_{0} and a generator σ of Gal(X_{2}/X_{0}). Let ζ be the primitive r th root of unity corresponding to 1 under the fixed isomorphism μ_{r} ≃ R. Kummer theory then establishes the existence of a unique g ∈ ℚ̄(X_{0})^{×}/ℚ̄(X_{0})^{×r} such that ℚ̄(X_{2}) = ℚ̄(X_{0})(g^{1/r}), and such that σ(g) = ζg. Since the covering is everywhere unramified the divisor div(g) = rD for some divisor of degree 0 D, and ũ_{0} = [D] is the class of this D in P ic^{0}(X_{0}) = J_{0}. It follows from this interpretation that ũ_{0} lies in the Shimura subgroup, and we have noticed before that the Shimura subgroup intersects the connected component of the Néron model trivially, and therefore maps injectively into Φ(J_{0}, ℤ_{p}). We now invoke Grothendieck’s “Théorème d’orthogonalité” ([SGA7], exposé IX, Thm. 2.4), which implies that Hom(M_{0}, μ_{r}) is orthogonal under the Weil pairing to J_{0}^{0}(𝔽̄_{p})[r], and is therefore dual (by counting) to J_{0}[r]/J_{0}^{0}(𝔽̄_{p})[r]. In particular, it follows that for a suitable x ∈ Hom(M_{0}, μ_{r}) ⊂ J_{0}[r]^{sub}, ⟨ũ_{0}, x⟩ is of order r, as desired.

Observe that

(2.46)  genus(X_{0}) = 2genus(X_{0}^{et}) + #(S) − 1

so that

(2.47)  m + 1 = 2genus(X_{0}^{et}) + #(C_{0}^{et}) + #(S) − 1 = rank_{R}(J_{0}^{#}[r]^{sub})^{∗}.

We choose our generators of Γ_{0} in such a way so that u_{0}, u_{1}…. , u_{m} project to a basis of W^{#} and u_{m+1}, …, u_{2m} form a basis of U^{#}. The same counting argument as in [dS1], Proposition 2.8, Step 3 (p.87-88), now shows that

(2.48)  W^{#} ≃ Λ/I ⊕ Λ^{m}, U^{#} ≃ Λ^{m},

and the sequence 0 → U^{#} → V^{#} → W^{#} → 0 splits as a sequence of Λ-modules. We summarize the discussion of this chapter in the following theorem.

Theorem 2.8. In the commutative diagram

(2.49)  \begin{aligned}0 & → & U^{#} & → & V^{#} & → & W^{#} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & U^{#} & → & V^{#} & → & W^{#} & → & 0 \end{aligned} ,

where U^{#} = (J_{1}^{#}[r, I′]^{quot})^{∗}, V^{#} = J_{1}^{#}[r, I′]^{∗}, W^{#} = (J_{1}^{#}[r, I′]^{sub})^{∗} and U^{#} = (J_{0}^{#}[r]^{quot})^{∗}, V^{#} = J_{0}^{#}[r]^{∗}, W^{#} = (J_{0}^{#}[r]^{sub})^{∗}

• The rows are exact and the vertical arrows are surjective.

• All six modules are stable under H and under Gal(ℚ̄_{p}/ℚ_{p}). The modules U^{#}, W^{#}, and W^{#} (but not U^{#}) are unramified for the action of the decomposition group. In fact Gal(ℚ̄_{p}/ℚ_{p}) acts on W^{#} through the character ϕ, and on U^{#} through ωϕ^{−1}, where ϕ(σ) = U_{p} if σ is the arithmetic Frobenius in Gal(ℚ_{p}^{nr}/ℚ_{p}), and ω(τ) = ⟨χ(τ)⟩_{Np} is the Teichmuller character.

• As a short exact sequence Λ-modules, the top row splits, and

(2.50)  W^{#} ≃ Λ/I ⊕ Λ^{m}, U^{#} ≃ Λ^{m}

The bottom row is identified with (the top row)⊗_{Λ}R.

• The map W^{#} → W^{#} is dual to the inclusion J_{0}^{#}[r]^{sub} ↪ J_{1}^{#}[r, I′]^{sub}. This inclusion fits in the following exact sequence

(2.51)  \begin{aligned}0 & → & Hom(M, μ_{r}) & → & J_{0}^{#}[r]^{sub} & → & J_{0}^{et#}[r] & → & 0 \\ & & || & & ↓ & & ↓ \\ 0 & → & Hom(M, μ_{r}) & → & J_{1}^{#}[r, I′]^{sub} & → & J_{2}^{et#}[r] & → & 0 \end{aligned} .

In fact J_{0}^{#}[r]^{sub} is the r-torsion in the generalized Jacobian of X_{0}^{et} = X_{1}(N)_{/𝔽_{p}} with respect to the reduced modulus C_{0}^{et} ∪ S of the cusps and supersingular points, and the same can be said for J_{1}^{#}[r, I′]^{sub} and the Igusa curve X_{2}^{et}. The degree r covering X_{2}^{et} → X_{0}^{et} is totally ramified over S and étale elsewhere (including over the cusps).

We complement the theorem with the following fact concerning V (as opposed to V^{#}).

Proposition 2.9. As H-modules, W is isomorphic to Hom(U, R) = U^{∗}.

Proof. This has already been noticed in the proof given to Proposition 2.7, using the twisted Weil pairing.

