Ki-Seng Tan

PREPRINT

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

www.math.ntu.edu.tw/ ~ mathlib/preprint/2013- 02.pdf

Selmer Groups Over Z _p ^d -Extensions

Ki-Seng Tan

January 30, 2013

KI-SENG TAN

Abstract. Consider an abelian variety A defined over a global field K and let L/K be a Z^dp- extension, unramified outside a finite set of places of K, with Gal(L/K) = Γ. Let Λ(Γ) := Zp[[Γ]]

denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the Λ(Γ)- module XL, the dual p-primary Selmer group, varies when L/K is replaced by a intermediate Z^ep-extension.

1. Main Results s:intro

Let A be a g-dimensional abelian variety defined over a global field K and let L/K be a Z^dp- extension, unramified outside a finite set of places of K, with Gal(L/K) = Γ. For each finite intermediate extension F/K of L/K, let Selp^∞(A/F ) denote the p-primary Selmer group (see

§2.5) and set

Sel_p^∞(A/L) = lim

−→F

Sel_p^∞(A/F ).

We endow Sel_p^∞(A/L) (resp. Sel_p^∞(A/F )) with the discrete topology and let X_L (resp. X_F) denote its Pontryagin dual group. The main aim of this paper is to study how the characteris- tic ideal of XL over Λ(Γ) := Z^p[[Γ]] (the Iwasawa algebra) varies, when L/K is replaced by an intermediate Z^ep-extension L⁰/K. Our result has many applications. In particular, it leads to a structure theorem of Z_L, the Pontryagin dual of lim

−→^FQp/Zp⊗ A(F ) (see §1.6).

su:ns

1.1. Notation. Let S denote the set of places of K ramified over L/K. For an algebraic extension F/K and a place w of F , let Fw denote the w-completion of F . If w is a non-archimedean place, let Ow, mwand F^w(or OF_w, mF_wand F^Fw) denote the ring of integers, the maximal ideal and the residue field of F_w. Also, denote q_w= |Fw|. We fix an algebraic closure K of K and let K^s⊂ K denote the separable closure of K, and the same for K_v.

For an abelian group D, let D_p(resp. D_div) denote the p-primary (resp. p-divisible) part of D_tor, the torsion subgroup. For a locally compact group G, let G^∨ denote its Pontryagin dual group.

In this paper, we always have G^∨= Hom_cont(G, Qp/Zp) as G will be either pro-p or p-primary. If O is the ring of integers of a finite extension Q of Qp and G is an O-module, we endow G^∨ with the O-module structure by setting a · ϕ(g) = ϕ(a · g), a ∈ O, ϕ ∈ G^∨, g ∈ G. As O-modules, G is cofinitely generated if and only if G^∨ is finitely generated, and denote corank_O(G) := rank_O(G^∨).

If G is a Z^p-module, write OG for O ⊗_ZpG. Then we can identify (OG)^∨ with OG^∨ by introducing a non-degenerate pairing [ , ] : OG × OG^∨ −→ Qp/Z^p as follow. First choose a generator δ ∈ O of the different of the filed extension Q/Q^p and set Tr^∗(x) = Tr_Q/Qp(δ⁻¹· x) for x ∈ Q. If a ∈ Q/O is the residue class of some y ∈ Q modulo O, let T^∗(a) ∈ Qp/Zp denote the residue class of Tr^∗(y) modulo Zp. Then Q : O × Q/O −→ Qp/Zp given by Q(x, a) := T^∗(xa) is a non-degenerate pairing. Let < , >: OG×OG^∨−→ Q/O be the O-pairing given by < g, φ >= φ(g),

Acknowledgement: This research was supported in part by the National Science Council of Taiwan, NSC97- 2115-M-002-006-MY2, NSC99-2115-M-002-002-MY3.

1

for g ∈ G, φ ∈ G^∨. Then define [α, β] = T^∗(< α, β >). Let e1, ..., em be a Z^p-basis of O. If β =P

iei⊗ φi, φi∈ G^∨, satisfies [OG, β] = 0, then for all x ∈ O, g ∈ G, 0 = [x ⊗ g, β] = T^∗(x ·X

i

ei⊗ < g, φi >) = Q(x,X

i

ei⊗ < g, φi>).

Since Q is non-degenerate,P

iei⊗ < g, φi>= 0, and hence < g, φi >= 0 for all g. Consequently, φi= 0 for every i, whence β = 0. Similarly, if α ∈ OG satisfies [α, OG^∨] = 0, then α = 0.

