Limit Mordell–Weil Groups and their p-Adic Closure

Haruzo Hida

Received: September 9, 2013 Revised: May 18, 2014

Abstract. This is a twin article of [H14b], where we study the projective limit of the Mordell–Weil groups (called pro Λ-MW groups) of modular Jacobians of p-power level. We prove a control theorem of an ind-version of the K-rational Λ-MW group for a number field K. In addition, we study its p-adic closure in the group of K_{𝔭}-valued points of the modular Jacobians for a 𝔭-adic completion K_{𝔭} for a prime 𝔭|p of K. As a consequence, if K_{𝔭} = ℚ_{p}, we give an exact formula for the rank of the ordinary/co-ordinary part of the closure.

2010 Mathematics Subject Classification: primary: 11F25, 11F32, 11G18, 14H40; secondary: 11D45, 11G05, 11G10 Keywords and Phrases: modular curve, Hecke algebra, modular deformation, analytic family of modular forms, Mordell–Weil group, modular jacobian

1. Introduction

Consider a p-adic ordinary family of modular eigenforms of prime-to-p level N. This is an irreducible scheme Spec(𝕀) which is finite torsion-free over the Iwasawa algebra ℤ_{p}[[T]], and whose points P of codimension one and not in the special fiber correspond to ordinary p-adic modular eigenforms f_{P}. Among those points, many corresponds to modular classical eigenforms of weight 2 and level Np^{r} (for variable r), and such points are Zariski dense in Spec(𝕀). An old, well-known, and fundamental construction of Eichler–Shimura attaches to any modular cuspidal eigenform f of weight 2 an abelian variety A_{f} defined over ℚ, of dimension the degree of the field generated by the coefficients of f over ℚ. For these abelian varieties A_{f}, one can consider the Mordell–Weil group A_{f}(ℚ) and more generally, A_{f}(k) for k a fixed number field, which are finitely generated abelian groups. Let us set Â_{f}(k) = A_{f}(k) ⊗_{ℤ} ℤ_{p}. We consider the following natural question: how does the Mordell–Weil group Â_{f}(k) varies as f varies among those cuspidal eigenforms of weight 2 in the family? We give a partial answer to this question in the form of control theorems (Theorems 1.1 and 6.6) for these Mordell–Weil groups. An analogous result is proved when the number field k is replaced by an l-adic field k_{𝔩}, and also a consequence concerning the image of A_{f}(k) in A_{f}(k_{𝔩}).

Fix a prime p. This article concerns the p-slope 0 Hecke eigen cusp forms of level Np^{r} for r > 0 and p ∤ N, and for small primes p = 2, 3, they exists only when N > 1; thus, we may assume Np^{r} ≥ 4. Then the open curve Y_{1}(Np^{r}) (obtained from X_{1}(Np^{r}) removing all cusps) gives the fine smooth moduli scheme classifying elliptic curves E with an embedding μ_{Np^{r}} ↪ E. Anyway for simplicity, we assume that p ≥ 3, although we indicate often any modification necessary for p = 2. A main difference in the case p = 2 is that we need to consider the level Np^{r} with r ≥ 2, and whenever the principal ideal (γ^{p^{r−1}} − 1) shows up in the statement for p > 2, we need to replace it by (γ^{p^{r−2}} − 1) (assuming r ≥ 2), as the maximal torsion-free subgroup of ℤ_{2}^{×} is 1+2^{2}ℤ_{2}. We applied in [H86b] and [H14a] the techniques of U(p)-isomorphisms to p-divisible Barsotti–Tate groups of modular Jacobian varieties of all p-power level (with a fixed prime-to-p level N) in order to get coherent control under diamond operators. In this article, we apply the same techniques to Mordell– Weil groups of the Jacobians and see what we can say. We hope to study U(p)-isomorphisms of the Tate–Shafarevich groups of the Jacobians in a future article. Let X_{r} = X_{1}(Np^{r})_{/ℚ} be the compactified moduli of the classification problem of pairs (E, ϕ) of elliptic curves E and an embedding ϕ : μ_{Np^{r}} ↪ E[Np^{r}] as finite flat group schemes. Since Aut(μ_{p^{r}}) = (ℤ/p^{r}ℤ)^{×}, z ∈ ℤ_{p}^{×} acts on X_{r} via ϕ ↦ ϕ ∘ \overline{z} for the image \overline{z} ∈ (ℤ/p^{r}ℤ)^{×}. We write X_{s}^{r} (s > r) for the quotient curve X_{s}/(1 + p^{r}ℤ_{p}). The complex points X_{s}^{r}(ℂ) contains Γ_{s}^{r}\ℌ as an open Riemann surface for Γ_{s}^{r} = Γ_{0}(p^{s}) ∩ Γ_{1}(Np^{r}). Write J_{r/ℚ} (resp. J_{s/ℚ}^{r}) for the Jacobian of X_{r} (resp. X_{s}^{r}) whose origin is given by the infinity cusp ∞ of the modular curves. We regard J_{r} as the degree 0 component of the Picard scheme of X_{r}. For a number field k, we consider the group of k-rational points J_{r}(k). The Hecke operator U(p) and its dual U^{∗}(p) act on J_{r}(k) and their p-adic limit e = lim_{n→∞} U(p)^{n!} and e^{∗} = lim_{n→∞} U^{∗}(p)^{n!} are well defined on the Barsotti–Tate group J_{r}[p^{∞}]. For a general abelian variety over a number field k, we put X̂(k) = X(k) ⊗_{ℤ} ℤ_{p} (though we give the definition of the sheaf X̂ in the following section for global and local field k and if k is local, X̂ may not be the tensor product as above). By Picard functoriality, we have injective limits J_{∞}(k) = \underrightarrow{lim}_{r} Ĵ_{r}(k) and J_{∞}[p^{∞}](k) = \underrightarrow{lim}_{r} J_{r}[p^{∞}](k), on which e acts. Here J_{r}[p^{∞}] is the p-divisible Barsotti–Tate group of J_{r} over ℚ). Write G = e(J_{∞}[p^{∞}]), which is called the Λadic Barsotti–Tate group in [H14a] and whose integral property was scrutinized there. We define the p-adic completion of J_{∞}(k):

J̌_{∞}(k) = \underleftarrow{lim}_{n} J_{∞}(k)/p^{n}J_{∞}(k).

These groups we call ind (limit) MW-groups. Since projective limit and injective limit are left-exact, the functor R ↦ J_{∞}(R) is a sheaf with values in abelian groups on the fppf site over ℚ (we call such a sheaf an fppf abelian sheaf). Adding superscript or subscript “ord” (resp. “co-ord”), we indicate the image of e (resp. e^{∗}). The compact cyclic group Γ = 1 + pℤ_{p} ⊂ ℤ_{p}^{×} acts on these modules by the diamond operators. In other words, we identify canonically Gal(X_{r}/X_{0}(Np^{r})) for modular curves X_{r} and X_{0}(Np^{r}) with (ℤ/Np^{r}ℤ)^{×}, and the group Γ acts on J_{r} through its image in Gal(X_{r}/X_{0}(Np^{r})). We study control of J̌_{∞}(k)^{ord} under diamond operators. A compact or discrete ℤ_{p}-module M is called an Iwasawa module if it has a continuous action of the multiplicative group Γ = 1 + pℤ_{p} with a topological generator γ = 1 + p. If M is given by a projective or an injective limit of naturally defined compact ℤ_{p}[Γ/Γ^{p^{r}}]-modules M_{r}, we say M has exact control if M_{r} = M/(γ^{p^{r}} −1)M in the case of a projective limit and M_{r} = M[γ^{p^{r}} −1] = {x ∈ M|(γ^{p^{r}} − 1)x = 0} in the case of an injective limit. If M is compact and M/(γ − 1)M is finite (resp. of finite type over ℤ_{p}), M is Λ-torsion (resp. of finite type over Λ), where Λ = ℤ_{p}[[Γ]] = \underleftarrow{lim}_{r} ℤ_{p}[Γ/Γ^{p^{r}}] (the Iwasawa algebra). When p = 2, we need to take Γ = 1 + p^{2}ℤ_{2} and γ = 1 + 4 = 5 ∈ Γ. In addition, we need to assume often s > r > 1 in place of s > r > 0 for odd primes. The big ordinary Hecke algebra h (whose properties we recall at the end of this section) acts on J̌_{∞}^{ord} and J_{∞}^{ord} as endomorphisms of functors. Let k be a number field or a finite extension of ℚ_{l} for a prime l. Write B_{P} for Shimura’s abelian variety quotient of J_{r} in [Sh73] and A_{P} for his abelian subvariety A_{P} ⊂ J_{r} [IAT, Theorem 7.14] associated to a Hecke eigenform f_{P} in an analytic family of slope 0 Hecke eigenforms {f_{P} |P ∈ Spec(𝕀)} (for an irreducible component Spec(𝕀) of Spec(h) for the big ordinary Hecke algebra h). Here we assume that f_{P} has weight 2 and is a p-stabilized new form of level Np^{r} with r = r(P) > 0. Let Spec(𝕋) ⊂ Spec(h) be the connected component containing Spec(𝕀). For any h-module, we write M_{𝕋} (or M^{𝕋}) for the 𝕋-eigen component 1_{𝕋} · M = M ⊗_{h} 𝕋 for the idempotent 1_{𝕋} of T in h. Suppose that P is a principal ideal generated by α ∈ 𝕋 (regarding as P ∈ Spec(𝕋)). This principality assumption holds most of the cases (see Proposition 5.1). Then we may assume that α = \underleftarrow{lim}_{s} α_{s} (as an endomorphism of the fppf abelian sheaf J_{∞}) for α_{s} ∈ End(J_{s}), B_{P} = J_{r}/α_{r}(J_{r}), and the abelian variety A_{P} is the connected component of J_{r}[α_{r}] = Ker(α_{r}). For a finite extension k of ℚ or ℚ_{l} (for a prime l), we show in Section 4 that the Pontryagin dual G_{𝕋}(k)^{∨} is often a finite module and at worst is a torsion Λ-module of finite type. In this paper as Proposition 6.4, we prove the following exact sequence:

(1.1)  Â_{P} (k)^{ord,𝕋} ⟶^{ι_{∞}} J̌_{∞}(k)^{ord,𝕋} ⟶^{α} J̌_{∞}(k)^{ord,𝕋},

where Ker(ι_{∞}) is finite and Coker(α) is a ℤ_{p}-module of finite type with free rank less than or equal to dim_{ℚ_{p}} B̂_{r}(k)⊗_{ℤ_{p}} ℚ_{p}. The main result Theorem 6.6 of this paper is basically the ℤ_{p}-dual version of Proposition 6.4 for J̌_{∞}(k)_{ord,𝕋}^{∗} := Hom_{ℤ_{p}}(J̌_{∞}(k)^{ord,𝕋}, ℤ_{p}). Here is a shortened statement of our main theorem (Theorem 6.6 in the text):

Theorem 1.1. The sequence ℤ_{p}-dual to the one in (1.1):

(1.2)  0 → Coker(α)_{𝕋}^{∗} → J̌_{∞}(k)_{ord,𝕋}^{∗} → J̌_{∞}(k)_{ord,𝕋}^{∗} → Â_{P} (k)_{ord,𝕋}^{∗} → 0

is exact up to finite error.

In Theorem 6.6, we give many control sequences similar to (1.2) for other incarnations of J̌_{∞}(k)_{ord,𝕋}^{∗}.

These modules J̌_{∞}(k)_{ord,𝕋}^{∗} are modules over the big ordinary Hecke algebra h. We cut down these modules to an irreducible component Spec(𝕀) of Spec(h). In other words, we study the following 𝕀-modules:

J̌_{∞}(k)_{𝕀}^{ord} := J̌_{∞}(k)^{ord} ⊗_{h} 𝕀.

We could ask diverse questions out of our control theorem. For example, when is A_{P} (κ) dense in A_{P} (κ_{𝔭}) for a prime 𝔭|p of a number field κ? We can answer this question for almost all P if κ_{𝔭} = ℚ_{p} and dim_{ℚ} A_{P_{0}}(κ) ⊗_{ℤ} ℚ > 0 for one sufficiently generic P_{0} (see Corollary 7.2). In [H14b], we extend the control result to the projective limit \underleftarrow{lim}_{r} Ĵ_{r}(k)_{𝕋}^{ord}. In a forthcoming paper [H14c], we prove “almost” constancy of the Mordell–Weil rank of Shimura’s abelian variety in a p-adic analytic family. Our point is that we have a control theorem of the limit Mordell–Weil groups (under mild assumptions) which is possibly smaller than the Selmer groups studied more often. We hope to discuss the relation of our result to the limit Selmer group studied by Nekovár in [N06] in our future paper. The control theorems for h proven for p ≥ 5 in [H86a] and [H86b] and in [GME, Corollary 3.2.22] for general p assert that, for p > 2, the quotient h/(γ^{p^{r−1}} − 1)h is canonically isomorphic to the Hecke algebra h_{r} (r > 0) in End_{ℤ_{p}}(J_{r}[p^{∞}]^{ord}) generated over ℤ_{p} by Hecke operators T (n) (while for p = 2, h/(γ^{p^{r−2}} − 1)h ≅ h_{r} for r ≥ 2). By this control result, we showed that h is a free of finite rank over Λ (see [GK13] for the treatment for p = 2).

We recall succinctly how these control theorems were proven in [H86b] (and in [H86a]) for p ≥ 5, as it gives a good introduction to the methods used in the present paper. The arguments in these papers work well for p = 2, 3 assuming that Np^{r} ≥ 4 (see [GK13] for details in the case of p = 2). We have a well known commutative diagram of U(p^{s−r})-operators:

(1.3)  \begin{matrix}J_{r,R} & ⟶^{π^{∗}} & J_{s,R}^{r} \\ ↓ u & ↙ u′ & ↓ u′′ \\ J_{r,R} & ⟶^{π^{∗}} & J_{s,R}^{r}, \end{matrix}

where the middle u′ is given by U_{r}^{s}(p^{s−r}) and u and u′′ are U(p^{s−r}). These operators comes from the double coset Γ ( \begin{matrix}1 & 0 \\ 0 & p^{s−r} \end{matrix} ) Γ′ for Γ = Γ_{s}^{r} = Γ_{0}(p^{s}) ∩

Γ_{1}(Np^{r}) and Γ′ = Γ_{s′}^{r′} for suitable s ≥ r, s′ ≥ r′. Note that U(p^{n}) = U(p)^{n}. Then the above diagram implies

(1.4)  J_{r/ℚ}[p^{∞}]^{ord} ≅ J_{s/ℚ}^{r}[p^{∞}]^{ord} and Ĵ_{r/ℚ}(k)^{ord} ≅ Ĵ_{s/ℚ}^{r}(k)^{ord}.

The commutativity of the diagram (1.3) and the level lowering (1.4) are universally true even when we replace the fppf abelian sheaf J_{r} by any fppf sheaf with reasonable U(p)-action compatible with the modular tower ⋯ → X_{r} → ⋯ → X_{1}. For computational purpose, in [H86b], we identified J(ℂ) with a subgroup of H^{1}(Γ, T) (for the Γ-module T := ℝ/ℤ with trivial Γ-action). Since Γ_{s}^{r} ▷ Γ_{1}(Np^{s}), we may consider the finite cyclic quotient group C := \frac{Γ_{s}^{r}}{Γ_{1}(Np^{s})} =

Γ^{p^{r−1}}/Γ^{p^{s−1}}. By the inflation restriction sequence, we have the following commutative diagram with exact rows, writing H^{•}(?, T) as H^{•}(?):

H^{1}(C) ⟶^{↪} H^{1}(Γ_{s}^{r}) ⟶ H^{1}(Γ_{1}(Np^{s}))^{γ^{p^{r−1}} =1} ⟶ H^{2}(C)
↑  ^{∪}↑  ↑^{∪}  ↑
? ⟶ J_{s}^{r}(ℂ) ⟶ J_{s}(ℂ)[γ^{p^{r−1}} − 1] ⟶ ?.

Since H^{2}(C, T) = 0 and U(p)^{s−r}(H^{1}(C, T)) = 0, we have the control of Barsotti–Tate groups (see [H86b] and more recent [H14a, §4–5]):

J_{s}[p^{∞}][γ^{p^{r−1}} − 1]_{/ℂ}^{ord} ≅ J_{r}[p^{∞}]_{/ℂ}^{ord}.

Out of this control by the Γ-action of the ordinary Barsotti–Tate groups J_{r}[p^{∞}]^{ord}, we proved the control of h (cited above) by the diamond operators. A suitable power of U(p)-operator killing the kernel and cokernel of the restriction maps in (1) should be also universally true not just over ℂ but over smaller rings. We will study almost the same diagram obtained by replacing H^{1}(?, T) for ? = Γ_{1}(Np^{s}) and Γ_{s}^{r} by H_{fppf}^{1}(X_{/ℚ}, O_{X}^{×}) = Pic_{X/ℚ} for X = X_{s} and X_{s}^{r}. In an algebro-geometric way, we verify that an appropriate power of the U(p)- operator kills the corresponding kernel and cokernel. Technical points aside, this is a key to the proof of Theorem 1.1. This principle should hold for more general sheaves (under a Grothendieck topology) with U(p)-action compatible with the modular tower, and the author plans to present many other examples of such in his forthcoming papers. We call a point P ∈ Spec(h)(\overline{ℚ}_{p}) an arithmetic point of weight 2 if P (γ^{p^{j}} −1) = 0 for some integer j ≥ 0. Though the construction of the big Hecke algebra is intrinsic, to relate an algebra homomorphism λ : h → \overline{ℚ}_{p} killing γ^{p^{r−1}} − 1 for sufficiently large r > 0 to a classical Hecke eigenform, we need to fix (once

and for all) an embedding \overline{ℚ} ⟶^{i_{p}} \overline{ℚ}_{p} of the algebraic closure \overline{ℚ} in ℂ into a fixed algebraic closure \overline{ℚ}_{p} of ℚ_{p}.

Contents

1. Introduction 221 2. Sheaves associated to abelian varieties 226 3. U(p)-isomorphisms 231 4. Structure of Λ-BT groups over number fields and local fields 239 5. Abelian factors of modular Jacobians 243 6. Structure of ind-Λ-MW groups over number fields and local field 246 7. Closure of the global Λ-MW group in the local one 259 References 263

2. Sheaves associated to abelian varieties

Let k be a finite extension of ℚ or the l-adic field ℚ_{l}. In this section, we set the notation used in the rest of the paper and present a general fact about an exact sequence of abelian varieties. Let 0 → A → B → C → 0 be an exact sequence of algebraic groups proper over the field k. We assume that B and C are abelian varieties. However A can be an extension of an abelian variety by a finite (étale) group. If k is a number field, let S be a finite set of places where all members of the above exact sequence have good reduction outside S; so, all archimedean places are included in S. Let K = k^{S} (the maximal extension unramified outside S). If k is a finite extension of ℚ_{l}, we put K = \overline{k} (an algebraic closure of k). A general field extension of k is denoted by κ. We consider the étale topology, the smooth topology and the fppf topology on the small site over Spec(k). Here under the smooth topology, covering families are made of faithfully flat smooth morphisms. We want to define p-adically completed sheaves X̂ for X = A, B, C as above defined over these sites. The word “p-adically completed” does not always mean X̂(R) is given by the projective limit \underleftarrow{lim}_{n} X(R)/p^{n}X(R), and the definition depends on the choice of k. For the moment, assume that k is a number field. In this case, for an extension X of abelian variety defined over k by a finite flat group scheme, we define X̂(F) := X(F) ⊗_{ℤ} ℤ_{p} for an fppf extension F over k. We may regard its p-adic “completion” 0 → Â → B̂ → Ĉ → 0 as an exact sequence of fppf/smooth/étale abelian sheaves over k (or over any subring of k over which B and C extend to abelian schemes). Here the word “completion” means tensoring with ℤ_{p} over ℤ. Indeed, for any ring R of finite type over k, R ↦ Ĉ(R) := C(R) ⊗_{ℤ} ℤ_{p} is an exact functor from the category of abelian fppf/smooth/étale sheaves into itself; therefore, the tensor construction gives Ĉ(κ) = \underleftarrow{lim}_{n} C(κ)/p^{n}C(κ) if κ is a field of finite type, since C(κ) is an abelian group of finite type by a generalized Mordell-Weil theorem (e.g., [RTP, IV]). Let ϵ denote the dual number. Then we have a canonical identification Lie(C)_{/κ} = Ker(C(κ[ϵ]) → C(κ)) (e.g. [EAI, §10.2.4]), and hence Lie(C) ⊗_{ℤ} ℤ_{p} = Ker(Ĉ(κ[ϵ]) → Ĉ(κ)) is the p-adic completion of the κ-vector space Lie(C) if κ is a finite extension of k. Since we find a complementary abelian subvariety C′ of B such that C′ is isogenous to C and B = A + C′ with finite A ∩ C′, adding the primes dividing the order |A ∩ C′| to S, the intersection A ∩ C′ ≅ Ker(C′ → C) extends to an étale finite group scheme outside S; so, C′(K) → C(K) is surjective. Thus we have an exact sequence of Gal(K/k)-modules

0 → A(K) → B(K) → C(K) → 0.

Note that Â(K) = A(K) ⊗_{ℤ} ℤ_{p} := ⋃_{F} Â(F) for F running over all finite extensions of k inside K. Then we have an exact sequence

(2.1)  0 → Â(K) → B̂(K) → Ĉ(K) → 0.