Note the difference between Hida’s theory of “ordinary” p-adic deformations, and our “tame Hida theory”. While Hida had to restrict to the ordinary part in order to have U_{p} acting invertibly, in our case this is automatic (of course, it is acting now on r-torsion!). Another difference is that in Hida’s situation the p-divisible groups had a 2-step filtration for the decomposition group at p, where the quotient was unramified, while in our case the submodule J_{1}^{#}[r, I′]^{sub} is the unramified piece, and the source of the ramification in J_{1}^{#}[r, I′]^{quot} is different, the so-called “geometric inertia group action”. Yet the formal structure of our theorem is very similar to Hida’s. In particular, the (almost) freeness of the deformation over Λ, and the fact that the Γ_{0} structures are the I-coinvariants of their deformations is an analogue of Hida’s control theorem.

3. The p-old and p-new parts of the deformation

3.1. A study of the new subvariety. The following results are due to Ribet. Let i and j = i ∘ w_{p} be the two degeneracy maps X_{0} → X_{1}(N), which correspond, in terms of the moduli problem, to the maps

(3.1)  i : [E, P_{N}, H] ↦ [E, P_{N}], j : [E, P_{N}, H] ↦ [E/H, P_{N} mod H].

Then

(3.2)  J_{0,old} = Im (i^{∗}, j^{∗}) : J_{1}(N)^{2} → J_{0}.

Here we write J_{1}(N)^{2} as column vectors of length 2, and i^{∗} acts on the first entry, j^{∗} on the second. If i_{∗} and j_{∗} are the maps J_{0} → J_{1}(N) induced by Albanese functoriality, then

(3.3)  J_{0,new} = Ker(^{t}(i_{∗}, j_{∗}) : J_{0} → J_{1}(N)^{2})^{0}.

Lemma 3.1. The endomorphism α = ^{t}(i_{∗}, j_{∗}) ∘ (i^{∗}, j^{∗}) ∈ End(J_{1}(N)^{2}) is an isogeny, represented by the matrix

(3.4)  ( \begin{matrix}p + 1 & T_{p}^{∗} \\ T_{p∗} & p + 1 \end{matrix} ) ,

and T_{p∗} = (⟨p⟩_{N}^{−1})^{∗} ∘ T_{p}^{∗}.

Proof. Clearly i_{∗} ∘ i^{∗} = deg(i) = p + 1 = deg(j) = j_{∗} ∘ j^{∗}. An easy computation shows that i_{∗} ∘ j^{∗} = T_{p}^{∗}, j_{∗} ∘ i^{∗} = T_{p∗}, and that T_{p∗} = ⟨p⟩_{N∗} ∘ T_{p}^{∗} = (⟨p⟩_{N}^{−1})^{∗} ∘ T_{p}^{∗} (see also [M-W], p.236).

Jacobians are self-dual abelian varieties, and ^{t}(i_{∗}, j_{∗}) is the dual of (i^{∗}, j^{∗}). Now it is a theorem of Ribet, based on results of Ihara, that (i^{∗}, j^{∗}) : J_{1}(N)^{2} → J_{0} is injective (see [Ri], theorem 4.1). Denoting by (−)^{∨} the dual abelian variety, we get, upon dualizing the isomorphism

(3.5)  J_{1}(N)^{2} ≃^{(i^{∗},j^{∗})} J_{0,old} ↪ J_{0},

that the map ^{t}(i_{∗}, j_{∗}) factors as

(3.6)  J_{0} = J_{0}^{∨} ↠ J_{0,old}^{∨} ≃ J_{1}(N)^{2},

and has a connected kernel (so in (3.3) we may dispose of the (−)^{0}).

We conclude that J_{0}^{old} = J_{0,old}^{∨} and likewise, dualizing

0 → J_{0,old} → J_{0} → J_{0}^{new} → 0

we get J_{0}^{new} = J_{0,new}^{∨}, and the exact sequence

0 → J_{0,new} → J_{0} → J_{0}^{old} → 0.

The two isogenies J_{0,new} → J_{0}^{new} and α̃ : J_{0,old} → J_{0}^{old} are therefore polarizations, being of the form φ^{∨} ∘ φ.

The matrix α breaks as the composition of three maps

(3.7)  α : J_{1}(N)^{2} →^{α_{0}} J_{0,old} →^{α̃} J_{0}^{old} →^{α_{0}^{∨}} J_{1}(N)^{2}.

As α_{0} is an isomorphism,

(3.8)  deg(α) = #(J_{0,old} ∩ J_{0,new}).

Let f_{1}, …, f_{g} (g = genus(X_{1}(N)) be a basis of S_{2}(Γ_{1}(N)) consisting of eigenforms of T_{p}. Let a_{p,i} be the eigenvalue of T_{p} on f_{i}. Then by the Eichler-Shimura isomorphism we can diagonalize the action of T_{p}^{∗} on V_{l}J_{1}(N) ⊗ ℚ̄_{l} (l any prime) with a_{p,1}, …, a_{p,g}, ā_{p,1}, …, ā_{p,g} on the diagonal. In the same basis the diagonal matrix representing T_{p∗} will be ā_{p,1}, …, ā_{p,g}, a_{p,1}, …, a_{p,g} (because T_{p∗} and T_{p}^{∗} are dual to each other under the Hermitian Petersson inner product). We conclude that

(3.9)  deg(α) = {∏_{i=1}^{g} ((p + 1)^{2} − a_{p,i}ā_{p,i})}^{2} .

The Weil estimates |a_{p,i}| ≤ 2\sqrt{p} give the Archimedean estimate

(p − 1)^{4g} ≤ deg(α) ≤ (p + 1)^{4g}.

On the other hand, we will be interested in the l-part of deg(α). Since l|p − 1 we have deg(α) ≡ {∏_{i=1}^{g}(4 − a_{p,i}ā_{p,i})}^{2} mod l.