If G is a Γ-module, let Γ acts on G^∨ by ^γϕ(g) := ϕ(^γ−1g). The identification (OG)^∨ = OG^∨ depends on the choice of δ. However, it alters neither the O-module structure nor the Γ-module structure (if exists) on (OG)^∨.

Let µp^mdenote the p^mth root of unity and write µp^∞ =S

mµp^m regarded as a discrete subgroup of Q^×p. Let bΓ denote the group of all continuous characters from Γ to µp^∞ and let Gal(Qp/Q^p) act on it via the action on µp^∞. Thus, bΓ = Γ^∨ as topological groups, while Gal(Qp/Q^p) acts non- trivially on bΓ but trivially on Γ^∨. If ω ∈ bΓ with the image Im(ω) = µp^m, write Oω= Z^p[µp^m] ⊂ Qp. If O contains Oωand G is an O-module with a continuous action of Γ, write (for the ω-eigenspace)

G^(ω):= {g ∈ G | ^γg = ω(γ) · g}.

For a finitely generated Λ(Γ)-module W , let χ_Λ(Γ)(W ) denote its characteristic ideal (see §2.1).

Denote Γ⁰ = Gal(L⁰/K), Λ(Γ⁰) = Z^p[[Γ⁰]]. Our result also covers the d = 1 case in which Γ⁰ = 0, Λ(Γ⁰) = Z^p, and we define χ_Λ(Γ⁰₎(W ) = χ_Zp(W ), the usual characteristic ideal of Z^p-module.

Let A[p^m] denote the kernel of the multiplication by p^m on A viewed as a sheaf on the flat topology of K and denote A[p^∞] =S

mA[p^m]. In particular, A[p^∞](K) = A(K)p. Let A^t denote the dual abelian variety.

su:localcondition

1.2. A local condition. It is well known that if K is a number field, then the following question has an affirmative answer (see below).

Is X_Lf initely generated over Λ(Γ)?

In general, the answer could be obtained via the following local criterion.

p:iwasawa Proposition 1.2.1. The Iwasawa module XL is finitely generated over Λ(Γ) if and only if at each place v ∈ S, the local cohomology group H¹(Γ_v, A(L_v)) is cofinitely generated over Zp.

The proof, based on results in [Tan10], is given in §3.5. The condition of the proposition holds if K is a number field (Corollary 2.4.2), or if at every ramified place, A has either good ordinary reduction or split-multiplicative reduction [Tan10, Theorem 5]. However, if char.(K) = p and the reduction of A at a place v ∈ S is an abelian variety without non-trivial p-torsion points, then the condition fails to hold (Theorem 3.6.1).

su:desda

1.3. The specialization data. For the rest of this paper except §3.6, we shall assume that every v ∈ S is either good ordinary or split-multiplicative, and hence XL is finitely generated over Λ(Γ). Also, for simplicity, we assume that char.(K) = p, if K is not a number field. Write ΘL= χ_Λ(Γ)(XL). Extend the canonical map Γ −→ Γ⁰to the continuous Z^p-algebra homomorphism (the specialization map) p_L/L⁰ : Λ(Γ) −→ Λ(Γ⁰). Then the following question arises:

W hat is the relation between p_L/L0(Θ_L) and Θ_L⁰?

To illustrate our answer, some simplification and notation are in order. First, by choosing a sequence L⁰ ⊂ L⁰⁰ ⊂ · · · ⊂ L⁽ⁱ⁾ ⊂ · · · ⊂ L^(d−e) ⊂ L with each Gal(L⁽ⁱ⁾/K) ' Z^e−1+ip , we can write p_L/L⁰ = p_L⁰⁰_/L⁰ ◦ · · · ◦ p_L(i+1)/L⁽ⁱ⁾ ◦ · · · ◦ p_L/L(d−e), and hence answer the question for p_L/L⁰ by answering that for every p_L(i+1)/L⁽ⁱ⁾. Therefore, without loss of generality, we may assume that e = d−1. We shall make such assumption and then fix a topological generator ψ of Ψ := Gal(L/L⁰).