Now assume that k is a finite extension of ℚ_{l}. We put K = \overline{k} (an algebraic closure of k). Suppose that F is a finite extension of k. Then A(F) = O_{F}^{dim A} ⊕ Δ_{F} for a finite group Δ_{F} and the l-adic integer ring O_{F} of F (see [M55] ot [T66]). Now suppose l ≠ p. For an fppf extension R_{/k}, we define again

Â(R) := A[p^{∞}](R) = \underrightarrow{lim}_{n} A[p^{n}] for A[p^{n}] := Ker(A(R) ⟶^{p^{n}} A(R)). Then we have Â(F) = \underleftarrow{lim}_{n} A(F)/p^{n}A(F) = Δ_{F,p} := Δ_{F} ⊗_{ℤ} ℤ_{p}, and we have Â(K) = \underrightarrow{lim}_{F} Â(F) = A[p^{∞}](K), and Â, B̂ and Ĉ are identical to the fppf/smooth/étale abelian sheaves A[p^{∞}], B[p^{∞}] and C[p^{∞}], where X[p^{∞}] := \underrightarrow{lim}_{n} X[p^{n}] as an ind finite flat group scheme with X[p^{n}] = Ker(p^{n} : X → X) for X = A, B, C. We again have the exact sequence (2.1) of Gal(\overline{k}/k)-modules:

0 → Â(K) → B̂(K) → Ĉ(K) → 0

and an exact sequence of fppf/smooth/étale abelian sheaves

0 → Â → B̂ → Ĉ → 0

whose value at finite extension κ/ℚ_{l} coincides with the projective limit X̂(κ) = \underleftarrow{lim}_{n} X(κ)/p^{n}X(κ) for X = A, B, C. Suppose l = p. For any module M, we define M^{(p)} by the maximal primeto-p torsion submodule of M. For X = A, B, C and an fppf extension R_{/k}, the sheaf R ↦ X^{(p)}(R) = \underrightarrow{lim}_{p∤N} X[N](R) is an fppf/smooth/étale abelian

sheaf. Then we define the fppf/smooth/étale abelian sheaf X̂ by the sheaf quotient X/X^{(p)}. Since X(F) = O_{F}^{dim X} ⊕ X[p^{∞}](F) ⊕ X^{(p)}(F) for a finite extension F_{/k}, on the étale site over k, X̂ is the sheaf associated to a presheaf R ↦ X(R)/X^{(p)}(R) = O_{F}^{dim X} ⊕ X[p^{∞}](R). If X has semi-stable reduction over O_{F}, we have X̂(F) = X^{∘}(O_{F}) + X[p^{∞}](F) ⊂ X(F) for the formal group X^{∘} of the identity connected component of the Néron model of X over O_{F}. Since X becomes semi-stable over a finite Galois extension F_{0}/k, in general X̂(F) = H^{0}(Gal(F_{0}F/F), X(F_{0}F)) for any finite extension F_{/K} (or more generally for each finite ’etale extension F_{/k}); so, F ↦ X̂(F) is a sheaf over the étale site over k. Thus by [ECH, II.1.5], the sheafication coincides over the étale site with the presheaf F ↦ \underleftarrow{lim}_{n} X(F)/p^{n}X(F). Thus we conclude X̂(F) = \underleftarrow{lim}_{n} X(F)/p^{n}X(F) for any étale finite extensions F_{/k}. Moreover X̂(K) = ⋃_{F} X̂(F). Applying the snake lemma to the commutative diagram with exact rows (in the category of fppf/smooth/étale abelian sheaves):

A^{(p)} ⟶^{↪} B^{(p)} ⟶^{↠} C^{(p)}
∩↓  ^{∩}↓  ^{∩}↓
A ⟶_{↪} B ⟶_{↠} C,

the cokernel sequence gives rise to an exact sequence of fppf/smooth/étale abelian sheaves over k:

0 → Â → B̂ → Ĉ → 0

and an exact sequence of Gal(\overline{k}/k)-modules

0 → Â(K) → B̂(K) → Ĉ(K) → 0.

In this way, we extended the étale sheaves Â, B̂, Ĉ defined on the étale site over Spec(k) to an abelian sheaves on the smooth, fppf and étale sites keeping the exact sequence Â ↪ B̂ ↠ Ĉ intact. However note that our way of defining X̂ for X = A, B, C depends on the base field k = ℚ, ℚ_{p}, ℚ_{l}. In summary, we have, for fppf algebras R_{/k}:

(S) X̂(R) = \begin{cases}X(R) ⊗_{ℤ} ℤ_{p} & if [k : ℚ] < ∞, \\ X[p^{∞}](R) & if [k : ℚ_{l}] < ∞ (l ≠ p), \\ (X/X^{(p)})(R) as a sheaf quotient & if [k : ℚ_{p}] < ∞. \end{cases}

Lemma 2.1. Let the notation be as above (in particular, X is an extension of an abelian variety over k by a finite étale group scheme). If κ is either an integral domain or a field of finite type over k and either k is a number field or a local field with residual characteristic l ≠ p, we have X̂(κ) = \underleftarrow{lim}_{n} X̂(κ)/p^{n}X̂(κ). If κ is an étale extension of finite type over k and k is a p-adic field, we again have X̂(κ) = \underleftarrow{lim}_{n} X̂(κ)/p^{n}X̂(κ).

Proof. First suppose that k is a number field. If κ is a field extension of finite type over k, by [RTP, IV], X(κ) is a ℤ-module of finite type; so, we have X̂(κ) = X(κ) ⊗_{ℤ} ℤ_{p} = \underleftarrow{lim} X(κ)/p^{n}X(κ). Here the first identity is just by the definition. More generally, if κ/k is a Krull domain of finite type over k, κ is a normal noetherian domain; and κ = ⋂_{V} V for discrete valuation ring V in Q(κ) containing κ. By projectivity of the abelian variety, we have X(V) = X(Q(κ) (by the valuative criterion of properness), which implies X(κ) = ⋂_{V} X(V) = X(Q(κ)) (so, X̂(κ) = X̂(Q(κ))) for the quotient field Q(κ) of κ. In particular, if κ is a smooth extension of finite type, an the result follows, Since the normalization κ̃ of κ in Q(κ) is a Krull domain, we find X̂(κ) ⊂ X̂(κ̃) = X̂(Q(κ)); so, X̂(κ) is an abelian group of finite type as long as κ is an integral domain of finite type over k (and hence is a reduced algbera of finite type over k). If k is local of residual characteristic l ≠ p, we have X̂ = X[p^{∞}]. If κ is an integral domain of finite type over k, then X̂(κ) is a finite p-group, and the result is obvious.

The case where k is local of residual characteristic p is already dealt with before the lemma. □

For a sheaf X under the topology ?, we write H_{?}^{•}(X) for the cohomology group H_{?}^{1}(Spec(κ), X) under the topology ?. If we have no subscript, H^{1}(X) means the Galois cohomology H^{•}(Gal(K/κ), X) for the Gal(K/κ)-module X.

Lemma 2.2. Let X be an extension of an abelian variety over k by a finite étale group scheme of order prime to p. For any intermediate extension K/κ/k, We have a canonical injection

\underleftarrow{lim}_{n} X̂(κ)/p^{n}X̂(κ) ↪ \underleftarrow{lim}_{n} H^{1}(X[p^{n}]).

Similarly, for any fppf, smooth or étale extension κ/k of finite type which is an integral domain, we have an injection

\underleftarrow{lim}_{n} X̂(κ)/p^{n}X̂(κ) ↪ \underleftarrow{lim}_{n} H_{?}^{1}(X[p^{n}])

for ? = fppf, sm or ét according as κ/k is an fppf extension or a smooth extension.

By Lemma 2.1, we have X̂(κ) = \underleftarrow{lim}_{n} X̂(κ)/p^{n}X̂(κ) in the following cases: (2.2)

[k : ℚ] < ∞ and κ is an integral domain of finite type over k

[k : ℚ_{l}] < ∞ with l ≠ p and κ is an integral domain of finite type over k [k : ℚ_{p}] < ∞ and κ is a finite algebraic extension over k.

Proof. We consider the sheaf exact sequence under the topology ? = fppf or sm or étale on Spec(κ)

0 → X[p^{n}] → X̂ ⟶^{p^{n}} X̂.

We want to show that the multiplication by p^{n} is surjective. If our cohomology theory is Galois cohomology (or equivalently ? = étale), we have an exact sequence

0 → X[p^{n}](K) → X(K) ⟶^{p^{n}} X(K) → 0.

Since X̂(K) = X(K) ⊗_{ℤ} ℤ_{p}, the desired exactness follows. Let κ be an fppf extension of k. Then for each x ∈ X(κ), we consider the Cartesian diagram

X_{x} ⟶ X
↓  ↓^{p^{n}}
Spec(κ) ⟶_{x} X.

Then X_{x} ≅ X[p^{n}] as schemes over κ; so, X_{x} = Spec(R) for an étale finite extension R of κ, which is obviously smooth and also fppf extension of κ. Thus over the covering R/κ, x is the image of the point given by Spec(R) ↪ X. Then

by [ECH, II.2.5 (c)], X ⟶^{p^{n}} X is an epimorphism of sheaves under étale, smooth and also fppf topology. If k is a number field, we have X̂(κ) = X(κ) ⊗_{ℤ} ℤ_{p}, we get the exactness of X[p^{n}] ↪ X̂ ↠ X̂ from the exactness of X[p^{n}] ↪ X ↠ X. If k is a finite extension of ℚ_{l} for l ≠ p, we can argue as above replacing X by X̂ = X[p^{∞}] and get the exactness of X[p^{n}] ↪ X̂ ↠ X̂. Suppose that k is a finite extension of ℚ_{p}. Then X̂ = X/X^{(p)} as a ?-sheaf. Take x ∈ X̂(κ). Then by definition, we have an ?-extension R of κ such that x is the image of y ∈ X(R). Then as above we can find a ?-extension R′/R such that y = p^{n}y′ for y′ ∈ X(R′). Then for the image x′ of y′ ∈ X(R′) in X̂(R′), we have p^{n}x′ = x.

Thus again X̂ ⟶^{p^{n}} X̂ is an epimorphism of sheaves under the topology ?. Thus we can apply Kummer theory to the sheaf exact sequence

0 → X[p^{n}] ↪ X̂ ⟶^{p^{n}} X̂ → 0

with respect to the topology given by ?, we have an inclusion X̂(κ)/p^{n}X̂(κ) ↪ H_{?}^{1}(X[p^{n}]). Passing to the limit with respect to n, we have δ : \underleftarrow{lim}_{n} X(κ)/p^{n}X(κ) → \underleftarrow{lim}_{n} H_{?}^{1}(X[p^{n}]). Since taking projective limit is a left exact functor, δ is injective as desired. □

Taking instead an injective limit, we get

Lemma 2.3. Let A be an abelian variety over k. For any intermediate extension K/κ/k, we have an exact sequence

0 → Â(κ) ⊗_{ℤ_{p}} 𝕋_{p} → H_{?}^{1}(A[p^{∞}]) → H_{?}^{1}(Â) → 0

for ? = fppf, sm or ét according as κ/k is an fppf extension, a smooth extension or an étale extension. In particular, the Pontryagin dual of H_{?}^{1}(Â) is a ℤ_{p}- module of finite type; so, H_{?}^{1}(Â) has the form (ℚ_{p}/ℤ_{p})^{j} ⊕Δ for some 0 ≤ j ∈ ℤ and a finite p-group Δ.

Proof. Since any smooth covering has finer étale covering, we have H_{sm}^{•}(Â) = H_{ét}^{•}(Â) (cf. [ECH, III.3.4 (c)]). Since an étale covering is covered by a finer étale finite coverings, H_{ét}^{q}(Â) and H^{q}(A) for q > 0 is a torsion module. This torsionness is well known for Galois cohomology (as the Galois group is profinite; see [CNF, (1.6.1)]). Pick any x ∈ Â(κ). We can find an étale finite extension κ′/κ such that p^{n}y = x for some y ∈ Â(κ′). Then y is unique modulo Â[p^{n}](κ′). Therefore, the sheaf quotient (Â/A[p^{∞}])(κ) is p-divisible and torsion-free; so, is a sheaf of ℚ_{p}-vector spaces. In other words, Â/A[p^{∞}] is isomorphic to the sheaf tensor product Â ⊗_{ℤ_{p}} ℚ_{p}. Thus we have an exact sequence

0 → A[p^{∞}] → Â → Â ⊗_{ℤ_{p}} ℚ_{p} → 0.

Since H_{?}^{1}(Â ⊗_{ℤ_{p}} ℚ_{p}) is a ℚ_{p}-vector space, the image in H_{?}^{1}(Â ⊗_{ℤ_{p}} ℚ_{p}) of the torsion module H^{1}(Â) vanishes. Thus we have an exact sequence

0 → Â(κ) ⊗_{ℤ_{p}} 𝕋_{p} → H_{?}^{1}(A[p^{∞}]) → H_{?}^{1}(Â) → 0.

Since 0 → Â(κ) ⊗_{ℤ_{p}} ℤ/pℤ → H_{?}^{1}(Â[p]) → H_{?}^{1}(Â)[p] → 0 is exact, by the finiteness of H^{1}(Â[p]) = H^{1}(A[p]) (see [ADT, I.5]), the last assertion for Galois cohomology follows. Then using the comparison theorem (cf. [ECH, III.3.4 (c) and III.3.9]), we conclude the same for other topologies. □

3. U(p)-isomorphisms

We recall the results in [H14b, §3] with detailed proofs for some results and a brief account for some others (as [H14b] is being written along with this paper). For Z[U]-modules X and Y, we call a ℤ[U]-linear map X ⟶^{f} Y a U-injection (resp. a U-surjection) if Ker(f) is killed by a power of U (resp. Coker(f) is killed by a power of U). If f is both U-injection and U-surjection, we call f is a U-isomorphism. Thus, f is a U-injection (resp. a U-surjection, a U-isomorphism) if after tensoring ℤ[U, U^{−1}], it becomes an injection (resp. a surjection, an isomorphism). In terms of U-isomorphisms, we describe briefly the facts we study in this article (and in later sections, we fill in more details in terms of the ordinary projector e). Let N be a positive integer prime to p. We assume Np^{r} ≥ 4 (without losing any generality as remarked in the introduction). We consider the (open) modular curve Y_{1}(Np^{r})_{/ℚ} which classifies elliptic curves E with an embedding ϕ : μ_{p^{r}} ↪ E[p^{r}] = Ker(p^{r} : E → E). Let R_{i} = ℤ_{(p)}[μ_{p^{i}}], K_{i} = ℚ[μ_{p^{i}}], R_{∞} = ⋃_{i} R_{i} ⊂ \overline{ℚ} and K_{∞} = ⋃_{i} K_{i} ⊂ \overline{ℚ}. For a valuation subring or a subfield R of K_{∞} over ℤ_{(p)} with quotient field K, we write X_{r/R} for the normalization of the j-line P(j)_{/R} in the function field of Y_{1}(Np^{r})_{/K}. The group z ∈ (ℤ/p^{r}ℤ)^{×} acts on X_{r} by ϕ ↦ ϕ∘z, as Aut(μ_{Np^{r}}) ≅ (ℤ/Np^{r}ℤ)^{×}. Thus Γ = 1+pℤ_{p} = γ^{ℤ_{p}} acts on X_{r} (and its Jacobian) through its image in (ℤ/Np^{r}ℤ)^{×}. Only in the following section, we need the result over a discrete valuation ring R. Hereafter, in most cases, we take U = U(p) for the Hecke-Atkin operator U(p) (though we take U = U^{∗}(p) sometimes for the dual U^{∗}(p) of U(p)). Let J_{r/R} = Pic_{X_{r}/R}^{0} be the connected component of the Picard scheme. We state a result comparing J_{r/R} and the Néron model of J_{r/K} over R. Thus we assume that R is a valuation ring. By [AME, 13.5.6, 13.11.4], X_{r/R} is regular; the reduction X_{r} ⊗_{R} 𝔽_{p} is a union of irreducible components, and the component containing the ∞ cusp has geometric multiplicity 1. Then by [NMD, Theorem 9.5.4], J_{r/R} gives the identity connected component of the Néron model of the Jacobian of X_{r/R}. We write X_{r/R}^{s} for the normalization of the j-line in the function field of the canonical ℚ-curve associated to the modular curve of the congruence subgroup Γ_{s}^{r} = Γ_{1}(Np^{r}) ∩ Γ_{0}(p^{s}) (for 0 < r ≤ s). The open curve Y_{s/ℚ}^{r} = X_{s/ℚ}^{r} − {cusps} classifies triples (E, C, ϕ : μ_{Np^{r}} ↪ E) with a cyclic subgroup C of order p^{s} containing the image ϕ(μ_{p^{r}}).

We denote Pic_{X_{s}^{r}/R}^{0} by J_{s/R}^{r}. Similarly, as above, J_{s/R}^{r} is the connected component of the Néron model of X_{s/K}^{r}. Note that

(3.1) Γ_{s}^{r}\Γ_{s}^{r} ( \begin{matrix}1 & 0 \\ 0 & p^{s−r} \end{matrix} ) Γ_{1}(Np^{r})

= \{( \begin{matrix}1 & a \\ 0 & p^{s−r} \end{matrix} ) |a mod p^{s−r}\} = Γ_{1}(Np^{r})\Γ_{1}(Np^{r}) ( \begin{matrix}1 & 0 \\ 0 & p^{s−r} \end{matrix} ) Γ_{1}(Np^{r}).

Write U_{r}^{s}(p^{s−r}) : J_{r/R}^{s} → J_{r/R} for the Hecke operator of Γ_{r}^{s}α_{s−r}Γ_{1}(Np^{r}) for α_{m} = ( \begin{matrix}1 & 0 \\ 0 & p^{m} \end{matrix} ). Strictly speaking, the Hecke operator induces a morphism of the generic fiber of the Jacobians and then extends to their connected components of the Néron models by the functoriality of the model (or equivalently by Picard functoriality). Then we have the following commutative diagram from the above identity, first over ℂ, then over K and by Picard functoriality over R:

(3.2)  \begin{matrix}J_{r/R} & ⟶^{π^{∗}} & J_{s/R}^{r} \\ ↓ u & ↙ u′ & ↓ u′′ \\ J_{r/R} & ⟶^{π^{∗}} & J_{s/R}^{r}, \end{matrix}

where the middle u′ is given by U_{r}^{s}(p^{s−r}) and u and u′′ are U(p^{s−r}). Thus

(u1) π^{∗} : J_{r/R} → J_{s/R}^{r} is a U(p)-isomorphism (for the projection π : X_{s}^{r} →

X_{r}). Taking the dual U^{∗}(p) of U(p) with respect to the Rosati involution associated to the canonical polarization on the Jacobians. We have a dual version of the above diagram for s > r > 0:

(3.3)  \begin{matrix}J_{r/R} & ⟵^{π_{∗}} & J_{s/R}^{r} \\ ↑ u^{∗} & ↗ u′^{∗} & ↑ u′′^{∗} \\ J_{r/R} & ⟵^{π_{∗}} & J_{s/R}^{r}. \end{matrix}

Here the superscript “∗” indicates the Rosati involution corresponding to the canonical divisor on the Jacobians, and u^{∗} = U^{∗}(p)^{s−r} for the level Γ_{1}(Np^{r}) and u′′^{∗} = U^{∗}(p)^{s−r} for Γ_{s}^{r}. Note that these morphisms come from the following double coset identity:

(3.4) Γ_{s}^{r}\Γ_{s}^{r} ( \begin{matrix}p^{s−r} & 0 \\ 0 & 1 \end{matrix} ) Γ_{1}(Np^{r})

= \{( \begin{matrix}p^{s−r} & a \\ 0 & 1 \end{matrix} ) |a mod p^{s−r}\} = Γ_{1}(Np^{r})\Γ_{1}(Np^{r}) ( \begin{matrix}p^{s−r} & 0 \\ 0 & 1 \end{matrix} ) Γ_{1}(Np^{r}).

From this, we get (u^{∗}1) π_{∗} : J_{s/R}^{r} → J_{r/R} is a U^{∗}(p)-isomorphism, where π_{∗} is the dual of π^{∗}.

In particular, if we take the ordinary and the co-ordinary projector e = lim_{n→∞} U(p)^{n!} and e^{∗} = lim_{n→∞} U^{∗}(p)^{n!} on J[p^{∞}] for J = J_{r/R}, J_{s/R}, J_{s/R}^{r}, noting U(p^{m}) = U(p)^{m}, we have

π^{∗} : J_{r/R}^{ord}[p^{∞}] ≅ J_{s/R}^{r,ord}[p^{∞}] and π_{∗} : J_{s/R}^{r,co-ord}[p^{∞}] ≅ J_{r/R}^{co-ord}[p^{∞}] where “ord” (resp. “co-ord”) indicates the image of the projector e (resp. e^{∗}). For simplicity, we write G_{r/R} := J_{r/R}^{ord}[p^{∞}]_{/R}.