3.2. The Néron models. Let J_{0,new} (resp. J_{0}^{new}) be the Néron model of J_{0,new} (resp. J_{0}^{new}) over ℤ_{p}. From [BLR], theorem 7.5/4 we deduce the exactness of the sequences

(3.10)  0 → J_{0,new} → J_{0} → J_{0}^{old} → 0,
(3.11)  0 → J_{0,old} → J_{0} → J_{0}^{new},

from which we get exact sequences for the identity components

(3.12)  0 → J_{0,new} ∩ J_{0}^{0} → J_{0}^{0} → J_{0}^{old} → 0,
(3.13)  0 → J_{0,old} → J_{0}^{0} → (J_{0}^{new})^{0} → 0.

Recall the exact sequence

(3.14)  0 → Hom(M_{0}, 𝔾_{m}) → J_{0/𝔽_{p}}^{0} → J_{0}^{et} × J_{0}^{μ} → 0

for the connected component of the special fiber, which came from Raynaud’s theorem. In the next proposition we put the three last exact sequences into one diagram.

Proposition 3.2. Consider the following diagram of group schemes over 𝔽_{p} :

(3.15) \begin{aligned}0 & & & & 0 \\ & ↘ & & & ↓ \\ & & Hom(M_{0}, 𝔾_{m}) & ⊂ & J_{0,new} ∩ J_{0}^{0} & ⊃ & J_{0,new}^{0} \\ & & & ↘ & ↓ \\ 0 & → & J_{0,old} & ⟶^{ι} & J_{0}^{0} & ⟶ & (J_{0}^{new})^{0} & ⟶ & 0 \\ & & & ↘_{α̃} & π ↓ & ↘^{ψ} \\ & & & & J_{0}^{old} & ←^{γ̃} − & J_{1}(N)^{2} \\ & & & & ↓ & & & ↘ \\ & & & & 0 & & & & 0 \end{aligned}

where we have used the following notation. The map ι (resp. π) is the inclusion of (resp. projection onto) the old subvariety (resp. the old quotient). Then α̃ = π ∘ ι is the isogeny denoted by this name before. The map ψ is the one obtained from Raynaud’s theorem, restricting a line bundle in J_{0/𝔽_{p}}^{0} = P ic_{𝔛_{0}/𝔽_{p}}^{0} to X_{0}^{et} and X_{0}^{μ}. The target of ψ is J_{0}^{et} × J_{0}^{μ}. This is identified with J_{1}(N)^{2} via the isomorphism

(3.16)  θ^{∗} : J_{1}(N)^{2} ≃ J_{0}^{et} × J_{0}^{μ}

which is derived from (3.17) θ : X_{0}^{et}×X_{0}^{μ} ≃ X_{1}(N)^{2}, θ([E′, P′_{N}, H′], [E, P_{N}, H]) = ([E′/H′, P′_{N} mod H′], [E, P_{N}]).

Note that θ^{∗} sends the first copy of J_{1}(N) to J_{0}^{et}, and the second to J_{0}^{μ}. The map γ̃ exists since Ker(ψ) ⊂ Ker(π) (as it is obviously new and connected). Then:

• Hom(M_{0}, 𝔾_{m}) = J_{0,new}^{0}.

• Let γ ∈ End(J_{1}(N)^{2}) be the map γ̃ followed by the canonical isomorphism α_{0}^{∨} : J_{0}^{old} → J_{1}(N)^{2}. Let β = ψ ∘ ι ∘ α_{0} ∈ End(J_{1}(N)^{2}). Then as matrices

(3.18)  β = ( \begin{matrix}⟨p⟩_{N}^{−1}V & 1 \\ 1 & V \end{matrix} ) , γ = ( \begin{matrix}⟨p⟩_{N}F & 1 \\ 1 & F \end{matrix} )

where F and V are the Frobenius and Verschiebung endomorphisms of J_{1}(N)_{/𝔽_{p}}. Consequently β and γ are dual to each other, and α = γ ∘ β.

• deg(γ) = [J_{0}^{0} ∩ J_{0,new} : J_{0,new}^{0}] = deg(β) and

(3.19)  [J_{0}^{0} ∩ J_{0,new} : J_{0,new}^{0}]^{2} = #(J_{0,old} ∩ J_{0,new}).

Proof. The first assertion is obvious, as J_{0,new/𝔽_{p}}^{0} is a torus, but Hom(M_{0}, 𝔾_{m}) is the maximal torus in J_{0}^{0}. Let γ = α_{0}^{∨} ∘ γ̃ as above. Consider a point

(3.20)  ^{t}(x, 0) = ^{t} (cl {∑ m_{i}[E_{i}, P_{N,i}]} , 0) ∈ J_{1}(N)(𝔽̄_{p})^{2},

(∑ m_{i} = 0, cl denotes the divisor class in P ic_{X_{1}(N)/𝔽̄_{p}}^{0}), with all the E_{i} ordinary elliptic curves over 𝔽̄_{p}, and represent x by the corresponding ∑ m_{i}[E′_{i}, P′_{N,i}, H′_{i}] ∈ J_{0}^{et}(𝔽̄_{p}), where H′_{i} = Ker(V er_{E′_{i}}), E_{i} = E′_{i}/H′_{i} and P_{N,i} = P′_{N,i} mod H′_{i}, so that the point [E′_{i}, P′_{N,i}, H′_{i}] lies on X_{0}^{et} (but is not in S). Here we denote by F rob_{E} : E → E^{(p)} and V er_{E} : E → E^{(p^{−1})} the Frobenius and Verschiebung of an elliptic curve over 𝔽̄_{p} (to distinguish them from F and V, which are the corresponding endomorphisms of J_{1}(N)). Now^{t}(x, 0) = ψ(cl{∑ m_{i}[E′_{i}, P′_{N,i}, H′_{i}]}), where now we consider the divisor as a divisor on 𝔛_{0/𝔽̄_{p}}, which is of degree 0 on each irreducible component, and therefore represents a point in J_{0}^{0}(𝔽̄_{p}). Thus (3.21) γ(^{t}(x, 0)) = α_{0}^{∨} ∘ π(cl{∑ m_{i}[E′_{i}, P′_{N,i}, H′_{i}]}) = ^{t}(i_{∗}, j_{∗})(cl{∑ m_{i}[E′_{i}, P′_{N,i}, H′_{i}]}).