1.3.1. The global factor. Let K⁰/K be a Z^p-extension and let σ be a topological generator of the Galois group. If 1, ..., lare eigenvalues, counted with multiplicities, of the action of σ on the Tate module TpA[p^∞](K⁰). Then the product Ql

j=1(1 − ⁻¹_j σ) ⊂ Z^p[[Gal(K⁰/K)]] is nothing but the characteristic ideal of TpA[p^∞](K⁰) over Z^p[[Gal(K⁰/K)]], and in particular, the ideal

w_K0/K :=

l

Y

j=1

(1 − ⁻¹_j σ)(1 − ⁻¹_j σ⁻¹)

is independent of the choice of σ (See Proposition 2.3.5). If K⁰/K is a Z^ep-extension with e ≥ 2, set w_K0/K = (1).

d:glfactor Definition 1.3.1. Define the Λ(Γ⁰)-ideal

%_L/L⁰ =

(w_L0/K if d ≥ 2;

|A[p^∞](K)|²

|A[p^∞](K)T A[p^∞](L)div|² if d = 1.

1.3.2. Local factors at unramified places.

d:badunram Definition 1.3.2. For each v, let Π_v denote the group of the connected components of the closed fiber (over Fv) of the N´eron model of A/K_v and let π_v denote the Zp-ideal (|Π^Gal(Fv ^v/Fv)|).

1.3.3. Local factors at good ordinary places. Suppose that A has good ordinary reduction ¯A at v. Then eigenvalues of the Frobenius endomorphism Fv : ¯A −→ ¯A over F^v are, counted with multiplicities,

α1, ..., αg, qv/α1, ..., qv/αg,

where α1, ..., αgare eigenvalues of the (twist) matrix u of the action on the Tate module of ¯A[p^∞] by the Frobenius substitution Frobv∈ Gal(Fv/F^v) ([Maz72, Corollary 4.37]).

d:goodord Definition 1.3.3. Suppose A has good ordinary reduction ¯A at v and L⁰/K is unramified at v with the Frobenius element [v]L⁰/K ∈ Γ⁰. Define

fL⁰,v:=

g

Y

i=1

(1 − α⁻¹_i · [v]_L⁰_/K) ×

g

Y

i=1

(1 − α_i⁻¹· [v]⁻¹_L0/K) ⊂ Λ(Γ⁰).

1.3.4. Local factors at split multiplicative places. Suppose A has split multiplicative reduction at v. This means there is a rank g lattice Ωv' Z × · · · × Z sitting inside the torus T = (Kv^×)^g so that T /Ωv is isomorphic to the rigid analytic space associated to A (see [Ger72]). In particular, e:desplit

e:desplit (1) A(Kv) ' (K^×_v)^g/Ωv.

Consider the composition Ωv−→ (K_v^×)^{g R}

g

−→ (Γv v)^gwhere Rv: K_v^×−→ Γvis the local reciprocity map, and extend it Z^p-linearly to

e:mcrv

e:mcrv (2) R_v: Zp⊗_ZΩ_v // (Γv)^g.

d:splitmul Definition 1.3.4. Define wv= χ_Zp(coker[Rv]).

su:maint

1.4. The main theorem. Here is our main theorem. Recall that Ψ = Gal(L/L⁰).

t:compatible Theorem 1. Suppose d ≥ 1 and assume the above notation. Then we have ΘL⁰ · ϑ_L/L⁰ = %_L/L⁰· p_L/L⁰(ΘL),

where ϑ_L/L0:=Q

vϑ_v with each ϑ_v an ideal of Λ(Γ⁰) defined by the following conditions:

(a) Suppose v 6∈ S. If Ψ_v6= 0, then ϑv= π_v; otherwise, ϑ_v = (1).

(b) Suppose v ∈ S and A has good ordinary reduction at v. If v is unramified over L⁰/K, then ϑv= fL⁰,v; otherwise ϑv = (1).

(c) Suppose v ∈ S and A has split-multiplicative reduction at v. Then

ϑv=







Λ(Γ⁰) · wv, if Γ⁰_v= 0;

(σ − 1)^g, if Ψv ' Zpand Γ⁰_vis topologically generated by σ;

(1), otherwise.

The proof will be completed in §5.2. The tools, local and global, for the proof will be established in §2, §3, and §4. See §1.5 for the application of the theorem to the Iwasawa Main Conjecture.

Here is an immediate application.

t:otr09 Theorem 2. Suppose char.(K) = p and L contains the constant Z^p-extension of K. Then XL is torsion over Λ(Γ).

See [MaRu07] for examples of non-torsion X_L in the number field case, while examples in characteristic p can be found in [LLTT13, Appendix].

Proof. Let F^q denote the constant field of K and let L0= KFq^p∞ be the constant Z^p-extension over K with Γ⁰ = Gal(L0/K) topologically generated by the Frobenius substitution Frobq : x 7→ x^q. The theorem is already proved in [OTr09] for the L = L0case. This means ΘL0 6= 0. By repeatedly applying Theorem 1 (d − 1 times), we deduce

p_L/L0(Θ_L) · w_L0/K = Θ_L0·Y

v

ϑ_v.