Suppose that we have morphisms of three noetherian schemes X ⟶^{π} Y ⟶^{g} S with f = g ∘ π. We look into

H_{fppf}^{0}(T, R^{1}f_{∗}𝔾_{m}) = R^{1}f_{∗}O_{X}^{×}(T) = Pic_{X/S}(T)

for S-scheme T and the structure morphism f : X → S (see [NMD, Chapter 8]). Suppose that f and g have compatible sections S ⟶^{s_{g}} Y and S ⟶^{s_{f}} X so that π ∘ s_{f} = s_{g}. Then we get (e.g., [NMD, Section 8.1])

Pic_{X/S}(T) = Ker(s_{f}^{1} : H_{fppf}^{1}(X_{T} , O_{X}^{×}) → H_{fppf}^{1}(T, O_{T}^{×}))
Pic_{Y/S}(T) = Ker(s_{g}^{1} : H_{fppf}^{1}(Y_{T} , O_{Y_{T}}^{×}) → H_{fppf}^{1}(T, O_{T}^{×}))

for any S-scheme T, where s_{f}^{q} : H_{fppf}^{q}(X_{T} , O_{X_{T}}^{×}) → H_{fppf}^{q}(T, O_{T}^{×}) and s_{g}^{n} : H_{fppf}^{n}(Y_{T} , O_{Y_{T}}^{×}) → H_{fppf}^{n}(T, O_{T}^{×}) are morphisms induced by s_{f} and s_{g}, respectively. Here we wrote X_{T} = X ×_{S} T and Y_{T} = Y ×_{S} T. We suppose that the functors Pic_{X/S} and Pic_{Y/S} are representable by smooth group schemes (for example, if X, Y are curves and S = Spec(k) for a field k; see [NMD, Theorem 8.2.3 and Proposition 8.4.2]). We then put J_{?} = Pic_{?/S}^{0} (? = X, Y). Anyway we suppose hereafter also that X, Y, S are varieties (in the sense of [ALG, II.4]). For an fppf covering U → Y and a presheaf P = P_{Y} on the fppf site over Y, we define via Čech cohomology theory an fppf presheaf U ↦ Ȟ^{q}(U, P) denoted by \underline{Ȟ}^{q}(P_{Y}) (see [ECH, III.2.2 (b)]). The inclusion functor from the category of fppf sheaves over Y into the category of fppf presheaves over Y is left exact. The derived functor of this inclusion of an fppf sheaf F = F_{Y} is denoted by \underline{H}^{•}(F_{Y}) (see [ECH, III.1.5 (c)]). Thus \underline{H}^{•}(𝔾_{m/Y})(U) = H_{fppf}^{•}(U, O_{U}^{×}) for a Y -scheme U as a presheaf (here U varies in the small fppf site over Y). Assuming that f, g and π are all faithfully flat of finite presentation, we use the spectral sequence of Čech cohomology of the flat covering π : X ↠ Y in the fppf site over Y [ECH, III.2.7]:

(3.5)  Ȟ^{p}(X_{T} /Y_{T} , \underline{H}^{q}(𝔾_{m/Y})) ⇒ H_{fppf}^{n}(Y_{T} , O_{Y_{T}}^{×}) ⟶_{ι}^{∼} H^{n}(Y_{T} , O_{Y_{T}}^{×})

for each S-scheme T. Here F ↦ H_{fppf}^{n}(Y_{T} , F) (resp. F ↦ H^{n}(Y_{T} , F)) is the right derived functor of the global section functor: F ↦ F (Y_{T}) from the category of fppf sheaves (resp. Zariski sheaves) over Y_{T} to the category of abelian groups. The canonical isomorphism ι is the one given in [ECH, III.4.9]. By the sections s_{?}, we have a splitting H^{q}(X_{T} , O_{X_{T}}^{×}) = Ker(s_{f}^{q}) ⊕ H^{q}(T, O_{T}^{×}) and H^{n}(Y_{T} , O_{Y_{T}}^{×}) = Ker(s_{g}^{n}) ⊕ H^{n}(T, O_{T}^{×}). Write \underline{H}_{Y_{T}}^{•} for \underline{H}^{•}(𝔾_{m/Y_{T}}) and Ȟ^{•}(\underline{H}_{Y_{T}}^{0} ) for Ȟ^{•}(Y_{T} /X_{T} , \underline{H}_{Y_{T}}^{0} ). Since

Pic_{X/S}(T) = Ker(s_{f,T}^{1} : H^{1}(X_{T} , O_{X_{T}}^{×}) → H^{1}(T, O_{T}^{×}))

for the morphism f : X → S with a section [NMD, Proposition 8.1.4], from this spectral sequence, we have the following commutative diagram with exact rows, writing Ȟ^{0}(\frac{X_{T}}{Y_{T}} ,?) as Ȟ^{0}(?) and H^{1}(T, O_{T}^{×}) as H^{1}(O_{T}^{×}): (3.6)

?_{1}  ⟶  Ȟ^{1}(\underline{H}_{Y_{T}}^{0} )  Ȟ^{1}(\underline{H}_{Y_{T}}^{0} )
↓  ↓  ↓^{∩}
Pic_{T} ⊕J_{Y} (T) ⟶^{↪} Pic_{T} ⊕ Pic_{Y/S}(T) ⟶^{∼} H^{1}(O_{T}^{×}) ⊕ Ker(s_{g,T}^{1} )
c↓  ^{b}↓  ^{a}↓
Pic_{T} ⊕Ȟ^{0}(J_{X}(T)) ⟶^{↪}  Ȟ^{0}(Pic_{Y} (T))  Ȟ^{0}(\underline{H}^{1}(𝔾_{m,Y}))
↓  ↓  ↓
?_{2}  ⟶  Ȟ^{2}(\underline{H}_{Y_{T}}^{0} )  Ȟ^{2}(\underline{H}_{Y_{T}}^{0} ),

where we have written J_{?} = Pic_{?/S}^{0} (the identity connected component of Pic_{?/S}). Here the vertical exactness at the right two columns follows from the spectral sequence (3.5) (see [ECH, Appendix B]).

We now recall the definition of the Čech cohomology: for a general S-scheme T and Čech cochain c_{i_{0},...,i_{q}} ∈ H^{0}(X_{T}^{(q+1)}, O_{X_{T}^{(q+1)}}^{×}),

(3.7) Ȟ^{q}(\frac{X_{T}}{Y_{T}} , \underline{H}^{0}(𝔾_{m/Y})) =

\frac{\{(c_{i_{0},...,i_{q}})| ∏_{j}(c_{i_{0}...ǐ_{j}...i_{q+1}} ∘ p_{i_{0}...ǐ_{j}...i_{q+1}})^{(−1)^{j}} = 1\}}{\{db_{i_{0}...i_{q}} = ∏_{j}(b_{i_{0}...ǐ_{j}...i_{q}} ∘ p_{i_{0}...ǐ_{j}...i_{q}})^{(−1)^{j}}|b_{i_{0}...ǐ_{j}...i_{q}} ∈ H^{0}(X_{T}^{(q)}, O_{X_{T}^{(q)}}^{×})\}}

where we agree to put H^{0}(X_{T}^{(0)}, O_{X_{T}}^{(0)} ) = 0 as a convention,

X_{T}^{(q)} = \overbrace{X ×_{Y} X ×_{Y} ⋯ ×_{Y} X}^{q} ×_{S}T,

O_{X_{T}^{(q)}} = \overbrace{O_{X} ×_{O_{Y}} O_{X} ×_{O_{Y}} ⋯ ×_{O_{Y}} O_{X}}^{q} ×_{O_{S}}O_{T} ,

the identity ∏_{j}(c∘p_{i_{0}...ǐ_{j}...i_{q+1}})^{(−1)^{j}} = 1 takes place in O_{X_{T}^{(q+2)}} and p_{i_{0}...ǐ_{j}...i_{q+1}} :

X_{T}^{(q+2)} → X_{T}^{(q+1)} is the projection to the product of X the j-th factor removed.

Since T ×_{T} T ≅ T canonically, we have X_{T}^{(q)} ≅ \overbrace{X_{T} ×_{T} ⋯ ×_{T} X_{T}}^{q} by transitivity of fiber product. Take a correspondence U ⊂ Y ×_{S} Y given by two finite flat projections π_{1}, π_{2} : U → Y of constant degree (i.e., π_{j,∗}O_{U} is locally free of finite rank deg(π_{j}) over O_{Y}). Consider the pullback U_{X} ⊂ X ×_{S} X given by the Cartesian diagram:

U_{X} = U ×_{Y ×_{S}Y} (X ×_{S} X) ⟶ X ×_{S} X
↓  ↓
U  ⟶^{↪} Y ×_{S} Y

Let π_{j,X} = π_{j} ×_{S} π : U_{X} ↠ X (j = 1, 2) be the projections. We describe the correspondence action of U on H^{0}(X, O_{X}^{×}) in down-to-earth terms. Consider α ∈ H^{0}(X, O_{X}). Then we lift π_{1,X}^{∗}α = α ∘ π_{1,X} ∈ H^{0}(U_{X}, O_{U_{X}}). Put α_{U} := π_{1,X}^{∗}α. Note that π_{2,X,∗}O_{U_{X}} is locally free of rank d = deg(π_{2}) over O_{X}, the multiplication by α_{U} has its characteristic polynomial P (T) of degree d with coefficients in O_{X}. We define the norm N_{U}(α_{U}) to be the constant term P (0). Since α is a global section, N_{U}(α_{U}) is a global section, as it is defined everywhere locally. If α ∈ H^{0}(X, O_{X}^{×}), N_{U}(α_{U}) ∈ H^{0}(X, O_{X}^{×}). Then define U(α) = N_{U}(α_{U}), and in this way, U acts on H^{0}(X, O_{X}^{×}). For a degree q Čech cohomology class [c] ∈ Ȟ^{q}(X_{/Y} , \underline{H}^{0}(𝔾_{m/Y})) of a Čech q-cocycle c = (c_{i_{0},...,i_{q}}), U([c]) is given by the cohomology class of the Čech cocycle U(c) = (U(c_{i_{0},...,i_{q}})), where U(c_{i_{0},...,i_{q}}) is the image of the global section c_{i_{0},...,i_{q}} under U. Indeed, (π_{1,X}^{∗}c_{i_{0},...,i_{q}}) plainly satisfies the cocycle condition, and (N_{U}(π_{1,X}^{∗}c_{i_{0},...,i_{q}})) is again a Čech cocycle as N_{U} is a multiplicative homomorphism. By the same token, we see that this operation sends coboundaries to coboundaries and obtain the action of U on the cohomology group.

Lemma 3.1. Let the notation and the assumption be as above. In particular, π : X → Y is a finite flat morphism of geometrically reduced proper schemes over S = Spec(k) for a field k. Suppose that X and U_{X} are proper schemes over a field k satisfying one of the following conditions:

(1) U_{X} is geometrically reduced, and for each geometrically connected component X^{∘} of X, its pull back to U_{X} by π_{2,X} is also connected; i.e.,

π^{0}(X) ⟶_{∼}^{π_{2,X}^{∗}} π^{0}(U_{X});

(2) (f ∘ π_{2,X})_{∗}O_{U_{X}} = f_{∗}O_{X}.

If π_{2} : U → Y has constant degree deg(π_{2}), the action of U on H^{0}(X, O_{X}^{×}) factors through the multiplication by deg(π_{2}) = deg(π_{2,X}).

Proof. By properness, under (1) or (2), H^{0}(U_{X}, O_{U_{X}}) =^{π_{2,X,∗}} H^{0}(X, O_{X})(=^{(1)} k^{π^{0}(X)}) for the set of connected components π^{0}(X) of X. In particular, we see α_{U} ∈ H^{0}(U_{X}, O_{U_{X}}) = H^{0}(X, O_{X}), which tells us that N_{U}(α_{U}) = α_{U}^{deg(π_{2})}, and the result follows. □

Consider the iterated product π_{i,X^{(q)}} = π_{i,X} ×_{Y} ⋯ ×_{Y} π_{i,X} : U_{X}^{(q)} → X^{(q)}

(i = 1, 2). Here U_{X}^{(q)} = \overbrace{U_{X} ×_{Y} U_{X} ×_{Y} ⋯ ×_{Y} U_{X}}^{q}. We plug in U_{X}^{(j)} in the first j slots of the fiber product (for 0 < j ≤ q) and consider

U_{X}^{(j−1)} ×_{Y} X^{(q−j+1)} ⟵^{π_{1,j}^{(q)}} U_{j} := U_{X}^{(j)} ×_{Y} X^{(q−j)} ⟶^{π_{2,j}^{(q)}} U_{X}^{(j−1)} ×_{Y} X^{(q−j+1)}

which induces a correspondence U_{j} in (U_{X}^{(j−1)} ×_{Y} X^{(q−j+1)}) ×_{Y} (U_{X}^{(j−1)} ×_{Y} X^{(q−j+1)}). Here π_{i,j} restricted to first j − 1-factors U_{X} is the identity id_{U_{X}}; the last q−j factors is the identity id_{X} and at the j-th factor, it is the projection π_{i} (i = 1, 2). For example, if q = 3 and i = 2, we have

U_{X} ×_{Y} U_{X} ×_{Y} U_{X} ⟶_{id_{U} × id_{U} ×π_{2}}^{π_{2,3}^{(q)}} U_{X} ×_{Y} U_{X} ×_{Y} X

⟶_{id_{U} ×π_{2}×id_{X}}^{π_{2,2}^{(q)}} U_{X} ×_{Y} X ×_{Y} X ⟶_{π_{2}×id_{X} × id_{X}}^{π_{2,1}^{(q)}} X ×_{Y} ×_{Y} X.

Naturally π_{2,X^{(q)}} factors through the following q consecutive coverings U_{q} ⟶^{ρ_{q}}

U_{q−1} ⟶^{ρ_{q−1}} ⋯ ⟶^{ρ_{1}} X^{(q)} for ρ_{j} = π_{2,j}^{(q)}. Note that the norm map N_{U_{q}} =

N_{π_{2,X^{(q)}}} : π_{2,X^{(q)},∗}O_{U_{q}}^{×} → O_{X^{(q)}}^{×} factors through the corresponding norm maps:

N_{U_{q}} = N_{q} ∘ N_{q−1} ∘ ⋯ ∘ N_{1},

where N_{j} is the norm map with respect to U_{j} → U_{j−1}. The last norm is essentially the product of N_{U} and the identity of X^{(q−1)} corresponding to U ×_{Y} X^{(q−1)} ↠ X^{(q)}. In particular, ρ_{1,∗}(O_{U_{1}}) = π_{2,X,∗}(O_{U_{X}}) ⊗_{O_{Y}} O_{X^{(q−1)}} and

(f ∘ ρ_{1})_{∗}(O_{U_{1}}) = (f ∘ π_{2,X})_{∗}(O_{U_{X}}) ⊗_{O_{Y}} \overbrace{f_{∗}O_{X} ⊗_{O_{Y}} ⋯ ⊗_{O_{Y}} f_{∗}O_{X}}^{q−1} .

Thus if the assumption (2) in Lemma 3.1 is satisfied, the corresponding assumption for U_{1} is satisfied. The assumption (1) implies (2) which is really necessary for the proof of Lemma 3.1. Applying the argument proving Lemma 3.1 to the correspondence U_{1} and the last factor N_{1} of the norm, we get

Corollary 3.2. Let the notation and the assumption be as in Lemma 3.1. Then the action of U^{(q)} on H^{0}(X, O_{X^{(q)}}^{×}) factors through the multiplication by deg(π_{2}) = deg(π_{2,X}).

Here is a main result of this section:

Proposition 3.3. Suppose that S = Spec(k) for a field k. Let π : X → Y be a finite flat covering of (constant) degree d of geometrically reduced proper varieties over k, and let Y ⟵^{π_{1}} U ⟶^{π_{2}} Y be two finite flat coverings (of constant degree) identifying the correspondence U with a closed subscheme U ↪^{π_{1}×π_{2}} Y ×_{S} Y. Write π_{j,X} : U_{X} = U ×_{Y} X → X be the base-change to X. Suppose one of the conditions (1) and (2) of Lemma 3.1 for (X, U). Then

(1) The correspondence U ⊂ Y ×_{S} Y sends Ȟ^{q}(\underline{H}_{Y}^{0} ) into deg(π_{2})(Ȟ^{q}(\underline{H}_{Y}^{0} )) for all q > 0.

(2) If d is a p-power and deg(π_{2}) is divisible by p, Ȟ^{q}(\underline{H}_{Y}^{0} ) for q > 0 is killed by U^{M} if p^{M} ≥ d.

(3) The cohomology Ȟ^{q}(\underline{H}_{Y}^{0} ) with q > 0 is killed by d.

Proof. The first assertion follows from Corollary 3.2. Indeed, by (3.7), U^{(q)} acts on each Čech q-cocycle, through an action factoring through the multiplication by deg(π_{2,X}) = deg(π_{2}) by Corollary 3.2. Now we regard X ⟶^{π} Y as a correspondence of Y (with multiplicity d) by the projection π_{1} = π_{2} = π : X → Y. Then [X](c) = dc for c ∈ Ȟ^{q}(X/Y, \underline{H}^{0}(𝔾_{m/Y})). On the other hand, by the definition of the correspondence action, [X] factors through Ȟ^{q}(X/X, \underline{H}^{0}(𝔾_{m/Y})) = 0, and hence dx = 0. This shows that if X/Y is a covering of degree d, Ȟ^{q}(X/Y, \underline{H}^{0}(𝔾_{m/Y})) is killed by d proving (3), and the assertion (2) follows from the first (1). □

We apply the above proposition to (U, X, Y) = (U(p), X_{s}, X_{s}^{r}) with U given by U(p) ⊂ X_{s}^{r} × X_{s}^{r} over ℚ. Indeed, U := U(p) ⊂ X_{s}^{r} × X_{s}^{r} corresponds to X(Γ) given by Γ = Γ_{1}(Np^{r}) ∩ Γ_{0}(p^{s+1}) and U_{X} is given by X(Γ′) for Γ′ = Γ_{1}(Np^{s}) ∩ Γ_{0}(p^{s+1}) both geometrically irreducible curves. In this case π_{1} is induced by z ↦ \frac{z}{p} on the upper complex plane and π_{2} is the natural projection of degree p. In this case, deg(X_{s}/X_{s}^{r}) = p^{s−r} and deg(π_{2}) = p.

Assume that a finite group G acts on X_{/Y} faithfully. Then we have a natural morphism ϕ : X × G → X ×_{Y} X given by ϕ(x, σ) = (x, σ(x)). In other words, we have a commutative diagram

X × G ⟶^{(x,σ)↦σ(x)} X
(x,σ)↦x↓  ↓
X  ⟶ Y,

which induces ϕ : X × G → X ×_{Y} X by the universality of the fiber product. Suppose that ϕ is surjective; for example, if Y is a geometric quotient of X by G; see [GME, §1.8.3]). Under this map, for any fppf abelian sheaf F, we have a natural map Ȟ^{0}(X/Y, F) → H^{0}(G, F (X)) sending a Čech 0-cocycle c ∈ H^{0}(X, F) = F (X) (with p_{1}^{∗}c = p_{2}^{∗}c) to c ∈ H^{0}(G, F (X)). Obviously, by the surjectivity of ϕ, the map Ȟ^{0}(X/Y, F) → H^{0}(G, F (X)) is an isomorphism (e.g., [ECH, Example III.2.6, page 100]). Thus we get

Lemma 3.4. Let the notation be as above, and suppose that ϕ is surjective. For any scheme T fppf over S, we have a canonical isomorphism: Ȟ^{0}(X_{T} /Y_{T} , F) ≅ H^{0}(G, F (X_{T})).

We now assume S = Spec(k) for a field k and that X and Y are proper reduced connected curves. Then we have from the diagram (3.6) with the exact middle two columns and exact horizontal rows:

0  ⟶  ℤ  ℤ  ⟶  0
↑  ^{deg}↑^{onto}  ^{deg}↑^{onto}  ↑
Ȟ^{1}(\underline{H}_{Y}^{0} ) ⟶ Pic_{Y/S}(T) ⟶^{b} Ȟ^{0}(\frac{X_{T}}{Y_{T}} , Pic_{Y/S}(T)) ⟶ Ȟ^{2}(\underline{H}_{Y}^{0} )
↑  ^{∪}↑  ↑^{∪}  ↑
?_{1} ⟶ J_{Y} (T) ⟶_{c} Ȟ^{0}(\frac{X_{T}}{Y_{T}} , J_{X}(T)) ⟶ ?_{2},

Thus we have ?_{j} = Ȟ^{j}(\underline{H}_{Y}^{0} ) (j = 1, 2). By Proposition 3.3, if q > 0 and X/Y is of degree p-power and p| deg(π_{2}), Ȟ^{q}(\underline{H}_{Y}^{0} ) is a p-group, killed by U^{M} for M ≫ 0. Taking (X, Y, U)_{/S} to be (X_{s/ℚ}, X_{s/ℚ}^{r}, U(p))_{/ℚ} for s > r ≥ 1 for p odd and s > r ≥ 2 for p = 2, we get for the projection π : X_{s} → X_{s}^{r}

Corollary 3.5. Let F be a number field or a finite extension of ℚ_{l} (for a prime l not necessarily equal to p). Then we have

(u) π^{∗} : J_{s/ℚ}^{r}(F) → Ȟ^{0}(X_{s}/X_{s}^{r}, J_{s/ℚ}(F)) =^{(∗)} J_{s/ℚ}(F)[γ^{p^{r−1}} − 1] is a U(p)-

isomorphism, where J_{s/ℚ}(F)[γ^{p^{r−1}} − 1] = Ker(γ^{p^{r−1}} − 1 : J_{s}(F) → J_{s}(F)).

From these, we got the following facts as [H14b, Lemma 3.7]

Lemma 3.6. We have morphisms

ι_{s}^{r} : J_{s/ℚ}[γ^{p^{r−1}} − 1] → J_{s/ℚ}^{r} and ι_{s}^{r,∗} : J_{s/ℚ}^{r} → J_{s/ℚ}/(γ^{p^{r−1}} − 1)(J_{s/ℚ})

satisfying the following commutative diagrams:

(3.8)  \begin{matrix}J_{s/ℚ}^{r} & ⟶^{π^{∗}} & J_{s/ℚ}[γ^{p^{r−1}} − 1] \\ ↓ u & ↙ ι_{s}^{r} & ↓ u′′ \\ J_{s/ℚ}^{r} & ⟶^{π^{∗}} & J_{s/ℚ}[γ^{p^{r−1}} − 1], \end{matrix}

and

(3.9)  \begin{matrix}J_{s/ℚ}^{r} & ⟵^{π_{∗}} & J_{s/ℚ}/(γ^{p^{r−1}} − 1)(J_{s/ℚ}) \\ ↑ u^{∗} & ↗ ι_{s}^{r,∗} & ↑ u′′^{∗} \\ J_{s/ℚ}^{r} & ⟵^{π^{∗}} & J_{s/ℚ}/(γ^{p^{r−1}} − 1)(J_{s/ℚ}), \end{matrix}

where u and u′′ are U(p^{s−r}) = U(p)^{s−r} and u^{∗} and u′′^{∗} are U^{∗}(p^{s−r}) = U^{∗}(p)^{s−r}. In particular, for an fppf extension T_{/ℚ}, the evaluated map at T:

(J_{s/ℚ}/(γ^{p^{r−1}} − 1)(J_{s/ℚ}))(T) ⟶^{π_{∗}} J_{s}^{r}(T) (resp. J_{s}^{r}(T) ⟶^{π^{∗}} J_{s}[γ^{p^{r−1}} − 1](T)) is a

U^{∗}(p)-isomorphism (resp. U(p)-isomorphism).