It is easily checked that

(3.22)  i_{∗}([E′_{i}, P′_{N,i}, H′_{i}]) = [E′_{i}, P′_{N,i}] = ⟨p⟩_{N}^{∗}F ([E_{i}, P_{N,i}])
(3.23)  and j_{∗}([E′_{i}, P′_{N,i}, H′_{i}]) = [E_{i}, P_{N,i}].

This gives the first column in the matrix of γ. Similar computations give the second column, and the matrix of β. But

(3.24)  γ ∘ β = α_{0}^{∨} ∘ γ̃ ∘ ψ ∘ ι ∘ α_{0} = α_{0}^{∨} ∘ π ∘ ι ∘ α_{0} = α_{0}^{∨} ∘ α̃ ∘ α_{0} = α.

Notice that we have re-proved the Eichler-Shimura congruence relations

(3.25)  T_{p}^{∗} = ⟨p⟩_{N}^{∗}F + V, T_{p∗} = F + ⟨p⟩_{N∗}V.

Finally the degrees of γ and β are easily computed from the big diagram and are each equal to \sqrt{deg(α)}, which was computed before. This concludes the proof.

3.3. Breaking the deformation into components. Although the deformation V^{#} → V^{#} can be studied as one piece, for applications (such as for elliptic curves uniformized by X_{0}), it is often necessary to break it into “components”. We shall now make certain assumptions which we keep until the end of this work.

Fix an orthogonal decomposition

(3.26)  S_{2}(Γ_{0}) = S ⊕ S^{⊥}

(with respect to the Petersonn inner product) where the two subspaces are rational over ℚ (with respect to to the ℚ-structure given by the q-expansions) and H-stable. It is well-known that S and S^{⊥} are the cotangent spaces of abelian subvarieties A, A^{⊥} ⊂ J_{0}, defined over ℚ, stable under H, whose intersection is finite, and whose sum is J_{0}. We shall assume the following three conditions:

• (A1) A is not l-Eisenstein (see Section 1.6).

• (A2) (#A ∩ A^{⊥}, l) = 1.

• (A3) A is isogenous to a subvariety of J_{0}(Np), and if l|N, then N_{l} = l and A is l-new.

The second assumption means that J_{0}[r] = A[r] ⊕ A^{⊥}[r], and (A1) and (A2) together imply that

(3.27)  J_{0}^{#}[r] = A[r] ⊕ A′[r],

where we have denoted by A′ the inverse image of A^{⊥} in J_{0}^{#} (see Section 1.6). Proposition 1.1 and lemma 1.5 will allow us to lift this decomposition uniquely in J_{1}^{#}[r, I′].

In the category of abelian varieties up to isogeny (defined over ℚ) we have

(3.28)  H(J_{0}) ⊗ ℚ ⊂ End_{H}(J_{0} ⊗ ℚ) ⊂ End_{H}(S_{2}(Γ_{0}, ℚ)) = H(J_{0}) ⊗ ℚ,

which shows that End_{H}(J_{0} ⊗ ℚ) = H(J_{0}) ⊗ ℚ. Since A is H-stable, the projector onto A⊗ℚ in the category of abelian varieties up to isogeny commutes with H, hence is a (rational) Hecke operator, which is an idempotent. We call it e_{A} ∈ H(J_{0}) ⊗ ℚ.

Let d be an integer which is divisible by #(A ∩ A^{⊥}), and satisfies d ≡ 1 mod r (its existence follows from (A2)). Note that de_{A} ∈ End(J_{0}), because it is the endomorphism which is equal to d_{A} on A and to 0 on A^{⊥}. As some rational multiple of de_{A} is a Hecke operator, de_{A} ∈ H′(J_{0}). We are in a position to apply Proposition 1.1, because if l|N, condition (A3) implies that U_{l} is invertible on A. Let 𝔐 be any maximal ideal of H_{l}(J_{0}), which is in the support of H_{l}(A). If l|N, 𝔐 is ordinary. We conclude from Proposition 1.1 that

(3.29)  e_{𝔐}de_{A} ∈ e_{𝔐}H′_{l}(J_{0}) = e_{𝔐}H_{l}(J_{0}) ⊂ H_{l}(J_{0}).

Summing over all the maximal 𝔐 in the support of H_{l}(A) we see that de_{A} ∈ H_{l}(J_{0}). Its image in H_{l}(J_{0}[r]) is an idempotent, projecting to A[r], because d ≡ 1 mod r. Since A[r] is not Eisenstein, we arrive at the following conclusion.

Lemma 3.3. There exists a unique idempotent e ∈ H(J_{0}^{#}[r]), which projects onto A[r], and kills A′[r].

By lemma 1.5, this idempotent lifts uniquely to an idempotent e ∈H(J_{1}^{#}[r, I′]). We shall be interested in the deformation

(3.30)  eV^{#} → eV^{#} = A[r]^{∗}.