Since Γ⁰_v 6= 0, for all v, the factor ϑv equals one of (1), (Frob^deg(v)−1)^g, or fL₀,v. In particular, ϑv6= 0, for all v. Therefore, pL/L₀(ΘL) 6= 0, and hence ΘL6= 0.  su:imc

1.5. The Iwasawa main conjecture. Possibly, Theorem 1 could be useful for determining an explicit generator of χ_Λ(Γ)(XL). Assume that an explicitly given element θ⁰_L ∈ Λ(Γ) is already known to be a generator of the characteristic ideal of a submodule X_L⁰ of XL, and we want to see if actually

e:actually

e:actually (3) ΘL = (θ_L⁰).

In addition, assume that there exists an intermediate Z^ep-extension L₀of L/K such that Θ_L0, the characteristic ideal of XL₀ over Λ(Gal(L0/K)), is explicitly given. Then by applying Theorem 1, we can obtain an explicit expression of p_L/L0(ΘL) in terms of ΘL₀ and other factors. Thus, by checking the explicit expressions, we would be able to determine if

e:able

e:able (4) p_L/L0(Θ_L) = (p_L/L0(θ⁰_L)).

The point is that Equations (3) and (4) are indeed equivalent. To see this, we only need to write ΘL= (θ⁰_L· θ⁰⁰), for some θ⁰⁰∈ Λ(Γ)

and observe that θ⁰⁰ is a unit of Λ(Γ) if and only if its image p_L/L0(θ⁰⁰) is a unit of Λ(Gal(L0/K)).

In the function field case, L0 could be taken to be the constant Z^p-extension, since an explicit expression of ΘL₀ is already given in [LLTT13] (for semi-stable A). We can also apply the theorem in the reverse direction: if (3) is already known then we can use the theorem together with (4) to determine an explicit expression of Θ_L0. In [LLTT13] this method is used in the case where char.(K) = p and A is a constant ordinary abelian variety.

su:zero

1.6. The zero set of ΘL and the structure of X_L⁰. Our theory is useful for determining the Λ(Γ)-modules structures of

YL:= (lim

−→F

Selp^∞(A/F )div)^∨, Z_L:= (lim

−→F

(Qp/Zp) ⊗ A(F ))^∨, and

X_L⁰ := ([

F

(Selp^∞(A/L)^{Gal(L/F )})div)^∨

as well. In general, we have the surjections X_L⁰ // // YL // // ZL due to the maps (Q^p/Z^p) ⊗ A(F ) // Selp^∞(A/F )div

resF // (Selp^∞(A/L)^{Gal(L/F )})div .

The above inclusion is from the Kummer exact sequence, it is an isomorphism if the p-primary part of the Tate-Shafarevich group of A over F is finite. Thus, if this holds for all F then ZL= YL. In contrast, by the control theorem (see e.g. [Tan10, Theorem 4]) if L/K only ramifies at good ordinary places then the restriction map resF is surjective for every F , and hence YL= X_L⁰. 1.6.1. The zero set. The structures of X_L⁰, Y_L and Z_L are related to the zero set of Θ_L. For θ ∈ Λ(Γ) define the zero set

∆θ:= {ω ∈ bΓ | pω(θ) = 0},

where pω : OωΛ(Γ) −→ Oω is the Oω-algebra homomorphism extending ω : Γ −→ O^×_ω. Note that for each ω ∈ bΓ, the eigenspace (OωSelp^∞(A/L))^(ω) is cofinitely generated over Oωas it is the Pontryagin dual of the finitely generated Oω-module OXL/ ker [pω] · XL.

d:rs Definition 1.6.1. For each ω ∈ bΓ, denote s(ω) := corank_Oω(OωSelp^∞(A/L))^(ω). We have the inclusions

((OωSelp^∞(A/L))^(ω))div⊂ ((OωSelp^∞(A/L)^ker[ω])div)^(ω)⊂ (Oω

[

F

(Selp^∞(A/L)^{Gal(L/F )})div)^(ω), where the left term is just the p-divisible part of the term. Hence,

e:news

e:news (5) s(ω) = corank_Oω(O_ω[

F

(Sel_p^∞(A/L)^{Gal(L/F )})_div)^(ω).

t:root Theorem 3. A character ω ∈ bΓ is contained in 4Θ_L if and only if s(ω) > 0.