Remark 3.7. Note here that the natural morphism:

\frac{J_{s}(T)}{(γ^{p^{r−1}} − 1)(J_{s}(T))} → (J_{s}/(γ^{p^{r−1}} − 1)(J_{s}))(T)

Documenta Mathematica · Extra Volume Merkurjev (2015) 221–264

may have non-trivial kernel and cokernel which may not be killed by a power of U^{∗}(p). In other words, the left-hand-side is an fppf presheaf (of T) and the right-hand-side is its sheafication. On the other hand, T ↦ J_{s}[γ^{p^{r−1}} − 1](T) is already an fppf abelian sheaf; so, J_{r}(T) ⟶^{π^{∗}} J_{s}[γ^{p^{r−1}} − 1](T) is a U(p)- isomorphism without ambiguity by the above Lemma 3.6 and Corollary 3.5 combined. Also, as remarked in the introduction, we need to replace γ^{p^{r−1}} − 1 in the above statement by γ^{p^{r−2}} − 1 if p = 2.

4. Structure of Λ-BT groups over number fields and local fields

Let G_{/R_{∞}} := \underrightarrow{lim}_{r} J_{r}[p^{∞}]_{/R_{∞}}^{ord}, which is a Λ-BT group in the sense of [H14a, Sections 3 and 5] with a canonical h-action. Here for an abelian variety A_{/R},

A[p^{n}] = Ker(A ⟶^{p^{n}} A) and A[p^{∞}]_{/R} = \underrightarrow{lim}_{n} A[p^{n}] (the p-divisible Barsotti– Tate group of A over R). For an h-algebra A, we put G_{A} = G ⊗_{h} A. Pick a reduced local ring 𝕋 of h and write a(l^{m}) for the image in 𝕋 of U(l^{m}) or T (l^{m}) for a prime l according as l|Np or l ∤ Np and 𝔪_{𝕋} for the maximal ideal of 𝕋. Since G_{𝕋} is a Λ-BT group in the sense of [H14a, Theorem 5.4, Remark 5.5], we have the connected-étale exact sequence over ℤ_{p}[μ_{p^{∞}}]:

0 → G_{𝕋}^{∘} → G_{𝕋} → G_{𝕋}^{ét} → 0,

where G_{𝕋}^{∘} is the connected component of the flat group G_{𝕋} and G_{𝕋}^{ét} is the quotient of G_{𝕋} by G_{𝕋}^{∘}. The étale group G_{𝕋/ℚ} over ℚ is a Λ-BT group over ℚ (in the sense of [H14a, §4]) on which ℤ_{p}^{×} act by diamond operators. The entire group G_{𝕋} extends to a Λ-BT group over ℤ_{p}[μ_{p^{∞}}] (see [H14a, Remark 5.5]). The \overline{ℚ}_{p}-points of this sequence descent to ℚ_{p} giving an exact sequence:

0 → G_{𝕋}^{∘}(\overline{ℚ}_{p}) → G_{𝕋}(\overline{ℚ}_{p}) → G_{𝕋}^{ét}(\overline{ℚ}_{p}) → 0

with G_{𝕋}^{ét}(\overline{ℚ}_{p}) = H_{0}(I_{p}, G_{𝕋}(\overline{ℚ}_{p})) for the inertia group I_{p} ⊂ Gal(\overline{ℚ}_{p}/ℚ_{p}). We know that G_{𝕋}^{∘} and G_{𝕋}^{ét} are well controlled, and the Pontryagin dual modules of G_{𝕋}^{∘}(\overline{ℚ}) and G_{𝕋}^{ét}(\overline{ℚ}) are Λ-free modules of (equal) finite rank (see [H86b, §9] or [H14a, Sections 4–5]). Here we equip these Λ-divisible modules with the discrete topology. Take a field k as a base field. Pick a 𝕋-ideal 𝔞. Write G_{𝕋}[𝔞] for the kernel of 𝔞:

G_{𝕋}[𝔞](R) = {x ∈ G_{𝕋}(R)|ax = 0 ∀a ∈ 𝔞},

where R is an fppf extension of k. Write a(p) for the image of U(p) in 𝕋. For the moment, assume that k is a finite extension k of ℚ_{p} with p-adic integer ring W. If the residual degree of k is f and a(p)^{f} ≢ 1 mod 𝔪_{𝕋} for the maximal ideal 𝔪_{𝕋} of 𝕋, we have

G_{𝕋}[𝔪_{𝕋}]^{ét}(k) = 0,

since the action of F rob_{p} on G_{𝕋}[𝔪_{𝕋}]^{ét}(\overline{ℚ}_{p}) is given by multiplication by a(p). On the other hand, the action of Gal(\overline{k}/k) on e · J_{∞}[p^{∞}]^{∘}(\overline{k}) ⊗_{h} 𝕋 factors through Gal(k[μ_{p^{∞}}]/k) ↪ ℤ_{p}^{×} → Λ^{×}, where the factor Γ = 1 + pℤ_{p} of ℤ_{p}^{×} = Γ × μ_{p−1} is embedded into Λ = ℤ_{p}[[Γ]] by natural inclusion and ζ ∈ μ_{p−1} is sent to ζ^{a} for some 0 ≤ a =: a(𝕋) = a_{k}(𝕋) < p − 1. Thus if a(𝕋) ≠ 0, we have G_{𝕋}^{∘}[𝔪_{𝕋}](k) = 0. We have a natural projection π = π_{s}^{r} : G_{s} := J_{s}[p^{∞}]_{/ℚ}^{ord} → G_{r} for s > r (see [H13a, Section 4] where π_{s}^{r} is written as N_{r}^{s}). This induces a projective system of Tate modules {T G_{s,𝕋} := T G_{s} ⊗_{h} 𝕋}_{s} and {T G_{s,𝕋}^{?}} for ? = ∘, ét. We put T G_{𝕋}^{?} = \underleftarrow{lim}_{s} T G_{s,𝕋}^{?}(\overline{ℚ}) for ? = nothing, ∘ or ét. They are Λ-free modules with a continuous action of Gal(\overline{ℚ}/ℚ). Write ρ_{𝕋} for the Galois representation realized on T G_{𝕋}, and put ρ_{P} = ρ_{𝕋} mod P acting on T G_{𝕋}/P T G_{𝕋} for P ∈ Spec(𝕋). In particular, we simply write \overline{ρ} = \overline{ρ}_{𝕋} = ρ_{𝔪_{𝕋}} for the maximal ideal 𝔪_{𝕋} of 𝕋. If 𝕋 is a Gorenstein ring, then for the Tate modules T G_{𝕋}, T G_{𝕋}^{∘} and T G_{𝕋}^{ét} as above, we have

T G_{𝕋} ≅ 𝕋^{2} and T G_{𝕋}^{∘} ≅ 𝕋 ≅ T G_{𝕋}^{ét}

as 𝕋-modules (e.g., [H13a, Section 4]), and if \overline{ρ}_{𝕋}(I_{p}) contains a non-trivial unipotent element for the inertia group I_{p} in Gal(\overline{ℚ}_{p}/k), again we have G_{𝕋}^{ét}[𝔪_{𝕋}](k) = 0. Thus we get

Lemma 4.1. Let k_{/ℚ_{p}} in \overline{ℚ}_{p} be a finite extension and 𝕋 be a reduced local ring of h. Assume that k has residual degree f and one of the following two conditions:

(1) a_{k}(𝕋) ≠ 0 and a(p)^{f} ≢ 1 mod 𝔪_{𝕋},

(2) 𝕋 is a Gorenstein ring, and \overline{ρ}_{𝕋}(I_{p}) has non-trivial unipotent element

for the inertia group I_{p} of Gal(\overline{ℚ}_{p}/k). Then we have G_{𝕋}(k) = 0.

Proof. Let V be the Λ-dual of T G_{𝕋}, which is also the Pontryagin dual of G_{𝕋}. Then we have H_{0}(k, V/𝔪_{𝕋}V) ≅ G_{𝕋}[𝔪_{𝕋}](k) = H^{0}(k, G_{𝕋}[𝔪_{𝕋}]). By the assumption (1) or (2), we have the vanishing G_{𝕋}[𝔪_{𝕋}](k) = 0. Look into the following exact sequence of sheaves

0 → G_{𝕋}[𝔪_{𝕋}] → G_{𝕋} ⟶^{φ} ⨁_{α∈I} G_{𝕋}

with φ(x) = (αx)_{α} for a finite set I = {α}_{α} of generators of 𝔪_{𝕋}. Taking the Gal(\overline{k}/k)-invariant, we get another exact sequence

0 → G_{𝕋}[𝔪_{𝕋}](k) → G_{𝕋}(k) ⟶^{φ_{k}} ⨁_{α∈I} G_{𝕋}(k).

Since Ker(φ_{k}) = G_{𝕋}(k)[𝔪_{𝕋}], we conclude (G_{𝕋}(k))[𝔪_{𝕋}] = G_{𝕋}[𝔪_{𝕋}](k) = 0. Taking the Pontryagin dual module written as M^{∨} for a compact or discrete module M, we have, setting V = G_{𝕋}(\overline{ℚ})^{∨},

H_{0}(k, V)/𝔪_{𝕋}H_{0}(k, V) ≅ (G_{𝕋}(k))^{∨}/𝔪_{𝕋}(G_{𝕋}(k))^{∨} = (G_{𝕋}(k)[𝔪_{𝕋}])^{∨} = 0,

which implies G_{𝕋}(k)^{∨} = H_{0}(k, V) = 0 by Nakayama’s lemma, and hence G_{𝕋}(k) = 0. This proves the assertion under (1) or (2). □

In the l ≠ p case, we remark the following fact:

Lemma 4.2. Let k_{/ℚ_{l}} in \overline{ℚ}_{l} be a finite extension for a prime l ≠ p and 𝕋 be a reduced local ring of h. If the semi-simplification of G_{𝕋}[𝔪_{𝕋}] as a representation of Gal(\overline{ℚ}_{l}/k) does not contain the identity representation, then G_{𝕋}(k) = 0. In general, G_{𝕋}(k)^{∨} is always a torsion Λ-module of finite type.

Proof. If the semi-simplification of G_{𝕋}[𝔪_{𝕋}] as a representation of Gal(\overline{ℚ}_{l}/k) does not contain the identity representation, we have H^{0}(k, G_{𝕋}[𝔪_{𝕋}]) = 0; so, H_{0}(k, V/𝔪_{𝕋}V) = 0 for V = G_{𝕋}(\overline{ℚ})^{∨}. Writing 𝔪_{𝕋} = (α_{i})_{i∈I} for α_{i} ∈ 𝕋 with a finite index set I, we have an exact sequence:

0 → G_{𝕋}[𝔪_{𝕋}](\overline{ℚ}) → G_{𝕋}(\overline{ℚ}) ⟶^{x↦(α_{i}x)_{i}} ⨁_{i∈I} G_{𝕋}(\overline{ℚ}).

Taking the Pontryagin dual we have another exact sequence of Galois modules:

0 ← V/𝔪_{𝕋}V ← V ⟵^{x↦(α_{i}x)_{i}} ∏_{i∈I} V.

Since Galois homology functor is right exact, the above exact sequence implies

H_{0}(k, V) ⊗_{𝕋} 𝕋/𝔪_{𝕋} = H_{0}(k, V/𝔪_{𝕋}V) = 0.

Then by Nakayama’s lemma, we get H_{0}(k, V) = 0, which implies G_{𝕋}(k) = 0. Let f be the residual degree of k as before. Consider the Hecke polynomial H_{f,l}(X) = X^{2} − A(l^{f})X + l^{f}⟨l⟩^{f}, where A(l^{f}) is determined by the following recurrence relation: A(l) = a(l) and A(l^{m}) = a(l^{m}) − l⟨l⟩a(l^{m−1}) for m ≥ 2. If l ∤ Np, G_{𝕋} is unramified over k. By the Eichler–Shimura congruence relation (e.g. [GME, Theorem 4.2.1]), if l ∤ Np, for the l-Frobenius element ϕ ∈ Gal(\overline{k}/k), the linear operator H_{f,l}(ϕ) annihilates G_{𝕋}. Thus if H_{f,l}(X) mod 𝔪_{𝕋} is not divisible by X −1, G_{𝕋}[𝔪_{𝕋}] as a representation of Gal(\overline{ℚ}_{l}/k) does not contain the identity representation. For an arithmetic prime P, H_{f,l}(X) mod P does not have a factor X − 1. Thus after the localization at P of the Pontryagin dual (G_{𝕋}(k)^{∨})_{P} is killed by H_{f,l}(ϕ) and ϕ − 1, and hence G_{𝕋}(k)^{∨} is a torsion Λ-module. Now assume that l|N. By the solution of the local Langlands conjecture (see [C86] and [AAG]), after replacing k by its finite extension, the Galois module G_{𝕋}[P] for an arithmetic point P becomes unramified unless ρ_{P} is Steinberg at l (i.e., is multiplicative type at l). Suppose that we have a non-Steinberg P. Then characteristic polynomial H(X) of ϕ modulo P is prime to X − 1 (as H(X) mod P has Weil numbers of weight f as its roots). Then by the same argument, we conclude the torsion property. Suppose that all arithmetic point of Spec(𝕋) is Steinberg at l (this often happens; see a remark below Conjecture 3.4 of [H11, §3]). Write ρ_{P} for the 2dimensional Galois representation realized on (G_{𝕋}(\overline{ℚ}_{l})^{∨}) ⊗_{𝕋} κ(P). Again by Langlands-Carayol, ρ_{P} (I_{l}) for the inertia group I_{l} ⊂ Gal(\overline{ℚ}_{l}/k) contains a non-trivial unipotent element. Thus ρ_{P} does not have a quotient on which I_{l} acts trivially. This shows again the Λ-torsion property. □

Let Spec(𝕀) ⊂ Spec(𝕋) be an irreducible component. Without assuming the Gorenstein condition, we have (T G_{𝕀})_{P} ≅ 𝕀_{P}^{2} for almost all height one primes P ∈ Spec(Λ); so, we have ρ_{𝕀} with values in GL_{2}(𝕀_{P}) for most of P. We call 𝕀 a CM component if ρ_{𝕀} ≅ Ind_{M}^{ℚ} Ψ for a Galois character Ψ : Gal(\overline{ℚ}/M) → 𝕀_{P}^{×} (for an imaginary quadratic field M). If 𝕀 is not a CM component, again for almost all P, by [Z14], ρ_{𝕋}(I_{p}) contains an unipotent element conjugate to ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) with non-zero-divisor u ∈ 𝕋_{P}^{×}. In this case, we have H_{0}(Gal(\overline{k}/k), T G_{𝕀})_{P} = 0; so, G_{𝕀}(k) is a co-torsion Λ-module.

Lemma 4.3. Let k_{/ℚ_{p}} in \overline{ℚ}_{p} be a finite extension with residual degree f and 𝕋 be a reduced local ring of h. Then the Pontryagin dual G_{𝕋}(k)^{∨} of G_{𝕋}(k) is a torsion Λ–module of finite type.

Proof. We may suppose either a(p)^{f} ≡ 1 mod 𝔪_{𝕋} or a_{k}(𝕋) = 0, as otherwise G_{𝕋}(k) = 0 by Lemma 4.1. Replacing 𝕋 by its irreducible component 𝕀, we only need to prove torsion-ness for G_{𝕀}(k)^{∨}. Write V for the Λ-torsion free quotient of T G_{𝕀}. Then for any P ∈ Spec(Λ)(\overline{ℚ}_{p}), we have V_{P} = (T G_{𝕀})_{P} (as the reflexive closure in [BCM, Chapter 7] of 𝕀 is Λ-free). If 𝕀 is not a CM component (i.e., ρ_{𝕀} is not an induced representation from the Galois group over an imaginary quadratic field), the assertion follows from the same argument proving Lemma 4.1 replacing 𝔪_{𝕋} by P 𝕋_{P} and 𝕋 by 𝕋_{P}. Indeed, taking an arithmetic point P of weight 2. Then by [Z14], we have u ∈ 𝕋_{P}^{×}. Then H^{0}(k, V_{P} /P V_{P}) is a submodule of H^{0}(I, V_{P} /P V_{P}) (for the inertia group I at p) killed by a(p)^{f} − 1. Since P is an arithmetic point of weight 1, we may choose P so that a(p) mod P is a Weil number of weight 1 (indeed, we only need to assume that the Neben character of f_{P} is non-trivial at p; see [MFM, Theorem 4.6.17]), and hence a(p)^{f} ≢ 1 mod P. Thus H^{0}(k, V_{P} /P V_{P}) = 0. This implies G_{𝕀}(k)[P] is a finite module; so, G_{𝕀}(k)^{∨} is a torsion Λ-module. Now assume that 𝕀 is a CM component with ρ_{𝕀} = Ind_{M}^{ℚ} Ψ. Define Ψ^{c}(σ) = Ψ(cσc^{−1}) for a complex conjugation c. In the imaginary quadratic field M, p splits into a product of two primes 𝔭\overline{𝔭} as ρ_{𝕀} is ordinary. For any arithmetic point 𝔓 ∈ Spec(𝕀)(\overline{ℚ}_{p}) Ψ_{𝔓} := Ψ mod 𝔓 ramifies at 𝔭 and its restriction to the inertia group at 𝔭 has infinite order, and Ψ^{c} is unramified at 𝔭 with infinite order Ψ^{c}(F rob_{p}) (from an explicit description of Ψ; cf, [H13a, §3]). Then we have V_{𝔓} = V ⊗_{𝕀} 𝕀_{𝔓} ≅ 𝕀_{𝔓}^{2}. Thus replacing k by the composite kM_{𝔭}, we have V_{𝔓} ≅ Ψ ⊕ Ψ^{c} over Gal(\overline{ℚ}_{p}/k). Since Ψ^{c} is unramified at 𝔭 and Ψ_{𝔓}^{c}(F rob_{p}) has infinite order. This shows that H^{0}(k, V_{𝔓}/𝔓V_{𝔓}) = 0, and again we find that G_{𝕀}^{∨}(k) is a torsion 𝕀-module and hence a torsion Λ-module. □

Corollary 4.4. If k is a number field or a finite extension of ℚ_{l}, the localization of G_{𝕋}(k)^{∨} at an arithmetic prime of weight 2 vanishes.

Proof. We only need to prove this for a finite extension k of ℚ_{l}. Write W for the integer ring of k. Replacing k by its finite extension, we may assume that A_{P} has semi-stable reduction over W for an arithmetic prime at P. If A_{P} has good reduction and l ≠ p, the l-Frobenius acts on T_{p}A_{P} by a Weil number of weight ≥ 1, and then A_{P} [p^{∞}](k) is finite; so, G_{𝕋}[P](k) is finite. If l = p, by [Z14], the inertia image in Aut(T_{p}A_{P}) contains a non-trivial unipotent element, and hence again A_{P} [p^{∞}](k) is finite, and the result follows. If A_{P} has multiplicative reduction, A_{P} [p^{∞}](k) is finite by a theorem of Tate–Mumford as the Tate period of A_{P} is non-trivial. This shows that G_{𝕋}[P](k) is finite, and hence the result follows. □

5. Abelian factors of modular Jacobians

Let h_{r}(ℤ) be the subalgebra generated by T (n) (including U(l) for l|Np) of End(J_{r/ℚ}). Then h_{r}(ℤ_{p}) = h_{r}(ℤ) ⊗_{ℤ} ℤ_{p} is canonically isomorphic to the ℤ_{p}- subalgebra of End(J_{r}[p^{∞}]) generated by T (n) (including U(l) for l|Np). Then h_{r} = h_{r}(ℤ_{p})^{ord} by the control theorems in [H86a] and [H86b]. As before, let k be a finite extension of ℚ inside \overline{ℚ} or a finite extension of ℚ_{l} inside \overline{ℚ}_{l}. Let A_{r} be a abelian subvariety of J_{r} defined over k. Write A_{s} (s > r) for the image of A_{r} in J_{s} under the morphism π^{∗} : J_{r} → J_{s} given by Picard functoriality from the projection π : X_{s} → X_{r}. If A_{r} is Shimura’s abelian subvariety attached to a Hecke eigenform f, we sometimes write A_{f,s} for A_{s} to indicate this fact. Hereafter we assume

(A) We have a coherent sequence α_{s} ∈ End(J_{s/ℚ}) (for all s ≥ r) having the limit α = \underleftarrow{lim}_{s} α_{s} ∈ End(J_{∞/ℚ}) such that

(a) A_{s} is the connected component of J_{s}[α_{s}] with J_{s} = A_{s} + α_{s}(J_{s}) so that the inclusion: A_{s}[p^{∞}] ≅ J_{s}[α_{s}][p^{∞}] is a U(p)-isomorphism,

(b) the restriction α_{s}|_{α_{s}(J_{s})} ∈ End(α_{s}(J_{s})) is a self-isogeny. Here for s′ > s, coherency of α_{s} means the following commutative diagram:

\begin{matrix}J_{s} & ⟶^{π^{∗}} & J_{s′} \\ ^{α_{s}}↓ & & ↓^{α_{s′}} \\ J_{s} & ⟶_{π^{∗}} & J_{s′} \end{matrix} .

The Rosati involution h ↦ h^{∗} and T (n) ↦ T^{∗}(n) (with respect to the canonical divisor on J_{r}) brings h_{r}(ℤ) to h_{r}^{∗}(ℤ) ⊂ End(J_{r/ℚ}). Define A_{s}^{∗} to be the identity connected component of J_{s}[α^{∗}]. The condition (A) is equivalent to

(B) The abelian quotient map J_{s} ↠ B_{s} = Coker(α_{s}) dual to A_{s}^{∗} ⊂ J_{s} induces an U(p)-isomorphism of Tate modules: T_{p}(J_{s}/α_{s}(J_{s})) → T_{p}B_{s} and α_{s} induces an automorphism of the ℚ_{p}-vector space T_{p}α_{s}(J_{s}) ⊗_{ℤ_{p}} ℚ_{p}.