Proposition 3.4. Under the assumptions (A1) and (A2) made above, eV^{#} = eV and eV^{#} = eV. As a Λ-module eV ≃ Λ^{2g} and eV = eV ⊗_{Λ} R ≃ R^{2g} where g is the dimension of A. Each of eU^{#} = eU and eW^{#} = eW is isomorphic to Λ^{g}, and as H-modules they are dual to each other. The decomposition group at p acts on eU and eW by the characters ωϕ^{−1} and ϕ, as described in Theorem 2.8.

Proof. By assumption (A1) the idempotent e is supported at non-Eisenstein primes. It follows from the uniqueness of the lifting of e to e that e too is supported at non-Eisenstien primes, hence eV^{#} = eV etc.

Consider the structure of J_{1}^{#}[r, I′]^{∗} = V^{#} over Λ as discussed in Section 2.6. The line Rv_{0} ≃ Λ/I ⊂ V^{#} is not canonical, but as explained in Section 2.6 its image Ru_{0} in J_{0}^{#}[r]^{∗} = V^{#} is canonical. Furthermore, it was computed there that u_{0} ∈ J_{0}[r]^{∗} and is represented, under the Weil pairing, by a unique ũ_{0} ∈ J_{0}[r], which belongs to the Shimura subgroup. The line Rũ_{0} maps injectively into the group of connected components Φ(J_{0}, ℤ_{p}). All these identifications respect the Hecke action. However, a theorem of Ribet and Edixhoven (see [E]) asserts that the Hecke action on Φ(J_{0}, ℤ_{p}) is Eisenstein. We conclude that the Hecke action on Ru_{0} is Eisenstein too, so (1−e)u_{0} = u_{0}. It follows that in the decomposition V^{#} = eV^{#} ⊕(1−e)V^{#}, the single copy of Λ/I survives in (1 − e)V^{#}, and eV^{#} is free over Λ. The rest follows from Theorem 2.8 and Proposition 2.9. In particular the equality of Λ-ranks of eU and eW follows from the fact that

(3.31)  eW ≃ Hom(eU, R)

and from the fact that both eU and eW are free over Λ.

4. The infinitesimal deformation of a p-new component, and its relationship with the p-adic period matrix

4.1. The infinitesimal deformation. Let Λ̃ = Λ/I^{2} ≃ R[ε] (ε^{2} = 0), the ring of dual numbers over R. To make the isomorphism concrete fix once and for all a primitive r-th root of unity ρ ∈ 𝔽_{p}^{×} and identify ρ − 1 mod I^{2} with ε, so that ρ^{k} − 1 ≡ kε mod I^{2}. For any Λ-module M we denote by M̃ the module M/I^{2}M. For any R = Λ/I module N we denote by N(1) the module N ⊗_{R} I/I^{2}. Since I/I^{2} = μ_{r} ≃ R non-canonically, N and N(1) are non-canonically isomorphic, but we can use our choice of ρ to identify them if we map ε to 1. For future reference note also that the identification of I/I^{2} with μ_{r} is via

(4.1)  ⟨u⟩_{p} − 1 mod I^{2} ↦ u_{R},

where u_{R} is the projection of u to μ_{r}.

The basic diagram

(4.2)  \begin{aligned}0 & → & eU & → & eV & → & eW & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & eU & → & eV & → & eW & → & 0 \end{aligned}

describing the deformation of eV = A[r]^{∗}, yields a diagram with exact rows and columns

(4.3)  \begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & eU(1) & → & eV (1) & → & eW (1) & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & eŨ & → & eṼ & → & eW̃ & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & eU & → & eV & → & eW & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned} .

We now twist the local Galois action by the character ϕ^{−1}, thereby trivializing the action on the right column of the diagram, and take the long exact sequences in cohomology. Consider the following portion of what is obtained:

(4.4)  \begin{aligned}H^{0}(ℚ_{p}, eW̃(ϕ^{−1})) & → & H^{1}(ℚ_{p}, eŨ(ϕ^{−1})) \\ ↓ & & ↓ \\ H^{0}(ℚ_{p}, eW (ϕ^{−1})) & →^{δ_{0}} & H^{1}(ℚ_{p}, eU(ϕ^{−1})) \\ & & ↓ δ_{1} \\ & & H^{2}(ℚ_{p}, eU(1)(ϕ^{−1})) \end{aligned} .

Since the vertical arrow between the H^{0} is surjective, and since H^{0}(ℚ_{p}, eW (ϕ^{−1})) = eW, we obtain the following result.

Proposition 4.1. In the situation described above, the map

(4.5)  δ_{1} ∘ δ_{0} : eW → H^{2}(ℚ_{p}, eU(1)(ϕ^{−1}))

is 0.

The rest of this work will be devoted to analyzing the relation δ_{1} ∘ δ_{0} = 0.

4.2. The case of a p-old subvariety. Suppose that A is p-old. In this case we have nothing interesting, because the short exact sequence 0 → eU → eV → eW → 0 is split. Indeed, the fact that A is p-old implies that A[r] ⊂ J_{0}^{0}[r]. In addition, J_{0,new} ⊂ A^{⊥} so Hom(M_{0}, μ_{r}) ⊂ A^{⊥}[r]. Since by our assumption A[r] and A^{⊥}[r] intersect trivially, A[r] ↪ J_{0}^{et}[r] × J_{0}^{μ}[r]. It is now clear that the filtration on A[r] corresponds to this decomposition and therefore splits, and dualizing, the filtration on eV splits. It follows that if A is p-old, already δ_{0} = 0.

4.3. The case of a p-new subvariety. Assume from now on that A is p-new. Since H(A) is a quotient of H(J_{0,new}), we get the relation U_{p} = −w_{p} on A, and therefore

(4.6)  U_{p}^{2} = ⟨−p⟩_{N}.