This theorem is proved in §5.3. Let θ ∈ Λ(Γ) be an element vanishing on ∆ΘL. By this, we mean that pω(θ) = 0 for every ω ∈ ∆Θ_L. Since the OωΛ(Γ)-structure of (OωSelp^∞(A/L))^(ω)factors through OωΛ(Γ) ^pω // Oω, we must have θ·(OωSelp^∞(A/L))^(ω) = pω(θ)·(OωSelp^∞(A/L))^(ω) = 0 for every ω ∈ ∆Θ_L. On the other hand, Theorem 3 says if ω 6∈ ∆Θ_L then (OωSelp^∞(A/L))^(ω) is finite. Thus, θ · (O_ωSel_p^∞(A/L))^(ω) is always finite for all ω ∈ bΓ.

For each finite intermediate extension F of L/K, denote Γ(F ) := Gal(F/K) and choose O so that it contains Oω for every ω ∈ bΓ(F ) := Hom(Γ(F ), µp^∞) regard as a finite subgroup of bΓ.

Consider the elements e_ω:=P

γ∈Γ(F )ω(g)⁻¹· g ∈ O[Γ(F )], ω ∈ bΓ(F ), which are |Γ(F )|-multiples of idempotents. Multiplying any finite O[Γ(F )]-module W by e_ω’s, we can form a homomorphism

M

ω∈bΓ(F )

W^(ω)−→ W

of finite kernel and cokernel. In particular, by taking W = ((O Selp^∞(A/L))^{Gal(L/F )})^∨and by the duality, we have a homomorphism

e:amorphism

e:amorphism (6) O Selp^∞(A/L)^{Gal(L/F )}−→ M

ω∈bΓ(F )

(O Sel_p^∞(A/L))^(ω)

of finite kernel and cokernel. Then by multiplying both sides of (6) by θ we see that e:thetazero

e:thetazero (7) θ · (Selp^∞(A/L)^{Gal(L/F )})div = 0

as the left-hand side of the equality is finite and p-divisible. By [Grn03, Proposition 3.3] (see [Tan10, Corollary 3.2.4] and the discussion in §3.3 of the paper for the characteristic p case), if

res_L/F : H¹(F, A[p^∞]) −→ H¹(L, A[p^∞])^{Gal(L/F )} denote the restriction map, then

e:kerreslf

e:kerreslf (8) | ker resL/F | < ∞,

and e:cokerreslf

e:cokerreslf (9) | coker resL/F | < ∞.

Then (7) and (8) imply θ · Selp^∞(A/F )div = 0 as it is also finite and p-divisible. We have proved:

c:vanishingtheta Corollary 1.6.2. If θ ∈ Λ(Γ) vanishes on ∆Θ_L, then θ annihilates (Selp^∞(A/L)^{Gal(L/F )})div, Selp^∞(A/F )div and (Q^p/Z^p) ⊗ A(F ), for all F . Hence θ · X_L⁰ = θ · YL= θ · ZL= 0.

1.6.2. A theorem of Monsky. Now we recall a theorem of Monsky ([Mon81, Lemma 1.5 and The- orem 2.6]). A subset T ⊂ bΓ is called a Z^p-flat of codimension k > 0, if there exist γ1, ..., γk ∈ Γ expandable to a Z^p-basis of Γ and ζ1, ..., ζk∈ µp^∞ so that

T = Tγ₁,...,γ_k;ζ₁,...,ζ_k := {ω ∈ bΓ | ω(γi) = ζi, i = 1, ..., k}.

Theorem 4. (Monsky) If θ ∈ Λ(Γ) is non-zero, then ∆_θ6= bΓ and is a finite union of Zp-flats.

Note that for a given θ ∈ Λ(Γ), if T ⊂ ∆θ then ^σT ⊂ ∆θ for all σ ∈ Gal(Qp/Q^p), as ∆θ is invariant under the action of the Galois group. Also, if Tγ,ζ ⊂ ∆θ with ζ ∈ O then γ − ζ divides θ in OΛ(Γ), and vice versa (see [LLTT13, Lemma 3.3.3] and its proof). In this case, γ −^σζ also divides θ.

d:simple Definition 1.6.3. An element f ∈ Λ(Γ) is simple, if there exist γ ∈ Γ − Γ^p and ζ ∈ µp^∞ so that f = fγ,ζ := Y

σ∈Gal(Qp(ζ)/Qp)

(γ −^σζ).