Again if A_{r} is Shimura’s abelian subvariety of J_{r} associated to a Hecke eigenform f, we sometimes write B_{f,s} for B_{s} as above. The condition (A) (and hence (B)) is a mild condition. Here are sufficient conditions for (α, A_{s}, B_{s}) to satisfy (A) (and (B)):

Proposition 5.1. Let Spec(𝕋) be a connected component of Spec(h) and Spec(𝕀) be an irreducible component of Spec(𝕋). Then the condition (A) holds for the following choices of (α, A_{s}, B_{s}):

(P1) Fix r > 0. Then α_{s} = α for a factor α|γ^{p^{r−1}} − 1 in Λ, A_{s} = J_{s}[α]^{∘}

(the identity connected component) and B_{s} = Pic_{A_{s}/ℚ}^{0} for all s ≥ r. (P2) Suppose that an eigen cusp form f = f_{P} new at each prime l|N belongs

to Spec(𝕋) and that 𝕋 = 𝕀 is regular (or more generally a unique factorization domain). Then writing the level of f_{P} as Np^{r}, the algebra homomorphism λ : 𝕋 → \overline{ℚ}_{p} given by f|T (l) = λ(T (l))f gives rise to the prime ideal P = Ker(λ). Since P is of height 1, it is principal generated by ϖ ∈ 𝕋. This ϖ has its image ϖ_{s} ∈ 𝕋_{s} = 𝕋 ⊗_{Λ} Λ_{s} for Λ_{s} = Λ/(γ^{p^{s−1}} −1). Since h_{s} = h⊗_{Λ} Λ_{s} = 𝕋_{s} ⊕X_{s} as an algebra direct sum, End(J_{s/ℚ}) ⊗_{ℤ} ℤ_{p} ⊃ h_{s}(ℤ_{p}) = 𝕋_{s} ⊕ Y_{s} with Y_{s} projecting down onto X_{s}. Then, we can approximate a_{s} = ϖ_{s} ⊕ 1_{s} ∈ h_{s}(ℤ_{p}) for the identity 1_{s} of Y_{s} by α_{s} ∈ h_{s}(ℤ) so that α_{s}h_{s}(ℤ_{p}) = a_{s}h_{s}(ℤ_{p}) (hereafter we call α_{s} “sufficiently close” to a_{s} if α_{s}h_{s}(ℤ_{p}) = a_{s}h_{s}(ℤ_{p})). For this choice of α_{s}, A_{s} := A_{f,s} and B_{s} := B_{f,s}.

(P3) More generally than (P2), we pick a general connected component Spec(𝕋) of Spec(h). Pick a (classical) Hecke eigenform f = f_{P} (of weight 2) for P ∈ Spec(𝕋). Assume that h_{s} (for every s ≥ r) is reduced and P = (ϖ) for ϖ ∈ 𝕋, and write ϖ_{s} for the image of ϖ in h_{s}(ℤ_{p}). Take the complementary direct summand Y_{s} of 𝕋_{s} in h_{s}(ℤ_{p}) and approximate a_{s} := ϖ_{s} ⊕ 1_{s} in h_{s}(ℤ_{p}) to get α_{s} sufficiently close to a_{s}. Then for this choice of α_{s}, A_{s} := A_{f,s} and B_{s} := B_{f,s}.

(P4) Suppose that 𝕋/(ϖ) for a non-zero divisor ϖ ∈ 𝕋 is a reduced algebra of characteristic 0 factoring through h_{r} := h/(γ^{p^{r−1}} − 1)h for some r > 0. Assume that 𝕋_{s} is reduced for every s ≥ r, and write ϖ_{s} for the image of ϖ in 𝕋_{s}. Then approximating a_{s} = ϖ_{s} ⊕ 1_{s} by α_{s} ∈ h_{s}(ℤ) sufficiently closely for each s ≥ r, we define A_{s} to be the connected component of J_{s}[α_{s}] and B_{s} to be its dual quotient.

Proof. We first prove (P4). Since α_{s} is sufficiently close to a_{s}, we have the identity α_{s}h_{s}(ℤ_{p}) = a_{s}h_{s}(ℤ_{p}) of ideals. By reducedness of 𝕋_{s}, we have an algebra product decomposition: h_{s}(ℚ_{p}) := h_{s}(ℤ_{p})⊗_{ℤ_{p}} ℚ_{p} = α_{s}(𝕋_{s})⊗_{ℤ_{p}} ℚ_{p}×Z_{s} for the complementary ℚ_{p}-subalgebra Z_{s}, which is given by (𝕋_{s}/(ϖ_{s})) ⊗_{ℤ_{p}} ℚ_{p}. Write the idempotent of Z_{s} as ϵ_{s} ∈ Z_{s}. Then ϵ_{s} + a_{s} is invertible in h_{s}(ℚ_{p}). For some positive integer M_{s}, β_{s} := U(p)^{M_{s}}ϵ_{s} ∈ h_{s}(ℤ) ⊂ End(J_{s}). Then by ϵ_{s} + a_{s} ∈ h_{s}(ℚ_{p})^{×}, the connected component A_{s} of J_{s}[α_{s}] is given by β_{s}(J_{s}), J_{s} = β_{s}(J_{s}) + α_{s}(J_{s}) = A_{s} + α(J_{s}), and the inclusion map A_{s} ↪ J_{s}[α_{s}] is an U(p)-isomorphism. Since α_{s} is invertible in α_{s}(h_{s}(ℚ_{p})), α_{s} induces a selfisogeny of α(J_{s}). Thus the triple satisfies (A). Since ϖ_{s′}h_{s′}(ℤ_{p}) surjects down to ϖ_{s}h_{s}(ℤ_{p}) for all s′ ≥ s, we can adjust α_{s} inductively to have a projective system \{α_{s} ∈ End(J_{s})\}_{s≥r}. Thus α = \underleftarrow{lim}_{s} α_{s} ∈ End(J_{∞}) does the job. This proves (P4). The assertions (P2) and (P3) are direct consequences of (P4). As for (P1), since α|(γ^{p^{r−1}} − 1)|(γ^{p^{s−1}} − 1) in the unique factorization domain Λ, factoring γ^{p^{s}} − 1 = α_{s}β_{s}, the ideals (α_{s}) and (β_{s}) are co-prime in the unique-factorization domain Λ. From this, we have J_{s} = β_{s}(J_{s}) + α(J_{s}) = A_{s} +α(J_{s}), and α|_{α(J_{s})} is a self isogeny of α(J_{s}) as α|_{α(J_{s})} is a non-zero-divisor in End(α(J_{s})). □

Remark 5.2. (i) Under (P2), all arithmetic points P of weight 2 in Spec(𝕀) satisfies (A). (ii) For a given weight 2 Hecke eigenform f, for density 1 primes 𝔭 of ℚ(f), f is ordinary at 𝔭 (i.e., a(p, f) ≢ 0 mod 𝔭; see [H13b, §7]). Except for finitely many primes 𝔭 as above, f belongs to a connected component 𝕋 which is regular (see [F02, §3.1]); so, (P2) is satisfied for such 𝕋. (iii) If N is square-free (as assumed for simplicity in the introduction), h_{s} is reduced [H13a, Corollary 1.3]; so, if an arithmetic prime P ∈ Spec(h_{r}) is principal, α_{s} as in (P3) satisfies (A).

If A_{r} = A_{f,r} is Shimura’s abelian subvariety associated to a primitive form f as in [IAT, Theorem 7.14], its dual quotient J_{r} ↠ B_{r} = B_{f,r} is also associated to f in the sense of [Sh73]. However, if A_{r} is not associated to a new form, the dual quotient may not be associated to the Hecke eigen form f. To clarify this point, we introduce an involution of J_{s}. We fix a generator ζ of the ℤ_{p}-module ℤ_{p}(1) = \underleftarrow{lim}_{n} μ_{p^{n}}(\overline{ℚ}); so, ζ is a coherent sequence of generators ζ_{p^{n}} of μ_{p^{n}}(\overline{ℚ}) (i.e., ζ_{p^{n+1}}^{p} = ζ_{p^{n}} for all n > 0). We also fix a generator ζ_{N} of μ_{N}(\overline{ℚ}), and put ζ_{Np^{r}} := ζ_{N}ζ_{p^{r}}. Identify the étale group scheme ℤ/Np^{n}ℤ_{/ℚ[ζ_{N} ,ζ_{p^{n}}]} with μ_{Np^{n}} by sending m ∈ ℤ to ζ_{Np^{n}}^{m}. Then for a couple (E, ϕ_{Np^{r}} : μ_{Np^{r}} ↪ E)_{/K} for a ℚ[μ_{Np^{r}}]-algebra K, let ϕ^{∗} : E[Np^{r}] ↠ ℤ/Np^{r}ℤ be the Cartier dual of ϕ_{Np^{r}}. Then ϕ^{∗} induces E[Np^{r}]/ Im(ϕ_{Np^{r}}) ≅ ℤ/Np^{r}ℤ. Define i : ℤ/p^{r}ℤ ≅ (E/ Im(ϕ_{Np^{r}}))[Np^{r}] by the inverse of ϕ^{∗}. Then we define φ_{Np^{r}} : μ_{Np^{r}} ↪

E/ Im(ϕ_{Np^{r}}) by φ_{Np^{r}} : μ_{Np^{r}} ≅ ℤ/Np^{r}ℤ ⟶^{i} (E/ Im(ϕ_{Np^{r}}])[p^{r}] ⊂ E/ Im(ϕ_{Np^{r}}).

This induces an involution w_{r} of X_{r} defined over ℚ[μ_{Np^{r}}], which in turn induces an automorphism w_{r} of J_{r/ℚ[ζ_{Np^{r}}]}. Let P ∈ Spec(h)(\overline{ℚ}_{p}) be an arithmetic point of weight 2. Then we have a pstabilized Hecke eigenform form f_{P} associated to P; i.e., f_{P} |T (n) = P (T (n))f_{P} for all n. Suppose f = f_{P} and write A_{f,r} = A_{P}. Then f_{P}^{∗} = w_{r}(f_{P}) is the dual common eigenform of T^{∗}(n). If f_{P} is new at every prime l|Np, f_{P}^{∗} is a constant multiple of the complex conjugate f_{P}^{c} of f_{P} (but otherwise, it could be different). Then the abelian quotient associated to f_{P}^{∗} is the dual abelian variety of A_{P}. Thus if f_{P}^{∗} is not constant multiple of f_{P}^{c}, B_{f,r} is not assocaited to f_{P}^{∗} (see a remark at the end of [H14b, §6] for more details of this fact). Pick an automorphism σ ∈ Gal(ℚ(μ_{Np^{r}})/ℚ) with ζ_{Np^{r}}^{σ} = ζ_{Np^{r}}^{z} for z ∈ (ℤ/Np^{r}ℤ)^{×}. Since w_{r}^{σ} is defined with respect to ζ_{Np^{r}}^{σ} = ζ_{Np^{r}}^{z}, we find w_{r}^{σ} = ⟨z⟩ ∘ w_{r}. By this formula, if x ∈ A_{P} (\overline{ℚ}) and σ ∈ Gal(\overline{ℚ}/ℚ) with ζ^{σ} = ζ^{z} for z ∈ ℤ_{p}^{×} × (ℤ/Nℤ)^{×} = \underleftarrow{lim}_{s}(ℤ/Np^{s}ℤ)^{×}, we have w_{r}(x)^{σ} = ⟨z⟩(w_{r}(x^{σ})). Thus w_{r}^{σ} = ⟨z⟩ ∘ w_{r} = w_{r} ∘ ⟨z^{−1}⟩ (see [MW86, page 237] and [MW84, 2.5.6]). Let π_{s,r,∗} : J_{s} → J_{r} for s > r be the morphism induced by the covering map X_{s} ↠ X_{r} through Albanese functoriality. Then we define π_{s}^{r} = w_{r} ∘ π_{s,r,∗} ∘ w_{s}. Then (π_{s}^{r})^{σ} = w_{r}⟨z^{−1}⟩π_{s,r,∗}⟨z⟩w_{s} = π_{s}^{r} for all σ ∈ Gal(ℚ(μ_{Np^{s}})/ℚ); thus, π_{s}^{r} is well defined over ℚ, and satisfies T (n) ∘ π_{s}^{r} = π_{s}^{r} ∘ T (n) for all n prime to Np and U(q) ∘ π_{s}^{r} = π_{s}^{r} ∘ U(q) for all q|Np (as w_{?} ∘ h ∘ w_{?} = h^{∗} for h ∈ h_{?}(ℤ) (? = s, r) by [MFM, Section 4.6]. Since w_{r}^{2} = 1, {J_{s}, π_{s}^{r}}_{s>r} form a Hecke equivariant projective system of abelian varieties defined over ℚ. We then define as described in (S) just above Lemma 2.1 an fppf abelian sheaf X̂ for any abelian variety quotient or subvariety X of J_{s/k} over the fppf site over k = ℚ and ℚ_{l} (note here the definition of X̂ depends on k). In general, for A_{s} in (A), we have A_{s}^{∗} = w_{s}(A_{s}) ⊂ J_{s} because T (n) ∘ w_{s} = w_{s} ∘ T^{∗}(n) for all n (see [MFM, Theorem 4.5.5]). Thus (B_{s}, π_{s}^{r}) in (B) gives rise to a natural projective system of abelian variety quotients of J_{s}.

6. Structure of ind-Λ-MW groups over number fields and local field

We return to the setting of Section 2; so, K/k is the infinite Galois extension defined there. In this section, unless otherwise mentioned, we often let κ denote an intermediate finite extension of k inside K (although the results in this section are valid for κ satisfying (2.2) unless otherwise mentioned). We assume (A) in Section 5 for (α_{s}, A_{s}, B_{s}). By (A), the inclusion A_{s}[p^{∞}] ↪ J_{s}[α_{s}][p^{∞}] is a U(p)-isomorphism; so, we have the identity of the ordinary parts: Â_{s}^{ord} = Ĵ_{s}^{ord}[α_{s}]. From the exact sequence

0 → J_{s}[α_{s}] → J_{s} ⟶^{α_{s}} J_{s} → B_{s} → 0,

we get the following exact sequence of sheaves:

(6.1)  0 → Â_{s}^{ord} → Ĵ_{s}^{ord} ⟶^{α_{s}} Ĵ_{s}^{ord} → B̂_{s}^{ord} → 0.

This is because tensoring ℤ_{p} (or taking the p-primary part X/X^{(p)} as in (S)) is an exact functor. Since taking injective limit is an exact functor, writing X_{∞}^{ord} = \underrightarrow{lim}_{s} X̂_{s}^{ord}, we get the following exact sequence of sheaves:

(6.2)  0 → A_{∞}^{ord} → J_{∞}^{ord} ⟶^{α} J_{∞}^{ord} → B_{∞}^{ord} → 0.

First, we shall describe A_{∞}^{ord} and B_{∞}^{ord} in terms of Â_{r} and B̂_{r}. The Picard functoriality induces a morphism π_{r,s}^{∗} : J_{r} → J_{s}. This gives a Hecke equivariant inductive system {J_{s}, π_{r,s}^{∗}}_{s>r} of abelian varieties defined over ℚ. Since the two morphisms J_{r} → J_{s}^{r} and J_{s}^{r} → J_{s}[γ^{p^{r−1}} − 1] (Picard functoriality) are U(p)-isomorphisms of fppf abelian sheaves by (u1) and Corollary 3.5 (see also Remark 3.7), we get the following two isomorphisms of fppf abelian sheaves:

(6.3)  A_{r}[p^{∞}]^{ord} ⟶_{π_{r,s}^{∗}}^{∼} A_{s}[p^{∞}]^{ord} and Â_{r}^{ord} ⟶_{π_{r,s}^{∗}}^{∼} Â_{s}^{ord},

since Â_{s}^{ord} is the isomorphic image of Â_{r}^{ord} ⊂ Ĵ_{r} in Ĵ_{s}[γ^{p^{r−1}} − 1]. Since w_{r} ∘ T (n) = T^{∗}(n)∘w_{r} (by [MFM, Theorem 4.5.5]), twisting Cartier duality pairing [·, ·] : J_{r}[p^{r}] × J_{r}[p^{r}] → μ_{p^{r}} coming from the canonical polarization, we get a perfect pairing (·, ·) : J_{r}[p^{r}] × J_{r}[p^{r}] → μ_{p^{r}} with (x|T (n), y) = (x, y|T (n))

(e.g., [H14a, Section 4]). By this w-twisted Cartier duality applied to the first identity of (6.3), we have

(6.4)  B_{s}[p^{∞}]^{ord} ⟶_{π_{s}^{r}}^{∼} B_{r}[p^{∞}]^{ord}.

Thus, by Kummer sequence, we have the following commutative diagram

B̂_{s}^{ord}(κ) ⊗ ℤ/p^{m}ℤ = (B_{s}(κ) ⊗ ℤ/p^{m}ℤ)^{ord} ⟶_{↪} H^{1}(B_{s}[p^{m}]^{ord})
π_{s}^{r} ↓  ^{≀}↓^{(6.4)}
B̂_{r}^{ord}(κ) ⊗ ℤ/p^{m}ℤ = (B_{r}(κ) ⊗ ℤ/p^{m}ℤ)^{ord} ⟶_{↪} H^{1}(B_{r}[p^{m}]^{ord})

This shows

B̂_{s}^{ord}(κ) ⊗ ℤ/p^{m}ℤ ≅ B̂_{r}^{ord}(κ) ⊗ ℤ/p^{m}ℤ.

Passing to the limit, we get

(6.5)  B̂_{s}^{ord} ⟶_{π_{s}^{r}}^{∼} B̂_{r}^{ord} and (B_{s} ⊗_{ℤ} 𝕋_{p})^{ord} ⟶_{π_{s}^{r}}^{∼} (B_{r} ⊗_{ℤ} 𝕋_{p})^{ord}

as fppf abelian sheaves. As long as κ is either a field extension of finite type of a number field or a finite extension of ℚ_{l} (l ≠ p) or a finite algebraic extension of ℚ_{p}, the projective limit of B_{?}(κ) ⊗ ℤ/p^{m}ℤ (with respect to m) is equal to B̂_{?} (by Lemma 2.1). In short, we get

Lemma 6.1. Assume κ to be given either by a field extension of finite type of k if k is a finite extension of ℚ or ℚ_{l} (l ≠ p) or by a finite algebraic extension of k if [k : ℚ_{p}] < ∞. Then we have the following isomorphism

Â_{r}(κ)^{ord} ⟶_{π_{s,r}^{∗}}^{∼} Â_{s}(κ)^{ord} and B̂_{s}(κ)^{ord} ⟶_{π_{s}^{r}}^{∼} B̂_{r}(κ)^{ord}

for all s > r including s = ∞.

By computation, we get π_{s}^{r} ∘ π_{r,s}^{∗} = p^{s−r}U(p^{s−r}). To see this, as Hecke operators, π_{r,s}^{∗} = [Γ_{s}^{r}], π_{r,s,∗} = [Γ_{r}]. Thus we have

(6.6) π_{s}^{r} ∘ π_{r,s}^{∗} = [Γ_{s}^{r}] · w_{s} · [Γ_{r}] · w_{r} = [Γ_{s}] · [w_{s}w_{r}] · [Γ_{r}]

= [Γ_{s}^{r} : Γ_{s}][Γ_{s}^{r} ( \begin{matrix}1 & 0 \\ 0 & p^{s−r} \end{matrix} ) Γ_{r}] = p^{s−r}U(p^{s−r}).

Then we have the commutative diagram of fppf abelian sheaves for s′ > s

(6.7)

Â_{s′}^{ord} ⟵_{π_{r,s′}^{∗}}^{∼} Â_{r}^{ord}
π_{s′}^{s} ↓  ↓_{p^{s′−s}U(p)^{s′−s}}
Â_{s}^{ord} ⟵_{π_{r,s}^{∗}}^{∼} Â_{r}^{ord}.

Note that A_{s} and B_{s} are mutually (w-twisted) dual as abelian varieties (see Section 5), and the w-twisted duality is compatible with Hecke operators. Thus B_{s}[p^{n}] is the w-twisted Cartier dual of A_{s}[p^{n}]. The w-twisted Cartier duality pairing in [H14a, Section 4] satisfies (x|X, y) = (x, y|X) for X = T (n), U(q), and π_{r}^{s} and π_{r,s}^{∗} are adjoint each other under this duality. Then we have the dual commutative diagram of fppf abelian sheaves:

(6.8)

B̂_{s′}^{ord} ⟶_{π_{s′}^{r}}^{∼} B̂_{r}^{ord}
π_{s,s′}^{∗} ↑  ↑_{p^{s′−s}U(p)^{s′−s}}
B̂_{s}^{ord} ⟶_{π_{s}^{r}}^{∼} B̂_{r}^{ord}.

By (6.7) and (6.8), we have the following four exact sequences of fppf abelian sheaves:

0 →A_{s}[p^{s−r}]^{ord} → A_{s}[p^{∞}]^{ord} ⟶^{π_{s}^{r}} A_{r}[p^{∞}]^{ord} → 0,
0 →B_{r}[p^{s−r}]^{ord} → B_{r}[p^{∞}]^{ord} ⟶^{π_{r,s}^{∗}} B_{s}[p^{∞}]^{ord} → 0

(6.9)

and

0 →A_{s}[p^{s−r}]^{ord} → Â_{s}^{ord} ⟶^{π_{s}^{r}} Â_{r}^{ord} → 0,
0 →B_{r}[p^{s−r}]^{ord} → B̂_{r}^{ord} ⟶^{π_{r,s}^{∗}} B̂_{s}^{ord} → 0.

(6.10)

Lemma 6.2. Let the notation and assumtions be as in Lemma 6.1. Then we have a canonical isomorphism

\underrightarrow{lim}_{s,π_{r,s}^{∗}} B̂_{s}^{ord}(κ) ≅ \underrightarrow{lim}_{s,p^{s−r}U(p)^{s−r}} B̂_{r}^{ord}(κ) ≅ B̂_{r}(κ)^{ord} ⊗_{ℤ_{p}} ℚ_{p}.

Proof. Identifying the left and the right column of (6.8), we have the cohomology exact sequence of the second exact sequence of (6.10):

(6.11) 0 → B_{r}[p^{s−r}]^{ord}(κ) ⟶^{π_{r}^{s}} B̂_{r}^{ord}(κ) ⟶^{π_{r,s}^{∗}} B̂_{s}^{ord}(κ) → H_{?}^{1}(B_{r}[p^{s−r}]^{ord}).