By (A3), the Hecke operators ⟨t⟩_{N} act trivially on A, and therefore U_{p}^{2} = 1. Furthermore, if A_{0} is the abelian subvariety of J_{0}(Np) isogenous to A, the isogeny A_{0} → A induces an isomorphism A_{0}[r] ≃ A[r], because the kernel of J_{0}(Np) → J_{0} is purely Eisenstein (see the proof of proposition 2.5, in particular diagram (2.29) and replace p by N in the arguments there. See also [M], Proposition 11.7), and A satisfies (A1). We may therefore regard A[r] as a submodule of J_{0}(Np).

4.4. The p-adic period pairing. Let ℚ_{p}(S) be the unramified extension of ℚ_{p} whose residue field 𝔽_{p}(S) is the minimal field of definition for all the supersingular points S of X_{1}(N)_{/𝔽_{p}}. Recall that we denoted by M_{0} = ℤ[S]_{0} the augmentation subgroup of ℤ[S]. The p-adic uniformization of J_{0,new} over ℚ_{p}(S) is obtained as follows (see [B]). The character group of the torus J_{0,new/𝔽_{p}}^{0} is M_{0} (see Proposition 3.2). Let L_{0} be the character group of the torus (J_{0/𝔽_{p}}^{new})^{0}, and recall that the abelian varieties J_{0,new} and J_{0}^{new} are dual to each other. Then there is a canonical isomorphism of rigid-analytic groups over ℚ_{p}(S)

(4.7)  J_{0,new} = Hom(M_{0}, 𝔾_{m}^{an})/L_{0}

where L_{0} injects as a discrete cocompact lattice in Hom(M_{0}, ℚ_{p}(S)^{×}). The pairing

(4.8)  Q : M_{0} × L_{0} → ℚ_{p}(S)^{×}

is called the p-adic period pairing of J_{0,new}. The dual abelian variety is canonically uniformized as

(4.9)  J_{0}^{new} = J_{0,new}^{∨} = Hom(L_{0}, 𝔾_{m}^{an})/M_{0}.

The principal polarization of J_{0} induces the polarization map J_{0,new} → J_{0}^{new}. The corresponding isogeny of connected components of special fibers of Néron models is

(4.10)  Hom(M_{0}, 𝔾_{m}) → Hom(L_{0}, 𝔾_{m})

and it therefore induces an inclusion h : L_{0} ↪ M_{0}. The pairing

(4.11)  Q_{h} : (l, l′) ↦ Q(h(l), l′), (l, l′ ∈ L_{0})

is symmetric and if ord denotes the valuation of ℚ_{p}(S), ord∘Q_{h} is positive definite. Note that h identifies L_{0} as the subgroup of M_{0} annihilating J_{0,old} ∩ J_{0,new}^{0}. The polarization map J_{0,new} → J_{0}^{new} is now induced from the map h via the uniformizations of these two abelian varieties. Its degree is [M_{0} : h(L_{0})]^{2}.

The uniformization of A is obtained as follows. Let M_{A} be the character group of the connected component of the special fiber of the Néron model of A. Then M_{A} is a quotient of M_{0}. Let L_{A} = L_{0} ∩ Hom(M_{A}, ℚ_{p}(S)^{×}). Then

(4.12)  A = Hom(M_{A}, 𝔾_{m}^{an})/L_{A}.

We denote by

(4.13)  Q_{A} : M_{A} × L_{A} → ℚ_{p}(S)^{×}

the corresponding pairing.

The Galois group Gal(ℚ̄_{p}/ℚ_{p}) acts on A(ℚ̄_{p}). It also acts on M_{A} and L_{A} via the unramified quotient Gal(ℚ_{p}(S)/ℚ_{p}), and the p-adic uniformization as well as the padic period pairing Q_{A} are compatible with this action. The endomorphism w_{p} acts on M_{A} and L_{A} via −σ, where σ is the Frobenius automorphism of Gal(ℚ_{p}(S)/ℚ_{p}) (see footnote preceeding Proposition 2.2).

Lemma 4.2. The values of the period pairing Q_{A} lie in ℚ_{p}^{×}.

Proof. This is a consequence of assumption (A3). Apply σ and compute:

(4.14) σQ_{A}(m, l) = Q_{A}(σm, σl) = Q_{A}(w_{p}m, w_{p}l) = Q_{A}(w_{p}^{2}m, l) = Q_{A}(m, l).

Alternatively, one can ”twist” the abelian variety A by the character

ϕ : Gal(ℚ̄_{p}/ℚ_{p}) → Aut(A)

(see Prop. 2.6, and recall that U_{p} is invertible). The twisted abelian variety has split multiplicative reduction over ℚ_{p}, because the Galois action on M_{0} has been trivialized. However, it has the same period pairing as A.

The principal polarization of J_{0} induces the polarization

(4.15)  A ⊂ J_{0,new} → J_{0}^{new} ↠ A^{∨}

on A, or the inclusion h_{A} : L_{A} ↪ M_{A}. By assumption (A2) the degree of this polarization is prime to l, and therefore

(4.16)  h_{A} : L_{A} ⊗ ℤ_{l} ≃ M_{A} ⊗ ℤ_{l}

is an isomorphism.

Lemma 4.3. The short exact sequence 0 → eU → eV → eW → 0 is dual to the short exact sequence

(4.17)  0 → Hom(M_{A}, μ_{r}) → A[r] → L_{A}/rL_{A} → 0

which is obtained from the p-adic uniformization.

Proof. We know that eJ_{0}[r] = A[r]. It is enough to prove that

eJ_{0}[r]^{sub} = Hom(M_{A}, μ_{r}).