If ζ is of order pⁿ⁺¹ and ti = γi − 1, i = 1, ..., d, where γ1, ..., γd is a Z^p-basis of Γ, then fγ₁,ζ is nothing but the polynomialPp−1

i=0(t1+ 1)^ipn that is irreducible in Z^p[t1]. Hence, a simple element is irreducible in Λ(Γ) = Z^p[[t1, ..., td]]. Obviously, ∆f_γ,ζ = S

σ

σTγ,ζ. In particular, two simple elements f and g divide each other if and only if ∆f = ∆g. On the other hand, if T = T_{γ1,...,γk;ζ1,...,ζk}, k ≥ 2, then we can find two relatively prime simple elements both vanishing on T , for example, f_γ1,ζ1 and f_γ2,ζ2.

1.6.3. The structure of X_L⁰. If W is a torsion Λ(Γ)-module then θ := χΛ(Γ)(W ) 6= 0 and there exists an pseudo-isomorphism

e:Msim

e:Msim (10) ι : (Λ(Γ)/(f₁^b1))^a1⊕ · · · ⊕ (Λ(Γ)/(f_l^bl))^al⊕

m

M

j=1

Λ(Γ)/(ξ_i) −→ W.

where a1, ..., al, b1, ..., blare positive integers, f1, ..., flare all the relatively prime simple factors of θ, and ξ1, ..., ξm ∈ Λ(Γ) are not divided by any simple element (l = 0 or m = 0 is allowed). The product φ = f1· · · flvanishes on every codimension one Z^p-flat of ∆θ. By the above argument, we can find two products ε = g1· · · gm and ε⁰ = g⁰₁· · · g_m⁰ 0, relatively prime to φ and to each other, of simple elements so that both ε and ε⁰vanish on every Z^p-flat of ∆θof codimension grater than 1. Then both φε and φε⁰ vanish on ∆θ. Note that ι is actually injective as its domain of definition contains no non-trivial pseudo-null submodule (see §2.1).

If X_L is torsion then by taking W = X_Lin (10) we obtain the exact sequence e:xlsim

e:xlsim (11) 0 // L^l_i=1(Λ(Γ)/(f_i^bi))^ai⊕Lm

j=1Λ(Γ)/(ξi) // XL // N // 0 , for some pseudo-null N . Let φε and φε⁰ be as above. Let ∼ denote pseudo-isomorphism.

t:x0 Theorem 5. Suppose XL is torsion over Λ(Γ) and assume the above notation. Then both φε and φε⁰ annihilate X_L⁰, YL, ZL and (Selp^∞(A/L)^{Gal(L/F )})div, Selp^∞(A/F )div, (Q^p/Z^p) ⊗ A(F ) for all finite intermediate extension F of L/K. Moreover, X_L⁰ is pseudo isomorphic to X_L/φ · X_L. Namely, if a₁, ..., a_l are as in (11), then

X_L⁰ ∼ (Λ(Γ)/(f1))^a1⊕ · · · ⊕ (Λ(Γ)/(fl))^al.

Proof. The first assertion follows from Corollary 1.6.2. Consequently, φ · X_L⁰ is pseudo-null, being annihilated by relatively prime ε and ε⁰. Thus, X_L⁰ ∼ X_L⁰/φ · X_L⁰. By taking W = X_L⁰ in (10) we obtain the exact sequence

0 // L^l

i=1(Λ(Γ)/(f_i))^{ci ι} // X_L⁰ // M // 0 ,

for some non-negative integers c1, ..., cland some pseudo-null M . By comparing this exact sequence with (11) using the fact that Λ(Γ)/(φ, ξi) is pseudo-null, we see that XL/φXL∼ X_L⁰ if and only if ci = ai for each i. We shall only show c1= a1, as the rest can be proved in a similar way.

First choose a ξ ∈ Λ(Γ) that annihilates M and is relatively prime to φ. Let Eωdenote the field OωQ^p and via Λ(Γ) ^pω // Oω ⊂ Eω we consider the map ιω:= Eω⊗_Λ(Γ)ι for an

ω ∈ ∆f₁− (∆f₂∪ · · · ∪ ∆f_l∪ ∆ξ∪ ∆ξ₁∪ · · · ∪ ∆ξ_m).