Passing to the inductive limit of {B_{r}[p^{s−r}]^{ord}, p^{s−r}U(p)^{s−r}}_{s}, {B̂_{r}(κ)^{ord}, p^{s−r}U(p)^{s−r}}_{s} and {B̂_{s}, π_{r,s}^{∗}}_{s}, we have the following commutative diagram with exact rows:

(6.12)

\underrightarrow{lim}_{s} B̂_{r}^{ord}(κ) → \underrightarrow{lim}_{s} B̂_{s}^{ord}(κ) → \underrightarrow{lim}_{s} H_{?}^{1}(B_{r}[p^{s−r}]^{ord})
∥ ∥ ≀ ↓
\underrightarrow{lim}_{s} B̂_{r}^{ord}(κ) → \underrightarrow{lim}_{s} B̂_{s}^{ord}(κ) → H_{?}^{1}(\underrightarrow{lim}_{s} B_{r}[p^{s−r}]^{ord}).

Here the last isomorphism comes from the commutativity of injective limit and cohomology. For a free ℤ_{p}-module F of finite rank, we suppose to have a commutative diagram:

F ⟶^{p^{n}} F
∥↓  ↓^{p^{−n}}
F ⟶_{↪} p^{−n}F.

Thus we have \underrightarrow{lim}_{n,x↦p^{n}x} F = \underrightarrow{lim}_{n,x↦p^{−n}x} p^{−n}F ≅ F ⊗_{ℤ_{p}} ℚ_{p}. If T is a torsion ℤ_{p}-module with p^{B}T = 0 for B ≫ 0, we have \underrightarrow{lim}_{n,x↦p^{n}x} T = 0. Thus for general M = F ⊕ T, we have \underrightarrow{lim}_{n,x↦p^{n}x} M ≅ M ⊗_{ℤ_{p}} ℚ_{p}. Applying this

consideration to M = B̂_{r}(κ), we get

\underrightarrow{lim}_{s,x↦p^{s}U(p)^{s}x} B̂_{r}(κ) ≅ B̂_{r}(κ) ⊗_{ℤ_{p}} ℚ_{p}.

Similarly, \underrightarrow{lim}_{n,x↦p^{n}U(p)^{n}x} B_{r}[p^{n}](κ) = \underrightarrow{lim}_{n,x↦p^{n}U(p)^{n}x} B_{r}[p^{n}](K) = 0. Thus from the above diagram (6.12), we conclude the lemma. □

Consider the composite morphism ϖ_{s} : A_{s} ↪ J_{s} ↠ B_{s} of fppf abelian sheaves. Since B_{s} = J_{s}/α_{s}(J_{s}) and J_{s} = A_{s} + α_{s}(J_{s}) with finite intersection J_{s} = A_{s} ×_{J_{s}} α_{s}(J_{s}), we have a commutative diagram with exact rows in the category of fppf abelian sheaves:

(6.13)  \begin{matrix}α(J_{s}) & ⟶^{↪} & J_{s} & ⟶_{ρ_{s}}^{↠} & B_{s} \\ ^{∪}↑ & & ^{∪}↑ & & ↑^{∥} \\ 0 → α(J_{s}) ×_{J_{s}} A_{s} & ⟶_{↪} & A_{s} & ⟶_{↠}^{ϖ_{s}} & B_{s}. \end{matrix}

We have this diagram over R_{s} := ℤ_{(p)}[μ_{p^{s}}] (not just over ℚ) by taking the connected components of the Néron models of J_{s}, A_{s} and B_{s}. The intersection α(J_{s}) ×_{J_{s}} A_{s} = Ker(ϖ_{s}) is an étale finite group scheme over ℚ. These abelian varieties are known to have semi-stable reduction over R_{s} by the good reduction theorem of Carayol–Langlands. If the character ℤ_{p}^{×} ∋ z ↦ ⟨z⟩ ∈ End(A_{s})^{×} is non-trivial, we may replace J_{s} by its complement J_{s}^{(0)} of the image of J_{s}^{0} in J_{s}. Under this circumstance, α(J_{s}) ×_{J_{s}} A_{s} = Ker(ϖ_{s}) is a finite flat group scheme over R_{s}. Since A_{s} and B_{s} has good reduction over R_{r}, Ker(ϖ_{s}) is a finite flat group scheme defined over R_{r}. We consider the exact sequence

0 → Ker(ϖ_{s}) → A_{s} ⟶^{ϖ_{s}} B_{s} → 0.

which is an exact sequence of fppf abelian sheaves over R_{r} (and smooth abelian sheaves over ℚ or ℤ[\frac{1}{p}]). From this, writing C_{s} for the p-primary part of Ker(ϖ_{s}), we have an exact sequence of fppf abelian sheaves over R_{r} (and smooth abelian sheaves over ℚ or ℤ[\frac{1}{p}]):

0 → C_{s} → Â_{s} → B̂_{s} → 0.

We have the following commutative diagram with exact rows:

A_{s}[p^{s−r}]^{ord} ⟶^{∼} A_{s}[p^{s−r}]^{ord} ⟶_{↠} 0
↓  ↓  ↓
C_{s}^{ord}  ⟶_{↪}  Â_{s}^{ord}  ⟶_{↠} B̂_{s}^{ord}
↓  ↓  ^{≀}↓
C_{r}^{ord}  ⟶_{↪}  Â_{r}^{ord}  ⟶_{↠} B̂_{r}^{ord}.

By the snake lemma applied to the right two exact columns of the above diagram, we get the following exact sequence:

(6.14)  0 → A_{r}[p^{s−r}]^{ord} → C_{s}^{ord} → C_{r}^{ord} → 0

with C_{s}^{ord} ↪ A_{s}[p^{∞}]^{ord} ⟵_{∼}^{π_{r,s}^{∗}} A_{r}[p^{∞}]^{ord}.

Proposition 6.3. We have the following exact sequence under the ?-topology over k, where ? = sm, étale, nothing and fppf:

(6.15)  0 → Â_{r}^{ord} → J_{∞}^{ord} ⟶^{α} J_{∞}^{ord} ⟶^{ρ_{∞}} B̂_{r}^{ord} ⊗_{ℤ_{p}} ℚ_{p} → 0

with Â_{r}^{ord}/Â_{r}^{ord}[p^{∞}] ≅ B̂_{r}^{ord} ⊗_{ℤ_{p}} ℚ_{p}.

Proof. By (6.13), C_{s}^{ord} is equal to Â_{s}^{ord} ∩ α̂(Ĵ_{s}^{ord}). Since A_{s} is the connected component of J_{s}[α] with U(p)-isomorphism A_{s} ↪ J_{s}[α_{s}], we have C_{s}^{ord} = α(Ĵ_{s}^{ord})[α]. Since α is an isogeny on α(J_{s}), we have an exact sequence of sheaves indexed by s under ?-topology

0 → C_{s}^{ord} → α(Ĵ_{s}^{ord}) ⟶^{α_{s}} α(Ĵ_{s}^{ord}) → 0.

Passing to the inductive limit of these exact sequences (and noting \underrightarrow{lim}_{s} C_{s}^{ord} = A_{r}[p^{∞}]^{ord} by (6.14)), we get another exact sequences:

0 → Â_{r}^{ord}[p^{∞}] → α(J_{∞}^{ord}) ⟶^{α} α(J_{∞}^{ord}) → 0.

Therefore by (6.14), we get the following exact sequences (indexed by s) of sheaves under ?-topology:

(6.16)  0 → C_{s}^{ord} → (Â_{s}^{ord} × α(Ĵ_{s}^{ord})) → Ĵ_{s}^{ord} → 0.

Passing again to the inductive limit of these exact sequences (and noting Â_{r}^{ord} ≅ Â_{s}^{ord} by π_{r,s}^{∗} and \underrightarrow{lim}_{s} C_{s}^{ord} = A_{r}[p^{∞}]^{ord}), we get the top and the bottom exact sequences of the following commutative diagram:

Â_{r}^{ord}[p^{∞}] ⟶^{↪} (Â_{r}^{ord} × α(J_{∞}^{ord})) ⟶^{↠} J_{∞}^{ord}
0↓  ^{α}↓  ^{α}↓
Â_{r}^{ord}[p^{∞}] ⟶^{↪} (Â_{r}^{ord} × α(J_{∞}^{ord})) ⟶^{↠} J_{∞}^{ord}.

Applying the snake lemma (noting that the connection map is the zero map), we get

Coker(J_{∞}^{ord} ⟶^{α} J_{∞}^{ord}) = Â_{r}^{ord}/Â_{r}^{ord}[p^{∞}].

Thus we have the following exact sequence of sheaves:

(6.17)  0 → Â_{r}^{ord} → J_{∞}^{ord} → J_{∞}^{ord} → Â_{r}^{ord}/Â_{r}^{ord}[p^{∞}] → 0.

There is another way to see (6.17). Passing to the inductive limit of the exact sequences of sheaves

0 → Â_{s}^{ord} → Ĵ_{s}^{ord} ⟶^{α_{s}} Ĵ_{s}^{ord} ⟶^{ρ_{s}} B̂_{s}^{ord} → 0,

we get the following exact sequence of sheaves:

0 → Â_{r}^{ord} → J_{∞}^{ord} ⟶^{α} J_{∞}^{ord} ⟶^{ρ_{∞}} \underrightarrow{lim}_{s,x↦p^{s−r}U(p)^{s−r}} B̂_{r}^{ord} → 0

as Â_{r}^{ord} ≅ Â_{s}^{ord} by π_{r,s}^{∗}. This combined with (6.17) and Lemma 6.2 proves the exact sequence in (6.15). By (6.16), we have Â_{s}^{ord} ∩ α(Ĵ_{s}^{ord}) ≅ C_{s}^{ord}; thus Ker(Â_{s}^{ord} → B̂_{s}^{ord}) ≅ C_{s}^{ord} with \underrightarrow{lim}_{s} C_{s}^{ord} = A_{r}[p^{∞}]^{ord}, passing to the inductive limit we again get the identity of sheaves:

\underrightarrow{lim}_{s,x↦p^{s−r}U(p)^{s−r}} B̂_{r}^{ord} ≅ Â_{r}^{ord}/Â_{r}[p^{∞}]^{ord} ≅ B̂_{r}^{ord} ⊗_{ℤ_{p}} ℚ_{p}.

This finishes the proof. □

We have two exact sequences of sheaves:

0 →Â_{r}^{ord} → J_{∞}^{ord} ⟶^{α} α(J_{∞}^{ord}) → 0,
0 →α(J_{∞}^{ord}) → J_{∞}^{ord} ⟶^{ρ_{∞}} B̂_{r}^{ord} ⊗_{ℤ_{p}} ℚ_{p} → 0.

(6.18)

These leave us to study the two error terms

E_{1}(κ) := α(J_{∞}^{ord})(κ)/α(J_{∞}^{ord}(κ)) and E_{2}(κ) := B̂_{r}^{ord}(κ) ⊗_{ℤ_{p}} ℚ_{p}/ρ_{∞}(J_{∞}^{ord}(κ)).

Let E_{1}^{s}(κ) := α(J_{s}^{ord})(κ)/α(J_{s}^{ord}(κ)) and E_{2}^{s}(κ) := B̂_{s}^{ord}(κ)/ρ_{s}(Ĵ_{s}^{ord}(κ)) = Coker(ρ_{s}) for ρ_{s} : J_{s}^{ord}(κ) → B̂_{s}^{ord}(κ). Note that

E_{1}^{s}(κ)(↪ H_{?}^{1}(Â_{r}^{ord}) = H_{?}^{1}(A_{r}^{ord}) ⊗_{ℤ} ℤ_{p})

and E_{2}^{s}(κ) = B_{s}^{ord}(κ)/ρ_{s}(Ĵ_{s}^{ord}(κ))(↪ H_{?}^{1}(α(Ĵ_{s}^{ord}))[α])

are p-torsion finite modules as long as s is finite. Note that α|_{α(J_{s})} is a self isogeny; so,

0 → α(J_{s})[α]^{ord} → α(Ĵ_{s}^{ord}) ⟶^{α_{s}} α(Ĵ_{s}^{ord}) → 0

is an exact sequence of sheaves. Since α(J_{s})[α]^{ord} = C_{s}^{ord}, we have another exact sequence:

0 → α(Ĵ_{s}^{ord})(κ)/α(α(Ĵ_{s}^{ord})(κ)) → H_{?}^{1}(Ĉ_{s}^{ord}) → H_{?}^{1}(α(Ĵ_{s}^{ord}))[α] → 0.

We have the following commutative diagram with exact rows and exact columns:

E_{1}^{s}(κ) ⟶^{↪} H_{?}^{1}(Â_{r}^{ord}) ⟶ H_{?}^{1}(Ĵ_{s}^{ord})
onto↑  ↑  ↑
\frac{α(Ĵ_{s}^{ord})(κ)}{α(α(Ĵ_{s}^{ord})(κ))} ⟶^{↪} H_{?}^{1}(C_{s}^{ord}) ⟶^{↠} H_{?}^{1}(α(Ĵ_{s}^{ord}))[α]
\overline{α}_{s}↑  ↑  ↑^{∪}
\frac{Ĵ_{s}^{ord}(κ)}{α(Ĵ_{s}^{ord})(κ)} ⟶_{↪}^{ρ_{s}} B̂_{s}^{ord}(κ) ⟶_{↠}  E_{2}^{s}(κ).

The left column is exact by definition. The middle column is the part of the long exact sequence attached to the short one C_{s}^{ord} ↪ Â_{s}^{ord} ↠ B̂_{s}^{ord}, and the right column is the same for α(Ĵ_{s}^{ord}) ↪ Ĵ_{s}^{ord} ↠ B̂_{s}^{ord}. Note \underrightarrow{lim}_{s} C_{s}^{ord} = A_{r}[p^{∞}]^{ord}. Passing to the limit, we have the limit commutative diagram with exact rows and exact columns:

(6.19)

E_{1}(κ) ⟶^{↪} H_{?}^{1}(Â_{r}^{ord}) ⟶ H_{?}^{1}(J_{∞}^{ord})
onto↑  ^{π_{B}} ↑  ↑
\frac{α(J_{∞}^{ord})(κ)}{α(α(J_{∞}^{ord})(κ))} ⟶^{↪} H_{?}^{1}(A_{r}[p^{∞}]^{ord}) ⟶^{↠} H_{?}^{1}(α(J_{∞}^{ord}))[α]
\overline{α}↑  ^{δ_{B}} ↑  ↑^{∪}
\frac{J_{∞}^{ord}(κ)}{α(J_{∞}^{ord})(κ)} ⟶_{↪}^{ρ_{∞}} B̂_{r}^{ord}(κ) ⊗_{ℤ_{p}} ℚ_{p} ⟶_{↠}  E_{2}(κ).

We have seen, Â_{r}^{ord}/A_{r}[p^{∞}]^{ord} ≅ B̂_{r} ⊗_{ℤ_{p}} ℚ_{p} as sheaves of ℚ_{p}-vector space; so, H_{?}^{1}(Â_{r}^{ord}/A_{r}[p^{∞}]^{ord}) is a ℚ_{p}-vector space. On the other hand, H_{?}^{1}(Â_{r}^{ord}) is a p-torsion module (e.g., Lemma 2.2). Therefore the natural map H_{?}^{1}(Â_{r}^{ord}) → H_{?}^{1}(Â^{ord}/A_{r}[p^{∞}]^{ord}) is the zero map. Thus by long exact sequence attached to 0 → A_{r}[p^{∞}]^{ord} → Â_{r}^{ord} → Â_{r}^{ord}/A[p^{∞}] → 0, the morphism π_{B} is onto. Since A_{r}(κ) ⊗_{ℤ} 𝕋_{p} = B_{r}(κ) ⊗_{ℤ} 𝕋_{p}, the map δ_{B} factors through the Kummer map A_{r}(κ) ⊗_{ℤ} 𝕋_{p} ↪ H^{1}(A_{r}[p^{∞}]^{ord}). Thus

Ker(δ_{B}) = Im(Â_{r}(κ) → B̂_{r}(κ) ⊗_{ℤ_{p}} ℚ_{p}) ≅ Ker(\overline{α}),

where the last identity follows from the snake lemma applied to the above diagram. Consider the following exact sequence:

E_{1}(κ)[p^{n}] = Tor_{1}^{ℤ}(E_{1}(κ), ℤ/p^{n}ℤ) ⟶^{i_{n}} α(J_{∞}^{ord}(κ)) ⊗ ℤ/p^{n}ℤ

→ α(J_{∞}^{ord})(κ) ⊗ ℤ/p^{n}ℤ → E_{1}(κ) ⊗ ℤ/p^{n}ℤ → 0, which produces the following commutative diagram with exact rows for n > m:

E_{1}(κ)[p^{n}] ⟶^{i_{n}} \frac{α(J_{∞}^{ord}(κ))}{p^{n}α(J_{∞}^{ord}(κ))} ⟶^{j_{n}} \frac{α(J_{∞}^{ord})(κ)}{p^{n}α(J_{∞}^{ord})(κ)} ⟶^{↠} E_{1}(κ) ⊗ ℤ/p^{n}ℤ
↓  ↓  ↓  ↓
E_{1}(κ)[p^{m}] ⟶^{i_{m}} \frac{α(J_{∞}^{ord}(κ))}{p^{m}α(J_{∞}^{ord}(κ))} ⟶^{j_{m}} \frac{α(J_{∞}^{ord})(κ)}{p^{m}α(J_{∞}^{ord})(κ)} ⟶_{↠} E_{1}(κ) ⊗ ℤ/p^{m}ℤ

This in turn produces two commutative diagrams with exact rows:

(6.20)

E_{1}(κ)[p^{n}] ⟶^{i_{n}} \frac{α(J_{∞}^{ord}(κ))}{p^{n}α(J_{∞}^{ord}(κ))} ⟶ Coker(i_{n}) = Im(j_{n}) → 0
↓  ↓  ↓
E_{1}(κ)[p^{m}] ⟶^{i_{m}} \frac{α(J_{∞}^{ord}(κ))}{p^{m}α(J_{∞}^{ord}(κ))} ⟶ Coker(i_{n}) = Im(j_{m}) → 0

and

(6.21)

0 → Ker(i_{n}) ⟶ E_{1}(κ)[p^{n}] ⟶^{i_{n}} Im(i_{n}) → 0
↓  ↓  ↓
0 → Ker(i_{m}) ⟶ E_{1}(κ)[p^{m}] ⟶^{i_{m}} Im(i_{m}) → 0.

Since the diagram of (6.21) is made of finite modules (as E_{1}(κ) ⊂ H^{1}(Â_{r}^{ord}); Lemma 2.3), projective limit is an exact functor (from the category of compact modules), and passing to the limit, we get

\underleftarrow{lim}_{n} Im(i_{n}) = Im(i_{∞} : \underleftarrow{lim}_{n} E_{1}(κ)[p^{n}] → \underleftarrow{lim}_{n} \frac{α(J_{∞}^{ord}(κ))}{p^{n}α(J_{∞}^{ord}(κ))}).

By the snake lemma (cf. [BCM, I.1.4.2 (2)]) applied to (6.20), Im(j_{n}) → Im(j_{m}) is a surjection for all n > m. Thus the projective system of the following exact sequences:

\{0 → Im(j_{n}) → \frac{α(J_{∞}^{ord})(κ)}{p^{n}α(J_{∞}^{ord})(κ)} → E_{1}(κ) ⊗ ℤ/p^{n}ℤ → 0\}_{n}

satisfies the Mittag–Leffler condition. Passing to the projective limit, we get the exact sequence

0 → α(J̌_{∞}^{ord}(κ)) = Im(j_{∞}) → α(J̌_{∞}^{ord})(κ) → \underleftarrow{lim}_{n} E_{1}(κ) ⊗ ℤ/p^{n}ℤ → 0.

Since E_{1}(κ) = (ℚ_{p}/ℤ_{p})^{R} ⊕ Δ ↪ H_{?}^{1}(Â_{r})^{ord} for a finite group Δ and an integer R ≥ 0 (by Lemma 2.3), \underleftarrow{lim}_{n} E_{1}(κ) ⊗ ℤ/p^{n}ℤ is a finite group isomorphic to the torsion subgroup Δ of E_{1}(κ). Thus

(6.22)  J̌_{∞}^{ord}(κ) ⟶^{α} α(J̌_{∞}^{ord})(κ) has finite cokernel Δ,

and Δ is isomorphic to the maximal torsion submodule of E_{1}(κ)^{∨}. Consider the “big” ordinary Hecke algebra h given by \underleftarrow{lim}_{s} h_{s} as in the introduction. For a Λ-algebra homomorphism h → R and an h-module M, we put M_{R} = M ⊗_{h} R. Take a connected component Spec(𝕋) of Spec(h) such that α in (A) restricted to Spec(𝕋) is a non-unit; so, Â_{s}^{ord}(K)_{𝕋} ≠ 0. Note that M_{𝕋} is a direct summand of M; so, the above diagrams and exactness are valid after tensoring 𝕋 over h (attaching subscript 𝕋). Note that α(J_{∞})^{ord}[p^{∞}](κ) ⊂ J_{∞}^{ord}[p^{∞}](κ) = G(κ). Since Im(ρ_{∞})_{𝕋} is a direct summand in J_{∞}(κ)_{𝕋} and α(J_{∞}^{ord}(κ)_{𝕋})[p^{n}] ≅ Tor_{1}^{ℤ_{p}}(α(J_{∞}^{ord}(κ)_{𝕋}), ℤ/p^{n}ℤ), we have the following exact sequences:

α(J_{∞}^{ord}(κ)_{𝕋})[p^{n}] → \frac{Â_{r}^{ord}(κ)_{𝕋}}{p^{n}Â_{r}^{ord}(κ)_{𝕋}} → \frac{J_{∞}^{ord}(κ)_{𝕋}}{p^{n}J_{∞}^{ord}(κ)_{𝕋}} ⟶^{α} \frac{α(J_{∞}^{ord}(κ)_{𝕋})}{p^{n}α(J_{∞}^{ord}(κ)_{𝕋})} → 0

0 →α(J_{∞}^{ord}(κ)_{𝕋}) ⊗ ℤ/p^{n}ℤ → J_{∞}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ → Im(ρ_{∞})_{𝕋} ⊗ ℤ/p^{n}ℤ → 0.