Pick e′ ∈ H(J_{0}) projecting onto e in H(J_{0}[r]), which maps J_{0} onto A. Let A be the Néron model of A over ℤ_{p}. Then e′ maps J_{0/𝔽_{p}}^{0} onto A_{/𝔽_{p}}^{0}, so we have (4.18)

eJ_{0}[r]^{sub} ⊆ eJ_{0/𝔽_{p}}^{0}[r] ⊆ A_{/𝔽_{p}}^{0}[r] = Hom(M_{A}, μ_{r}) ⊆ eHom(M_{0}, μ_{r}) ⊆ eJ_{0}[r]^{sub}.

This proves the lemma.

4.5. The map δ_{0}. We shall now analyze the map δ_{0} : eW → H^{1}(ℚ_{p}, eU(ϕ^{−1})). Since, by the last lemma,

(4.19)  eW = M_{A}/rM_{A}(−1), eU = Hom(L_{A}, R), hence also
H^{1}(ℚ_{p}, eU(ϕ^{−1})) = H^{1}(ℚ_{p}, Hom(L_{A}, R)(ϕ^{−1})) = Hom(L_{A}, H^{1}(ℚ_{p}, R))

(note that the twist by ϕ^{−1} has the affect of trivializing the quadratic, unramified Galois action on L_{A}), the map δ_{0} is a map from M_{A}/rM_{A}(−1) to the group Hom(L_{A}, H^{1}(ℚ_{p}, R)), or, tensoring with μ_{r} (which has a trivial Galois action), it is equivalent to a map

(4.20)  δ_{0}(1) : M_{A} ⊗ R → Hom(L_{A}, ℚ_{p}^{×} ⊗ R).

Here we have used Hilbert’s theorem 90, saying that H^{1}(ℚ_{p}, μ_{r}) = ℚ_{p}^{×} ⊗ R.

Proposition 4.4. The map δ_{0}(1) is the map obtained from the pairing Q_{A} ⊗ R.

Proof. This is an easy application of Kummer theory. See [dS1], lemma 3.4.

4.6. The map δ_{1}. A similar analysis holds for δ_{1}(1). Its domain is

(4.21)  Hom(L_{A}, ℚ_{p}^{×} ⊗ R) = Hom(L_{A}, R) ⊗ ℚ_{p}^{×},

and its range is the group

(4.22)  H^{2}(ℚ_{p}, Hom(L_{A}, μ_{r})(ϕ^{−1})) ⊗ μ_{r} = Hom(L_{A}, μ_{r}),

where we have used local class field theory to identify H^{2}(ℚ_{p}, μ_{r}) with R.

For every σ ∈ Gal(ℚ̄_{p}/ℚ_{p}) consider σ − 1 acting on eŨ(ϕ^{−1}). Since ϕ^{2} = 1 on eU = Hom(L_{A}, R) (or on eU(1) = Hom(L_{A}, μ_{r})), σ − 1 induces a map

(4.23)  P (σ) : Hom(L_{A}, R) → Hom(L_{A}, μ_{r}),

and the map σ ↦ P (σ) is a homomorphism. (With respect to suitable R-bases the matrix of σ is

(4.24)  ( \begin{matrix}1 & P (σ) \\ 0 & 1 \end{matrix} ) .)

It therefore factors through Gal(ℚ_{p}^{ab}/ℚ_{p}).

Proposition 4.5. The map

(4.25)  δ_{1}(1) : Hom(L_{A}, R) ⊗ ℚ_{p}^{×} → Hom(L_{A}, μ_{r})

is described as follows. Let t ∈ ℚ_{p}^{×}, and denote by σ_{t} ∈ Gal(ℚ_{p}^{ab}/ℚ_{p}) the Artin symbol of t. Then

(4.26)  δ_{1}(1)(x ⊗ t) = P (σ_{t})(x).

Proof. (Compare [G-S], theorem 3.11). Abbreviate G = Gal(ℚ̄_{p}/ℚ_{p}). Consider Tate’s local duality pairing

(4.27) Hom(G, R) × (ℚ_{p}^{×} ⊗ R) = H^{1}(ℚ_{p}, R) × H^{1}(ℚ_{p}, μ_{r}) →^{∪} H^{2}(ℚ_{p}, μ_{r}) = R,

which takes the pair (h, t ⊗ 1) to h(σ_{t}). (Although R and μ_{r} are non-canonically isomorphic as Galois modules, we distinguish between them because they play dual roles). It may be “fattened” to a perfect pairing

(4.28)  (L_{A} ⊗ Hom(G, R)) × Hom(L_{A}, ℚ_{p}^{×} ⊗ R) → R,

and here L_{A} ⊗ Hom(G, R) = H^{1}(ℚ_{p}, eU(ϕ^{−1})(1)^{∨}) and Hom(L_{A}, ℚ_{p}^{×} ⊗ R) = H^{1}(ℚ_{p}, eU(ϕ^{−1})(1)), where we have denoted by (−)^{∨} = Hom(−, μ_{r}) the Cartier dual.

Similarly, Tate’s local duality gives the obvious pairing

(4.29)  R(−1) × μ_{r} = H^{0}(ℚ_{p}, R(−1)) × H^{2}(ℚ_{p}, μ_{r}(1)) →^{∪} H^{2}(ℚ_{p}, μ_{r}) = R,

which may be “fattened” to a perfect pairing

(4.30)  (L_{A} ⊗ R(−1)) × Hom(L_{A}, μ_{r}) → R.

Here L_{A}⊗R(−1) = H^{0}(ℚ_{p}, eU(ϕ^{−1})(2)^{∨}) and Hom(L_{A}, μ_{r}) = H^{2}(ℚ_{p}, eU(ϕ^{−1})(2)).