Now the E_ω-vector space E_ω⊗_Λ(Γ)M = 0 as it is annihilated by p_ω(ξ) 6= 0. Similarly, as p_ω(f_i) 6= 0 for i ≥ 2, E_ω⊗_Λ(Γ)Λ(Γ)/(f_i) = 0 . On the other hand, as p_ω(f₁) = 0, E_ω⊗_Λ(Γ)Λ(Γ)/(f₁) = E_ω. Also, ker[ι_ω] = 0 as it is annihilated by p_ω(ξ). Therefore, ι_ω is an isomorphism between E_ω^c1 and Eω⊗_Λ(Γ)X_L⁰. Hence

rank_OωOω⊗Λ(Γ)X_L⁰ = dim_EωE_ω⊗Λ(Γ)X_L⁰ = c₁. Then we deduce s(ω⁻¹) = c1by using (5) together with the fact that

(Oω

[

F

(Selp^∞(A/L)^{Gal(L/F )})div)^(ω−1)= (OωX_L⁰/ ker[pω] X_L⁰)^∨' (Oω⊗Λ(Γ)X_L⁰)^∨. Similarly, by tensoring the exact sequence (11) with E_ω, we get a₁= s(ω⁻¹), whence a₁= c₁. 

By Theorem 5 there are non-negative integers a⁰₁, ..., a⁰_l, a⁰⁰₁, ..., a⁰⁰_l with a⁰⁰_i ≤ a⁰_i ≤ ai, so that Y_L∼ (Λ(Γ)/(f1))^a01⊕ · · · ⊕ (Λ(Γ)/(fl))^a0l and Z_L ∼ (Λ(Γ)/(f1))^a001 ⊕ · · · ⊕ (Λ(Γ)/(fl))^a00l.

su:algfun

1.7. Algebraic functional equations. Let ^] : Λ(Γ) −→ Λ(Γ), x 7→ x^], denote the Z^p-algebra isomorphism induced by the involution γ 7→ γ⁻¹, γ ∈ Γ. For each Λ(Γ)-module W , let W^] denote the Λ(Γ)-module with the same underlying abelian group as W , while Λ(Γ) acting via the isomorphism ^]. For a simple element f we have (Λ(Γ)/(f ))^] = Λ(Γ)/(f ). Thus, if X_L is torsion, the we have the functional equations X_L^0] ∼ X_L⁰, YL] ∼ YL and Z_L^] ∼ ZL as well. Taking the projective limit over F of the dual of

0 // Selp^∞(A/F )div // Selp^∞(A/F ) //X(A/F )^p/X(A/F )^div // 0, where X(A/F ) denote the Shafarevich-Tate group, we obtain the exact sequence

0 // aL // XL // YL // 0, where

aL := lim

←−F

(X(A/F )^p/X(A/F )^div)^∨.

Then by using the Cassels-Tate pairing on each X(A/F )^p× X(A^t/F )p, one can actually prove the pseudo-isomorphisms a^]_L ∼ aL and X_L^] ∼ XL. The proof is given in [LLTT13], in which the content of Theorem 5 actually plays a key role.

1.7.1. aLis torsion. The following is proved in [LLTT13] by using Theorem 3, while if every v ∈ S is a good ordinary place, then it can be proved by the control theorem.

t:a1 Theorem 6. The module a_L is finitely generated and torsion over Λ(Γ).

su:when

1.8. When is XM torsion? For convenience, call an intermediate extension M of L/K simple, if Gal(M/K) ' Z^cp, for some c. For such M , by repeatedly applying Theorem 1, we deduce e:d-c

e:d-c (12) p_L/M(ΘL) · % = ΘM· ϑ,

where % 6= 0 and ϑ is a product of local factors obtained from those ϑ_v’s in Theorem 1. It is easy to see that ϑ 6= 0 unless M is fixed by the decomposition subgroup Γ_v of some split-multiplicative place v ∈ S. Thus, the following theorem is proved by taking

T= {L^Γv | v ∈ S is a split-multiplicative place}.

t:nontor Theorem 7. Suppose X_L is non-torsion. If L/K only ramifies at good ordinary places, then X_M is non-torsion, for every simple intermediate extension M . In general, there is a finite set T consisting of proper simple intermediate extensions of L/K, such that XM is non-torsion unless M ⊂ Mj for some Mj∈ T.

By (12), if ΘM = 0, then p_L/M(ΘL) = 0. Put TM := {ω ∈ bΓ | ω(γ) = 1, for all γ ∈ Gal(L/M )}, the Z^p-flat of codimension d − c determined by M . Then p_L/M(ΘL) = 0 if and only if TM ⊂ ∆Θ_L, or equivalently, TM ⊂ Tj for some j if ∆Θ_L = Sν

j=1Tj. Let Mj be the maximal simple intermediate extension of L/K, so that T_Mj ⊂ T_j. Then the following theorem is proved by setting

T= {M_j | j = 1, ..., ν}.