(6.23)

The module α(J_{∞}^{ord}(κ)_{𝕋})[p^{n}] is killed by the annihilator 𝔞 of G_{𝕋}(κ)^{∨} in Λ which is prime to γ^{p^{r}} − 1 (note that γ^{p^{r}} − 1 kills Â_{r}^{ord}(κ)). Thus the image of α(J_{∞}^{ord}(κ)_{𝕋})[p^{n}] in Â_{r}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ is killed by 𝔄 = 𝔞 + (γ^{p^{r−1}} − 1) ⊂ Λ. Since Λ/𝔄 is a finite ring and G_{𝕋}^{∨} is a Λ-module of finite type, we get

(6.24)  | Ker(Â_{r}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ → J_{∞}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ)| < B

for a constant B > 0 independent of n. Applying the snake lemma to the following commutative diagram with exact rows:

p^{n}Â_{r}^{ord}(κ)_{𝕋} ⟶^{↪} Â_{r}^{ord}(κ)_{𝕋} ⟶^{↠} Â_{r}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ
↓  ↓  ↓
p^{n}J_{∞}^{ord}(κ)_{𝕋} ⟶_{↪} J_{∞}^{ord}(κ)_{𝕋} ⟶_{↠} J_{∞}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ,

we have an isomorphism, for F_{n} := p^{n}J_{∞}^{ord}(κ)_{𝕋} ∩ Â_{r}^{ord}(κ)_{𝕋},

F_{n}/p^{n}Â_{r}^{ord}(κ)_{𝕋} ≅ Ker(Â_{r}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ → J_{∞}^{ord}(κ)_{𝕋} ⊗ ℤ/p^{n}ℤ)

whose right-hand-side is finite with bounded order independent of n by (6.24). Consider the two filters on Â_{r}^{ord}(κ)_{𝕋}:

F := {F_{n} = (p^{n}J_{∞}^{ord}(κ)_{𝕋} ∩ Â_{r}^{ord}(κ)_{𝕋})}_{n} and {p^{n}Â_{r}^{ord}(κ)_{𝕋}}_{n}

with F_{n} ⊃ p^{n}Â_{r}^{ord}(κ)_{𝕋}. On the free quotient Â_{r}^{ord}(κ)_{𝕋}/Â_{r}^{ord}[p^{∞}](κ)_{𝕋}, the two filters induce the same p-adic topology. Writing Ã_{r}^{ord}(κ)_{𝕋} for the completion of Â_{r}^{ord}(κ)_{𝕋} with respect to F, therefore we find (6.25)

the natural surjective morphism: Â_{r}^{ord}(κ)_{𝕋} ↠ Ã_{r}^{ord}(κ)_{𝕋} has finite kernel.

This shows that the following sequence is exact by [CRT, Theorem 8.1 (ii)]:

(6.26)  0 → Ã_{r}^{ord}(κ)_{𝕋} → J̌_{∞}^{ord}(κ)_{𝕋} ⟶^{α} α(J̌_{∞}^{ord}(κ)_{𝕋}) → 0.

By this sequence combined with finiteness of Ker(Ã_{r}^{ord}(κ)_{𝕋} → Â_{r}^{ord}(κ)_{𝕋}), we get Proposition 6.4. Take a connected component Spec(𝕋) of Spec(h) with Â_{s,𝕋}^{ord} ≠ 0. Then we have the following exact sequence:

0 → Ã_{r}^{ord}(κ)_{𝕋} → J̌_{∞}^{ord}(κ)_{𝕋} ⟶^{α} J̌_{∞}^{ord}(κ)_{𝕋},

where Coker(α) is a ℤ_{p}-module of finite type with dim_{ℚ_{p}} Coker(α) ⊗_{ℤ_{p}} ℚ_{p} ≤ dim_{ℚ_{p}} B̂_{r}(κ)_{𝕋} ⊗_{ℤ_{p}} ℚ_{p}. Moreover we have a natural surjection: Â_{r}^{ord}(κ)_{𝕋} → Ã_{r}^{ord}(κ)_{𝕋} with finite kernel. If G_{𝕋}(κ) = 0, then Â_{r}^{ord}(κ)_{𝕋} ≅ Ã_{r}^{ord}(κ)_{𝕋}

We will see that the torsion submodule of Coker(α) is isomorphic to the maximal p-torsion submodule of E_{1}(κ)^{∨}.

Proof. The second sequence of (6.18) evaluated at κ produces the following exact sequence:

0 → α(J_{∞}^{ord})(κ)_{𝕋} → J_{∞}^{ord}(κ)_{𝕋} ⟶^{ρ_{∞}} B̂_{r}^{ord}(κ)_{𝕋} ⊗_{ℤ_{p}} ℚ_{p}.

By the exact sequence of the bottom row in the diagram (6.19), the image Im(ρ_{∞}) is embedded into (B̂_{r} ⊗ ℚ_{p})(κ)_{𝕋}, and thus Im(ρ_{∞}) ≅ ℚ_{p}^{i} ⊕ ℤ_{p}^{j} with i + j ≤ dim B̂_{r}(κ)_{𝕋} ⊗_{ℤ_{p}} ℚ_{p}. Thus we get the following exact sequences indexed by n:

0 = Im(ρ_{∞})[p^{n}] ≅ Tor_{1}^{ℤ_{p}}(Im(ρ_{∞}), ℤ/p^{n}ℤ) → α(J_{∞}^{ord})(κ)_{𝕋} ⊗_{ℤ_{p}} ℤ/p^{n}ℤ

→ J_{∞}^{ord}(κ)_{𝕋} ⊗_{ℤ_{p}} ℤ/p^{n}ℤ → Im(ρ_{∞})_{𝕋} ⊗_{ℤ_{p}} ℤ/p^{n}ℤ ≅ (ℤ/p^{n}ℤ)^{j} → 0.

Since these sequences satisfy the Mittag–Leffler condition, passing to the limit, we get another exact sequence:

0 → α(J̌_{∞}^{ord})(κ)_{𝕋} → J̌_{∞}^{ord}(κ)_{𝕋} ⟶^{ρ_{∞}} ℤ_{p}^{j} → 0.

Then the assertion follows from (6.22). We can check the last assertion by scrutinizing our computation, but here is a short cut. Since Ker(Â_{r}^{ord}(κ)_{𝕋} → Ã_{r}^{ord}(κ)_{𝕋}) is a submodule of A[p^{∞}]^{ord}(κ) ⊂ G_{𝕋}(κ) = 0. Thus the morphism has to be an isomorphism. □

Lemma 6.5. Let κ be as in Lemma 6.1. Then the maximal torsion submodule of J̌_{∞}(κ)_{𝕋}^{ord} is equal to G_{𝕋}(κ) if G_{𝕋}(κ) is finite. Otherwise, it is killed by p^{B} for some 0 < B ∈ ℤ.

Proof. By definition, the maximal torsion submodule of Ĵ_{s}(κ)_{𝕋}^{ord} = (J_{s}(κ) ⊗_{ℤ} ℤ_{p})_{𝕋}^{ord} for finite s is given by G_{s}(κ)_{𝕋} := J_{s}[p^{∞}](κ)_{𝕋}^{ord}. For s = ∞, the maximal torsion submodule of J_{∞}(κ)_{𝕋} = \underrightarrow{lim}_{s} Ĵ_{s}(κ)_{𝕋}^{ord} is given by G(κ)_{𝕋}. Thus we have an exact sequence for finite s:

0 → G_{s}(κ)_{𝕋} → Ĵ_{s}(κ)_{𝕋}^{ord} → F_{s} → 0

for the maximal ℤ_{p}-free quotient F_{s} := Ĵ_{s}(κ)_{𝕋}^{ord}/G_{s}(κ)_{𝕋}. This is a split exact sequence as the right term Ĵ_{s}(κ)_{𝕋}^{ord}/G_{s}(κ)_{𝕋} is ℤ_{p}-free. By taking p-adic completion: M ↦ M̌ = \underleftarrow{lim}_{n} M/p^{n}M, we get a split exact sequence for finite s:

0 → Ǧ_{s}(κ)_{𝕋} → Ĵ_{s}(κ)_{𝕋}^{ord} → F_{s} → 0.

This shows Ǧ_{s}(κ)_{𝕋} = G_{s}(κ)_{𝕋} for finite s, and Ǧ_{s}(κ)_{𝕋} is a finite module if κ is as in Lemma 6.1. Since F_{s} is ℤ_{p}-flat for all s ≥ r, F = \underrightarrow{lim}_{s} F_{s} is a ℤ_{p}-flat module. For s = ∞, we have the limit exact sequence (noting G(κ)_{𝕋} = G_{∞}(κ)_{𝕋})

0 → G(κ)_{𝕋} → J_{∞}(κ)_{𝕋}^{ord} → F → 0,

and F = J_{∞}(κ)_{𝕋}^{ord}/G(κ)_{𝕋}. By ℤ_{p}-flatness of F, after tensoring ℤ/p^{n}ℤ over ℤ_{p}, we still have an exact sequence (cf. [BCM, I.2.5]) indexed by 0 < n ∈ ℤ:

0 → G(κ)_{𝕋}/p^{n}G(κ)_{𝕋} → J_{∞}(κ)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℤ/p^{n}ℤ → F/p^{n}F → 0,

which obviously satisfies the Mittag–Leffler condition (with respect to n). Passing to the projective limit with respect to n, we get the limit exact sequence:

0 → Ǧ(κ)_{𝕋} → J̌_{∞}(κ)_{𝕋}^{ord} → F̌ → 0,

Since F is ℤ_{p}-flat, F̌ is torsion-free (and hence ℤ_{p}-flat by [BCM, I.2.4]). Indeed, we have the following commutative diagram with exact rows:

Tor_{ℤ_{p}}(F/pF, ℤ/p^{n}ℤ) ⟶^{↪} F/p^{n}F ⟶^{x↦px} F/p^{n}F ⟶^{↠} F/pF
≀↓  ^{∥}↓  ^{∥}↓  ↓^{∥}
F/pF  ⟶_{↪} F/p^{n}F ⟶^{x↦px} F/p^{n}F ⟶_{↠} F/pF.

Regard this as a projective system of exact sequences indexed by 0 < n ∈ ℤ. Then the transition maps of F/pF at the extreme right end is the identity and at the extreme left end is multiplication by p (i.e., the zero map). Passing to the limit, from left exactness of projective limit, we get an exact sequence

0 = \underleftarrow{lim}_{x↦px} F/pF → F̌ ⟶^{f↦pf} F̌,

and hence F̌ is p-torsion-free. If G(κ) is killed by p^{B} for some 0 < B ∈ ℤ, we still have Ǧ(κ)_{𝕋} = G(κ)_{𝕋}. Otherwise, for some 0 < j ∈ ℤ, G(κ)_{𝕋} fits into the following split exact sequence by Lemmas in Section 4,

0 → (ℚ_{p}/ℤ_{p})^{j} → G(κ)_{𝕋} → G(κ)_{𝕋}^{tor} → 0

for G(κ)_{𝕋}^{tor} killed by p^{B} for some 0 < B ∈ ℤ. Thus Ǧ(κ)_{𝕋} = G(κ)_{𝕋}^{tor}, which is the maximal torsion submodule of J̌_{∞}(κ)_{𝕋}^{ord}. □

We put M^{∗} = Hom_{ℤ_{p}}(M, ℤ_{p}) for a ℤ_{p}-module M and

X̌_{s,𝕋}(k)_{ord}^{∗} := Hom_{ℤ_{p}}(X̌_{s}(k)_{𝕋}^{ord}, ℤ_{p}) and X̂_{s,𝕋}(k)_{ord}^{∗} := Hom_{ℤ_{p}}(X̂_{s}(k)_{𝕋}^{ord}, ℤ_{p})

with s = r, r + 1, …, ∞ for X = J, A, B. The algebra h acts on J̌_{∞} naturally. As before, we write for an h-algebra R, J̌_{∞}(k)_{R}^{ord} = J̌_{∞}(k)^{ord} ⊗_{h} R and J̌_{∞}(k)_{ord, R}^{∗} = J̌_{∞}(k)_{ord}^{∗} ⊗_{h} R.

Assume the condition (A) in Section 5 for (α, A_{s}, B_{s}). Take a connected component Spec(𝕋) of Spec(h) in which the image of α is non-unit. Replacing α by 1_{𝕋}α for the idempotent 1_{𝕋} of 𝕋, we may assume that α ∈ 𝕋 as in the setting of (P4) in Proposition 5.1. Recall G_{𝕋}(k)_{tor}^{∨} is the maximal ℤ_{p}-torsion submodule of G_{𝕋}(k)^{∨}. We now state the principal result of this paper:

Theorem 6.6. Let k be either a number field or a finite extension of ℚ_{l} for a prime l. Then we get

(1) Consider the following sequence ℤ_{p}-dual to the one in Proposition 6.4:

0 → Coker(α)_{𝕋}^{∗} → J̌_{∞}(k)_{ord,𝕋}^{∗} ⟶^{α^{∗}} J̌_{∞}(k)_{ord,𝕋}^{∗} ⟶^{ι_{∞}^{∗}} Â_{r}(k)_{ord,𝕋}^{∗} → 0.

Then

(a) If G_{𝕋}(k) = 0, the sequence is exact except that Ker(ι_{∞}^{∗})/ Im(α^{∗}) is finite;

(b) If G_{𝕋}(k)_{tor}^{∨} = 0, the sequence is exact except that Ker(ι_{∞}^{∗})/ Im(α^{∗}) and Coker(ι_{∞}^{∗}) are both finite;

(c) If G_{𝕋}(k)_{tor}^{∨} ≠ 0, the sequence is exact up to finite error.

(d) The module G_{𝕋}(k)_{tor}^{∨} is killed by p^{B} for some finite 0 ≤ B ∈ ℤ, and the cokernel Coker(ι_{∞}^{∗}) is finite and is killed by p^{B}. In particular, after localizing the sequence by any prime divisor P ∈ Spec(Λ), the sequence is exact.

(2) After tensoring ℚ_{p} with the sequence (1), the following sequence

0 → Coker(α)_{𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} → J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}

→ J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} → Â_{r}(k)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} → 0

is an exact sequence of p-adic ℚ_{p}-Banach spaces (with respect to the Banach norm having the image of J̌_{∞}(k)_{ord,𝕋}^{∗} in J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} as its closed unit ball).

(3) The compact module J̌_{∞}(k)_{ord,𝕋}^{∗} is a Λ-module of finite type, and J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} is a Λ[\frac{1}{p}]-module of finite type.

Proof. We prove the exactness of the sequence (1). Since Â_{r}^{ord}(κ)_{𝕋} → Ã_{r}^{ord}(κ)_{𝕋} has finite kernel and is an isomorphism if G_{𝕋}(k) = 0 by Proposition 6.4, we only need to prove the various exactness of (1). By Proposition 6.4, the following sequence is exact:

0 → Ã_{r}^{ord}(k)_{𝕋} ⟶^{ι_{∞}} J̌_{∞}(k)_{𝕋}^{ord} ⟶^{α} J̌_{∞}(k)_{𝕋}^{ord} ⟶^{π_{∞}^{∗}} X → 0

for X = Coker(α). We consider the short exact sequence:

0 → Ã_{r}^{ord}(k)_{𝕋} ⟶^{ι_{∞}} J̌_{∞}(k)_{𝕋}^{ord} → Coker(ι_{∞}) → 0

and another exact sequence:

0 → Coker(ι_{∞}) ⟶^{α} J̌_{∞}(k)_{𝕋}^{ord} ⟶^{π_{∞}^{∗}} X → 0.

Applying the dualizing functor: M ↦ M^{∗} := Hom_{ℤ_{p}}(M, ℤ_{p}), we get the following exact sequences:

0 → Coker(ι_{∞})^{∗} → J̌_{∞}(k)_{ord,𝕋}^{∗} ⟶^{ι_{∞}^{∗}} Ã_{r}(k)_{ord,𝕋}^{∗} → Ext_{ℤ_{p}}^{1}(Coker(ι_{∞}), ℤ_{p}),

X^{∗} ↪ J̌_{∞}(k)_{ord,𝕋}^{∗} ⟶^{α^{∗}} Coker(ι_{∞})^{∗} → Ext_{ℤ_{p}}^{1}(X, ℤ_{p}) → Ext_{ℤ_{p}}^{1}(J̌_{∞}(k)^{ord,𝕋}, ℤ_{p}).

Thus Ext_{ℤ_{p}}^{1}(X, ℤ_{p}) contains Ker(ι_{∞}^{∗})/ Im(α^{∗}). Computing Ext_{ℤ_{p}}^{1}(M, ℤ_{p}) by the injective resolution 0 → ℤ_{p} → ℚ_{p} ⟶^{π} ℚ_{p}/ℤ_{p} → 0 (see [MFG, (4.10)]), we find

Ext_{ℤ_{p}}^{1}(M, ℤ_{p}) = Coker(Hom_{ℤ_{p}}(X, ℚ_{p}) ⟶^{π_{∗}} Hom_{ℤ_{p}}(X, ℚ_{p}/ℤ_{p})) = M[p^{∞}]^{∨}.

Since X is a ℤ_{p}-module of finite type by Proposition 6.4, Ext_{ℤ_{p}}^{1}(X, ℤ_{p}) = X[p^{∞}]^{∨} is finite. Similarly Ext_{ℤ_{p}}^{1}(J̌_{∞}(k)^{ord,𝕋}, ℤ_{p}) = J̌_{∞}(k)^{ord,𝕋}[p^{∞}]^{∨} = G_{𝕋}(k)_{tor}^{∨} and hence if G_{𝕋}(k)_{tor}^{∨} = 0, Ext_{ℤ_{p}}^{1}(X, ℤ_{p}) = Ker(ι_{∞}^{∗})/ Im(α^{∗}). Anyway, Ker(ι_{∞}^{∗})/ Im(α^{∗}) is finite. We have Coker(ι_{∞}) ↪ J̌_{∞}(k)_{𝕋}^{ord}. Again, we get, as Λ-modules,

Ext_{ℤ_{p}}^{1}(Coker(ι_{∞}), ℤ_{p}) ≅ Coker(ι_{∞})[p^{∞}]^{∨}

which is a quotient of G_{𝕋}(k)_{tor}^{∨} (see Lemmas 4.2, 4.3 and 6.5). Indeed, assuming finiteness of G_{𝕋}(k), the torsion part of J̌_{∞}(k)_{𝕋}^{ord} is isomorphic to a submodule of G_{𝕋}(k) by Lemma 6.5; in particular, it has finite torsion (this proves (1a)). Without assuming finiteness of G_{𝕋}(k), the p-torsion part of Coker(ι_{∞}) is a Λ-submodule of a bounded p-torsion Λ–module G_{𝕋}(k)_{tor} by Lemma 6.5. Thus Ext_{ℤ_{p}}^{1}(Coker(ι_{∞}), ℤ_{p}) is is a quotient of G_{𝕋}(k)_{tor}^{∨} and killed by p^{B} for some 0 ≥ B ∈ ℤ (and this proves (1b)). In addition, Coker(i_{∞}^{∗}) = Coker(J̌_{∞}(k)_{ord,𝕋}^{∗} → Ã_{r}(k)_{ord,𝕋}^{∗}) factors through the ℤ_{p}-module Ã_{r}(k)_{ord,𝕋}^{∗} of finite type, which lands in the bounded p-torsion module Coker(ι_{∞})[p^{∞}]^{∨} (by Lemma 6.5); so, Coker(i_{∞}^{∗}) must have finite order (this shows (1c)). Therefore, the error term Coker(i_{∞}^{∗}) is a pseudo-null Λ-module, it is killed after localization at prime divisors of Spec(Λ). Thus we get all the assertions in (1). The exact sequence in (1) tells us that J̌_{∞}(k)_{ord, 𝕋}^{∗}/α(J̌_{∞}(k)_{ord, 𝕋}^{∗}) is isomorphic (up to finite modules) to the ℤ_{p}-module Â_{r}(k)_{ord, 𝕋}^{∗} of finite type, which is a torsion Λ-module of finite type. Then by Nakayama’s lemma, J̌_{∞}(k)_{ord, 𝕋}^{∗} is a Λ-module of finite type. This proves the assertion (3). The extension modules appearing in the above proof of (1) is p-torsion Λmodule of finite type. Thus the sequence

0 → X^{∗} → J̌_{∞}(k)_{ord, 𝕋}^{∗} → J̌_{∞}(k)_{ord, 𝕋}^{∗} → Â_{r}(k)_{ord, 𝕋}^{∗} → 0

is exact up to p-torsion error. By tensoring ℚ over ℤ, we get the exact sequence (2):

X^{∗} ⊗_{ℤ_{p}} ℚ_{p} ↪ J̌_{∞}(k)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} → J̌_{∞}(k)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} ↠ Â_{r}(k)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}.

The above sequence is the p-adic Banach dual sequence of the following exact sequence obtained from the sequence in (1) by tensoring ℚ:

Â_{r}(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℚ_{p} ⟶_{↪}^{ι_{∞}} J̌_{∞}(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℚ_{p} ⟶^{α} J̌_{∞}(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℚ_{p} ⟶_{↠}^{π_{∞}^{∗}} X ⊗_{ℤ_{p}} ℚ_{p}.