With respect to these two pairings the dual δ_{1}(1)^{∗} of δ_{1}(1) is the connecting homomorphism

(4.31)  δ_{1}(1)^{∗} : H^{0}(ℚ_{p}, eU(ϕ^{−1})(2)^{∨}) → H^{1}(ℚ_{p}, eU(ϕ^{−1})(1)^{∨})

associated to the short exact sequence

(4.32)  0 → eU(ϕ^{−1})(1)^{∨} → eŨ(ϕ^{−1})(1)^{∨} → eU(ϕ^{−1})(2)^{∨} → 0.

What we have to prove (dualized) is that

(4.33)  δ_{1}(1)^{∗}(l)(σ) = P (σ)^{∗}(l), l ∈ L_{A} ⊗ R(−1), σ ∈ G.

This is obvious from the definition of P (σ) (perhaps up to a sign, depending on how one normalizes connecting homomorphisms in long exact sequences).

4.7. The meaning of the relation δ_{1} ∘ δ_{0} = 0.

Theorem 4.6. Let A be an abelian subvariety of J_{0} satisfying (A1), (A2), and (A3) above. Let M_{A} and L_{A} be the lattices defined above, and

(4.34)  q_{A,R} : M_{A} ⊗ R → Hom(L_{A}, R) ⊗ ℚ_{p}^{×}

the map obtained from the p-adic period pairing of A, taken modulo r. Let

(4.35)  δ : Hom(L_{A}, R) ⊗ ℚ_{p}^{×} → Hom(L_{A}, μ_{r})

be the map δ(l′ ⊗ t) = {(ωϕ^{−2})(σ_{t}) − 1}(l̃′) where l̃′ ∈ eŨ lifts l′ ∈ eU = Hom(L_{A}, R). Then δ ∘ q_{A,R} = 0.

Proof. As we have seen, the maps q_{A,R} and δ are none others but δ_{0}(1) and δ_{1}(1), respectively. The characterization of the second relies also on our knowledge of the Galois action on eŨ(ϕ^{−1}), which is via the character ωϕ^{−2} (see Proposition 3.4).

To write the last theorem in a completely explicit form, decompose every t ∈ ℚ_{p}^{×} as

(4.36)  t = u(t)p^{ν(t)},

where u(t) is a unit, and ν(t) ∈ ℤ. Recall that ϕ(σ_{t}) = U_{p}^{ν(t)} and ω(σ_{t}) = ⟨p⟩_{N}^{ν(t)}⟨u(t)^{−1}⟩_{p}. Here we must recall that the p-adic and N-adic cyclotomic characters satisfy χ_{p}(σ_{t}) = u(t)^{−1} and χ_{N}(σ_{t}) = p^{ν(t)}. Note that even under assumption (A3) we do not know that in the deformation eU the operators ⟨t⟩_{N} act trivially, because the deformation is taken within J_{1}(Np).

Now, if l̃′ ∈ eŨ,

(4.37)  {(ωϕ^{−2})(σ_{t}) − 1}(l̃′) = (ν_{R}(t)(U_{p}^{−2}⟨p⟩_{N} − 1) − u_{R}(t)) l̃′,

where ν_{R}(t) = ν(t) ⊗ 1 ∈ ℤ ⊗ R = R and u_{R}(t) = u(t) ⊗ 1 ∈ ℤ_{p}^{×} ⊗ R = μ_{R}.

Let m_{1}, …, m_{g} be a basis of M_{A}, and l_{1}, …, l_{g} a basis of L_{A} (g = dim A). Let l′_{1}, …, l′_{g} be the dual basis in Hom(L_{A}, R). Then

(4.38)  q_{A,R}(m_{i} ⊗ 1) = ∑_{j} l′_{j} ⊗ q_{ij}, q_{ij} = Q_{A}(m_{i}, l_{j}).

Applying δ we obtain from the relation δ ∘ q_{A,R}(m_{i} ⊗ 1) = 0 that

(4.39)  ∑_{j} ν_{R}(q_{ij})(U_{p}^{−2}⟨p⟩_{N} − 1)(l̃′_{j}) = ∑ u_{R}(q_{ij})l̃′_{j}.

In terms of matrices, if [U_{p}^{−2}⟨p⟩_{N} − 1] is the matrix representing U_{p}^{−2}⟨p⟩_{N} − 1 in a basis {l̃′_{j}} of eŨ lifting the basis {l′_{j}} of Hom(L_{A}, R), we have the relation

(4.40)  [ν_{R}(q_{ij})] [U_{p}^{−2}⟨p⟩_{N} − 1] = [u_{R}(q_{ij})] .

Let us consider the case of an elliptic curve A satisfiying (A1)-(A3). Then there is only one period q, the Tate period of A. In this case eU = R, eU = Λ, and the action of U_{p}^{−2}⟨p⟩_{N} on eU is given by 1 + λ, with λ ∈ I. The last formula reads then

(4.41)  ν_{R}(q)λ mod I^{2} = u_{R}(q).

Notice that in case (l, ν_{R}(q)) = 1, the quantity u_{R}(q)/ν_{R}(q) (we write μ_{r} additively!) may be called the refined L-invariant of A (similar to log_{p}(q)/ν(q)). Our main result therefore relates it to the “derivative” of U_{p}^{−2}⟨p⟩_{N} in the deformation eU of eU.

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Department of mathematics, University of Massachussettes, Amherst MA

   E-mail address: goldberger@math.umass.edu

Institute of Mathematics, Hebrew University, Giv’at-Ram 91904 Jerusalem, ISRAEL

   E-mail address: deshalit@math.huji.ac.il