Indeed, equipping J̌_{∞}(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℚ_{p} with the Banach p-adic norm so that the closed unit ball is given by the image of J̌_{∞}(k)_{𝕋}^{ord} in J̌_{∞}(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} ℚ_{p}, the sequence is continuous (the first and the last term are finite dimensional ℚ_{p}- vector spaces; so, there is a unique p-adic Banach space structure on them). The dual space of bounded functionals of each term is given by the ℚ_{p}-dual of the corresponding space before tensoring ℚ, which is given by Y ⊗_{ℤ_{p}} ℚ_{p} for Y = Â_{r}(k)_{ord, 𝕋}^{∗}, J̌_{∞}(k)_{ord,𝕋}^{∗} and X^{∗}, respectively. This proves (2). □

Corollary 6.7. Let the notation be as in (1) of Theorem 6.6. Consider the set Ω ⊂ 𝕋 of prime factors (in Λ) of γ^{p^{n}} − 1 for n = 0, 1, 2, …, ∞. Except for finitely many α ∈ Ω, we have Coker(α)_{𝕋}^{∗} ⊗_{Λ} Λ_{P} = 0 for P = (α) ∈ Spec(Λ), where Λ_{P} is the localization of Λ at P.

Proof. Note that α ∈ Ω (regarded as α ∈ 𝕋) satisfies the assumption (A) by Proposition 5.1 (P1) and that Λ[\frac{1}{p}] is a principal ideal domain (as Λ is a unique factorization domain of dimension 2; see [CRT, Chapter 7]). Pick an isomorphism J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{ℤ} ℚ ≅ Λ[\frac{1}{p}]^{R} ⊕ X_{k}^{∗} of Λ[\frac{1}{p}]-modules with torsion Λ[\frac{1}{p}]-module X_{k}^{∗}. Then for P outside the support of the Λ[\frac{1}{p}]-module X_{k}^{∗}, by Theorem 6.6 (2),

K := Ker(α : J̌_{∞}(k)_{ord,𝕋}^{∗} → J̌_{∞}(k)_{ord,𝕋}^{∗})

is killed by some p-power. Then by the assertion (1) of the above theorem, K is a ℤ_{p}-module of finite type; hence K is finite. This shows the result. □

7. Closure of the global Λ-MW group in the local one

Let κ be a number field and k = κ_{𝔭} be the 𝔭-adic completion of κ for a prime 𝔭|p of κ. Write W for the p-adic integer ring of k, and let Q be the quotient field of Λ. By [M55] or [T66], for an abelian variety A_{/k} of dimension g, Â(k) = A(k) ⊗_{ℤ} ℤ_{p} has torsion free part Â(k)_{f} isomorphic to the additive group W^{g}, and the torsion part Â(k)_{tor} is a finite group. Write F = κ or k. Recall the 𝕋-component

J̌_{∞}(F)_{ord,𝕋}^{∗} := J̌_{∞}(F)_{ord}^{∗} ⊗_{h} 𝕋

for a connected component Spec(𝕋) of Spec(h). By Theorem 6.6 (3), J̌_{∞}(F)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} is a 𝕋 ⊗_{ℤ_{p}} ℚ_{p}-module of finite type. For simplicity, write 𝕁_{𝕋}(F) := J̌_{∞}(F)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}. Let the notation be as in Corollary 6.7; in particular, Ω is the set of prime factors (in Λ) of γ^{p^{n}} − 1 for n = 1, 2, …, ∞. Note that α ∈ Ω ⊂ 𝕋 satisfies the condition (A) by Proposition 5.1 (P1). Then by Theorem 6.6 (2), this implies

𝕁_{𝕋}(F)/α(𝕁_{𝕋}(F)) ≅ Â_{r}^{ord}(F)_{𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}.

Further localizing at each arithmetic point P ∈ Spec(h)(\overline{ℚ}_{p}) with P |(γ^{p^{r−1}} −1), we get, for 𝕁_{𝕋_{P}} (F) = 𝕁_{𝕋}(F) ⊗_{𝕋} 𝕋_{P} for the localization 𝕋_{P} at P,

𝕁_{𝕋_{P}} (F)/α(𝕁_{𝕋_{P}} (F)) ≅ Â_{r}^{ord}(F)_{𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}.

Since Λ[\frac{1}{p}] is a principal ideal domain, 𝕁_{𝕋}(F) is isomorphic to Λ[\frac{1}{p}]^{m_{F}} ⊕ X′_{F} for a torsion Λ[\frac{1}{p}]-module X′_{F}. Put X_{F} := X′_{F} ⊕ G_{?}(k)^{∨} and decompose X_{F} = ⨁_{𝔓} Λ[\frac{1}{p}]/𝔓^{e_{F} (𝔓)} for maximal ideals 𝔓 of Λ[\frac{1}{p}]. Put Char_{Λ[ \frac{1}{p}]}(X_{F}) =

∏_{𝔓} 𝔓^{e_{F} (𝔓)}. If P ∈ Spec(Λ)(\overline{ℚ}_{p}) is prime to Char_{Λ[ \frac{1}{p}]}(X_{F}),

𝕁_{𝕋}(F)/α(𝕁_{𝕋}(F)) ≅ (Λ/P)^{m_{F}} ⊗_{ℤ_{p}} ℚ_{p}.

In other words, the Λ_{P} /P -dimension of 𝕁_{𝕋}(F)/α(𝕁_{𝕋}(F)) is independent of P for most of P. We formulate this fact for F = k as follows:

Theorem 7.1. Let the notation be as above. Write W for the 𝔭-adic integer ring of k and Q for the quotient field of Λ. Then the Q-vector space J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{Λ} Q has dimension equal to g = rank_{ℤ_{p}} W · rank_{Λ} 𝕋.

Proof. We use the notation introduced in Corollary 6.7. Pick α ∈ Ω, and let A ⊂ J_{r}[α] be the identity connected component. Define J_{r} ↠ B to be the dual quotient of A ↪ J_{r}. By the control Theorem 6.6 (2), we have 𝕁_{𝕋}(k)/α𝕁_{𝕋}(k) ≅ Ǎ(k)_{ord,𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} for all α ∈ Ω. Moreover, we have dim_{κ(P)} 𝕁_{𝕋}(k)/α𝕁_{𝕋}(k) = m_{k} outside a finite set S ⊂ Ω. The set S is made of prime factors in Ω of Char_{Λ[ \frac{1}{p}]}(X_{k}^{∗}). Note that m_{k} = rank_{Λ[ \frac{1}{p}]} 𝕁_{𝕋}(k) = dim_{Q} J̌_{∞}(k)_{ord,𝕋}^{∗} ⊗_{Λ} Q; so, we compute rank_{Λ[ \frac{1}{p}]} 𝕁_{𝕋}(k).

By [M55] or [T66], we have Â(k) ≅ W^{dim A} × Δ for a finite p-abelian group Δ. Regarding A(k) as a p-adic Lie group, we have a logarithm map log : A(k) → Lie(A_{/k}). For a ring R, write h(R) (resp. h_{r}(R)) for the scalar extension to R of

ℤ[T (n)|n = 1, 2, …] ⊂ End(A_{/ℚ}) ≅ End(B_{/ℚ})

(resp. ℤ[T (n)|n = 1, 2, …] ⊂ End(J_{r/ℚ})).

The Lie algebra Lie(A_{/ℚ_{p}}) is the dual of Ω_{B/ℚ_{p}}. Note that Ω_{J_{r}/ℚ} ≅ Ω_{X_{r}/ℚ} (e.g., [GME, Theorem 4.1.7]). By q-expansion at the infinity cusp, we have an embedding i : Ω_{X_{r}/ℚ} ↪ ℚ[[q]] sending ω to i(ω)\frac{dq}{q}. Writing i(ω) = ∑_{n=1}^{∞} a(n, ω)q^{n}, we have a(m, ω|T (n)) = ∑_{0<d|(m,n),(d,Np)=1} d· a(\frac{mn}{d^{2}}, ω|⟨d⟩) for the diamond operator ⟨d⟩ associated to d ∈ (ℤ/Np^{r}ℤ)^{×}. From this, the pairing ⟨·, ·⟩ : h_{r}(ℚ) × Ω_{X_{r}/ℚ} → ℚ given by ⟨H, ω⟩ = a(1, ω|H) is nondegenerate (see [GME, §3.2.6]). Thus we have

Ω_{J_{r}/k} ≅ Hom_{k}(h_{r}(k), k) and Ω_{A/k} ≅ Hom_{k}(h(k), k) as modules over h_{r}(k),

since h(k) is naturally a quotient of h_{r}(k) and B = J_{r}/α(J_{r}) for (α) = Ker(h_{r}(ℤ_{p}) ↠ h(ℤ_{p})) in h_{r}(ℤ_{p}). By the duality between Lie(A_{/k}) and Ω_{A_{/}k}, we have

Lie(A_{/k}) ≅ h(k) as an h(k)-module.

This leads to an isomorphism of h-modules:

Â(k)_{𝕋}^{ord} ⊗_{ℤ_{p}} k ⟶_{∼}^{log} Lie(A_{/k})_{𝕋} ≅ (𝕋/(α)𝕋) ⊗_{ℤ_{p}} k

as 𝕋/(α)𝕋 is canonically isomorphic to a ring direct summand h(ℤ_{p})_{𝕋}^{ord} of h(ℤ_{p})^{ord} as ℤ_{p}-algebras by the control theorem (cf. [GME, §3.2.6]). Thus

rank_{W} Â(k)_{𝕋}^{ord} = [k : ℚ_{p}] rank_{Λ/(α)} 𝕋/(α)𝕋 = [k : ℚ_{p}] rank_{Λ} 𝕋.

This proves the desired assertion, as [k : ℚ_{p}] = rank_{ℤ_{p}} W. □

We have a natural Λ-linear map

J̌_{∞}(κ)_{𝕋}^{ord} ⟶^{ι} J̌_{∞}(k)_{𝕋}^{ord} and J̌_{∞}(k)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p} ⟶^{ι^{∗}} J̌_{∞}(κ)_{ord, 𝕋}^{∗} ⊗_{ℤ_{p}} ℚ_{p}.

We would like to study their kernel and cokernel. Take a reduced irreducible component Spec(𝕀) ⊂ Spec(𝕋). Let 𝕀̃ be the normalization of 𝕀, and write Q(𝕀) for the quotient field of 𝕀. Then 𝕁_{F} := J̌_{∞}(F)_{ord,𝕀̃}^{∗} ⊗_{ℤ_{p}} ℚ_{p} is a 𝕀̃[\frac{1}{p}]-module of finite type for F = κ, k. Note that

𝕀̃[\frac{1}{p}] is a Dedekind domain. This we can decompose 𝕁_{F} = L_{F} ⊕X_{F} for a locally

free 𝕀̃[\frac{1}{p}]-module L_{F} of finite constant rank and a torsion module X_{F} isomor-

phic to ⨁_{𝔓} 𝕀̃[\frac{1}{p}]/𝔓^{e_{F} (𝔓)} for finitely many maximal ideals 𝔓 of 𝕀̃[\frac{1}{p}]. We put Char_{𝕀}(X_{F}) = ∏_{𝔓} 𝔓^{e_{F} (𝔓)}. For an abelian variety A over κ, write \overline{A(κ)} ⊂ Â(k) for the p-adic closure of the image of A(κ) in Â(k). Pick an arithmetic point P ∈ Spec(h)(\overline{ℚ}_{p}) of weight 2. Suppose that the abelian variety A_{P} is realized in J_{r} and satisfies the condition (A). By Theorem 6.6 (2), the natural map

𝕁_{F} /P 𝕁_{F} → Â_{P} (F)_{ord, 𝕀̃}^{∗} ⊗_{ℤ_{p}} ℚ_{p}

is an isomorphism. Thus as long as P ∤ Char(X_{k}) · Char(X_{κ}), we have a surjective linear map

(7.1)  Â_{P} (k)_{ord, 𝕀̃}^{∗} ⊗_{ℤ_{p}} ℚ_{p} ↠ \overline{A_{P} (κ)}_{ord, 𝕀̃}^{∗} ⊗_{ℤ_{p}} ℚ_{p} ≅ ι_{P}^{∗} (𝕁_{k}/P 𝕁_{k})

ℚ_{p}-dual to the inclusion

\overline{A_{P} (κ)}ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p} ⊂ Â_{P} (k))ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p},

where ι_{P}^{∗} = ι^{∗} ⊗ id : 𝕁_{k} ⊗_{h} h/P = 𝕁_{k}/P 𝕁_{k} → Â_{P} (κ)_{ord, 𝕀̃} ⊗_{ℤ_{p}} ℚ_{p} induced by ι^{∗}. Put

r_{k}(F; 𝕀) := dim_{Q(𝕀)} 𝕁_{F} ⊗_{𝕀̃[ \frac{1}{p}]} Q(𝕀) = rank_{𝕀̃[ \frac{1}{p}]} 𝕁_{F}

for the quotient field Q(𝕀) of 𝕀̃. We now assume

(a) Taking r = r(P) and A_{r} to be A_{P}, the condition (A) holds for A_{P} for

almost all arithmetic points P ∈ Spec(𝕀) of weight 2. By Proposition 5.1 (P2), the condition (A) holds for “all” arithmetic points P ∈ Spec(𝕀) of weight 2 if 𝕋 = 𝕀 and p is unramified in 𝕋/P for one arithmetic point P ∈ Spec(𝕋)(\overline{ℚ}_{p}). Indeed, as shown in [F02, Theorem 3.1], 𝕋 is regular under this assumption (and the regularity guarantees the validity of (A) by Proposition 5.1 (P2)). Pick a base arithmetic point P_{0} ∈ Spec(𝕀)(\overline{ℚ}_{p}) of weight 2. The point P_{0} gives rise to f = f_{P_{0}} ∈ S_{2}(Γ_{1}(Np^{r+1})) with B_{0} = B_{P_{0}} and A_{0} = A_{P_{0}} satisfying f|T (n) = P_{0}(T (n))f for all n > 0. By Theorem 6.6 (2), we have for F = k, κ,

(ct) 𝕁_{F} /P_{0}𝕁_{F} is isomorphic to the ℚ_{p}-dual of Â_{0}(F)ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p}.

Choosing P_{0} outside Char_{𝕀}(X_{κ}) · Char_{𝕀}(X_{k}), we may assume the following condition for F = k, κ: (dim) dim_{ℚ_{p}(f)} 𝕁_{F} /P_{0}𝕁_{F} = r_{k}(F; 𝕀). Here ℚ_{p}(f) is the quotient field of 𝕀/P_{0}𝕀 and is generated by P_{0}(T (n)) for all n over ℚ_{p}. Since A_{0}(κ) ⊗_{ℤ} ℚ is a ℚ(f) vector space, if A_{0}(κ) ⊗ ℚ ≠ 0, we have dim_{ℚ(f)} A_{0}(κ) ⊗ ℚ > 0, which implies that \overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p} ≠ 0. Suppose

k = ℚ_{p} and (Â_{0}(κ)ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p}) ≠ 0.

Then (\overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p})ord ; 𝕀̃ is a finite dimensional vector subspace over ℚ_{p} of Â_{0}(k) ⊗_{ℤ_{p}} ℚ_{p} stable under T (n) for all n. Let us identify P_{0}(T (n)) ∈ \overline{ℚ}_{p} with a system of eigenvalues of T (n) occurring on Â_{0}(κ)ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p}. Then (\overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p})ord ; 𝕀̃ and Â_{0}(k)ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p} are ℚ_{p}(f)-vector spaces. Thus we conclude

0 < dim_{ℚ_{p}(f)}(\overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p})ord ; 𝕀̃ ≤ dim_{ℚ_{p}(f)}(Â_{0}(k) ⊗_{ℤ_{p}} ℚ_{p})ord ; 𝕀̃ = 1,

which implies

0 < dim_{ℚ_{p}(f)}(\overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p})ord ; 𝕀̃ = dim_{ℚ_{p}(f)} Â_{0}(k)ord ; 𝕀̃ ⊗_{ℤ_{p}} ℚ_{p} = 1.

By (7.1), we get

dim_{ℚ_{p}} ι_{P_{0}}^{∗}(𝕁_{κ}/P_{0}𝕁_{κ})

= dim_{ℚ_{p}}(\overline{A_{0}(κ)} ⊗_{ℤ_{p}} ℚ_{p}) = dim_{ℚ_{p}}(Â_{0}(ℚ_{p}) ⊗_{ℤ_{p}} ℚ_{p}) = dim_{ℚ_{p}} ℚ_{p}(f).

In other words, by Theorem 6.6 (2), the kernel of the map ι^{∗}: K := Ker(ι^{∗} : 𝕁_{k} → 𝕁_{κ}) for k = ℚ_{p} is a torsion 𝕀̃[\frac{1}{p}]-module. Now we move weight 2 arithmetic points P ∈ Spec(𝕀)(\overline{ℚ}_{p}) ⊂ Spec(𝕋)(\overline{ℚ}_{p}). Then K_{P} = K/P K covers surjectively

Ker(ι_{P}^{∗} : 𝕁_{k}/P 𝕁_{k} → 𝕁_{κ}/P 𝕁_{κ}).

By 𝕀̃[\frac{1}{p}]-torsion property of K, K/P K = 0 for almost all points in Spec(𝕀)(\overline{ℚ}_{p}), and we get

Corollary 7.2. Let the notation and the assumption be as above. Suppose the condition (a), (dim), k = κ_{𝔭} = ℚ_{p} and

dim_{ℚ(f)} A_{0}(κ) > 0.

Then except for finitely many arithmetic points of Spec(𝕀)(\overline{ℚ}_{p}) weight 2, we have dim_{ℚ(f_{P})} A_{P} (κ) > 0 and

dim_{ℚ_{p}(f_{P})}(\overline{A_{P} (κ)}^{ord} ⊗_{𝕀/P} ℚ_{p}(f_{P})) = dim_{ℚ_{p}} ℚ_{p}(f_{P}).

For general abelian variety A_{/ℚ}, an estimate of dim_{ℚ_{p}} \overline{A(ℚ)} ⊗_{ℤ_{p}} ℚ_{p} relative to dim_{ℚ} A(ℚ) ⊗_{ℤ} ℚ and a conjecture is given in [W11]. Here we studied the dimension over a family and showed its co-ordinary (or ordinary) part stays maximal for most of members of the family if one is maximal.

References

Books

[AAG] S. S. Gelbart, Automorphic Forms on Adele Groups, Annals of Math. Studies 83, Princeton University Press, Princeton, NJ, 1975.

[ADT] J. S. Milne, Arithmetic Duality Theorems, Second edition, 2006, BookSurge, LLC

[ALG] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.

[AME] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Studies 108, Princeton University Press, Princeton, NJ, 1985.

[BCM] N. Bourbaki, Algèbre Commutative, Hermann, Paris, 1961–1998.

[CNF] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer Springer Grundlehren der mathematischen Wissenschaften Bd. 323, 2000

[CRT] H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge Univ. Press, New York, 1986.

[EAI] H. Hida, Elliptic curves and arithmetic invariants, Springer Monographs in Mathematics, Springer, 2013

[ECH] J. S. Milne, Étale Cohomology, Princeton University Press, Princeton, NJ, 1980.

[GME] H. Hida, Geometric Modular Forms and Elliptic Curves, second edition, World Scientific, Singapore, 2012.

[IAT] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, and Iwanami Shoten, Tokyo, 1971.

[MFG] H. Hida, Modular Forms and Galois Cohomology, Cambridge Studies in Advanced Mathematics 69, Cambridge University Press, Cambridge, England, 2000.

[MFM] T. Miyake, Modular Forms, Springer, New York-Tokyo, 1989.

[NMD] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Springer, New York, 1990.

[RTP] G. Faltings and G. Wüstholtz et al, Rational Points, Aspects of Mathematics E6, Friedr. Vieweg & Sohn, Braunschweig, 1992.

Articles

[C86] H. Carayol, Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ec. Norm. Sup. 4-th series, 19 (1986), 409–468.

[F02] A. Fischman, On the image of Λ-adic Galois representations. Ann. Inst. Fourier (Grenoble) 52 (2002), 351–378.

[GK13] E. Ghate and N. Kumar, Control theorems for ordinary 2-adic families of modular forms, Automorphic representations and L-functions, 231–261, Tata Inst. Fundam. Res. Stud. Math., 22, Tata Inst. Fund. Res., Mumbai, 2013.

[H86a] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ec. Norm. Sup. 4th series 19 (1986), 231–273.

[H86b] H. Hida, Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545–613.

[H11] H. Hida, Hecke fields of analytic families of modular forms, J. Amer. Math. Soc. 24 (2011), 51–80

[H13a] H. Hida, Image of Λ-adic Galois representations modulo p, Inventiones Math. 194 (2013) 1–40

[H13b] H. Hida, Local indecomposability of Tate modules of non CM abelian varieties with real multiplication, J. Amer. Math. Soc. 26 (2013), 853–877

[H14a] H. Hida, Λ-adic Barsotti–Tate groups, Pacific J. Math. 268 (2014), 283–312

[H14b] H. Hida, Control of Λ-adic Mordell–Weil groups, preprint, 2013,

posted at http://www.math.ucla.edu/~hida/LMW.pdf

[H14c] H. Hida, On the rank of Mordell–Weil groups of abelian varieties of GL(2)-type, in preparation, 2014

[M55] A. Mattuck, Abelian varieties over p-adic ground fields. Ann. of Math. (2) 62, (1955) 92–119.

[MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of Q. Inventiones Math. 76 (1984), 179–330

[MW86] B. Mazur and A. Wiles, On p–adic analytic families of Galois representations, Compositio Math. 59 (1986), 231–264

[N06] J. Nekovár, Selmer complexes. Astérisque 310 (2006), viii+559 pp.

[Sh73] G. Shimura, On the factors of the Jacobian variety of a modular function field. J. Math. Soc. Japan 25 (1973), 523–544.

[T66] J. Tate, p-divisible groups, Proc. Conf. on local filds, Driebergen 1966, Springer 1967, 158–183.

[W11] M. Waldschmidt, On the p-adic closure of a subgroup of rational points on an Abelian variety. Afr. Mat. 22 (2011), 79–89.

[Z14] B. Zhao, Local indecomposability of Hilbert modular Galois representations, Ann. Inst. Fourier (Grenoble) 64 (2014), 1521–1560 (posted on web: http://aif.cedram.org/item?id=AIF 2014 64 4 1521 0).

Haruzo Hida

Department of Mathematics UCLA Los Angeles, CA 90095-1555 U.S.A. hida@math.ucla.edu