Theorem 8. Suppose XL is torsion. Then there is a finite set T consisting of proper simple intermediate extensions of L/K, such that for each simple intermediate extension M outside T, XM is torsion.

Hence, if d = 2 and X_L is non-torsion (resp. torsion), then X_M is non-torsion (resp. torsion) for almost all intermediate Zp-extensions M .

su:mw

1.9. The growth of s_n. Let K_n denote the nth layer of L/K and write I_n for the kernel of Λ(Γ) // //Zp[Γ(K_n)] . Then Selp^∞(A/L)^Gal(L/Kn) is the Pontryagin dual of X_L/I_nX_L, whence cofinitely generated over Z^p. Let sn denote its corank. Theorem 9 below gives an asymptotic formula of sn. The following lemma as well as its proof is by I. Longhi. Denote En = Q^p(µpⁿ), En= p^n(d−2), for d ≥ 2; En= 1, for d = 1. Let J be an ideal of Λ(Γ).

Lemma 1.9.1. (Longhi) If J = (f_γ,ζ^m) for some positive integer m and δζ := [Q^p(ζ) : Q^p] then l:longhi

e:assytoticf

e:assytoticf (13) rank_ZpΛ/(I_n+ J ) = δ_ζ · p^n(d−1) for n  0 .

If J = (f ), f not divided by any simple element, or (f, g) ⊂ J , for some relatively prime f , g, then e:assytoticfg

e:assytoticfg (14) rank_ZpΛ/(In+ J ) = O(En).

Proof. Write Γn= Γ(Kn) and Vn := En⊗_Zp(Λ/In). Then

rank_ZpΛ/(In+ J ) = dimE_nEn⊗_Zp Λ/(In+ J )
= dimE_n(Vn/J Vn) . One has a decomposition of E_n-vector spaces

Vn= ⊕_ω∈bΓ

nV_n^(ω).

Moreover dimE_nVn^(ω)= 1 because Vn ' En[Γn] is the regular representation. Obviously J V_n^(ω)=

(0 if pω(J ) = 0 Vn^(ω) if pω(J ) 6= 0 . Denote ∆J= {ω ∈ bΓ | pω(J ) = 0}. Then

dim_En(V_n/J V_n) = |{ω ∈ bΓ_n: p_ω(J ) = 0}| = |∆_J∩ bΓ_n| = |∆J[pⁿ]|

(the last equality comes from bΓn = bΓ[pⁿ]). Here G[pⁿ] denotes the pⁿ torsion subgroup of G.

Monsky’s theorem yields ∆J=S Tj, where the Tj’s are Z^p-flats. Besides, by [Mon81, Lemma 1.6],

|Tj[pⁿ]| = p^n(d−kj) for n  0,

where k_j denotes the codimension of T_j. If J = (f_γ,ζ^m), then every T_i is of codimension 1. Hence dimE_n(Vn/(f_γ,ζ^m)Vn) = |∆_(f^m

γ,ζ)[pⁿ]| = δζ· p^n(d−1) for n  0 .

To show the second assertion we observe that every T_i should be of codimension greater than 1, because if some Tj = Tγ;ζ, then ∆f_γ,ζ⊂ ∆J, whence fγ,ζ divides all elements of J . Thus,

dimE_n(Vn/J Vn) = |∆J[pⁿ]| = O(En).

 t:mw Theorem 9. There exists a non-negative integer κ₁ such that

(15) sn= κ1p^nd+ O(p^n(d−1)).

X_L is torsion if and only if κ₁= 0, in this case there exists a non-negative integer κ₂ such that e:kappa2

e:kappa2 (16) sn= κ2p^n(d−1)+ O(En).

X_L⁰ is pseudo-null if and only if s_n= O(E_n).

Proof. Suppose W is a finitely generated Λ(Γ)-module. Then (10) gives rise to the exact sequence 0 // L^l_i=1(Λ(Γ)/(f_i^bi))^a1⊕Lm

j=1Λ(Γ)/(ξ^ι _i) // W // M // 0

for some pseudo-null M . Then we tensor the exact sequence with En⊗_ZpΛ(Γ)/In. Since M is annihilated by some relatively prime f and g, both M/I_nM and Tor_Λ(Γ)(Λ(Γ)/I_n, M ) are quotients of some direct sums of finite copies of V_n/(f, g)V_n. Hence, formulae (13) and (14) imply

e:assytoticw

e:assytoticw (17) rank_ZpW/InW = dimE_nEn⊗_ZpW/InW =X

i

δip^n(d−1)+ O(En),

where δi = δζ_i if fi = fγ_i,ζ_i. If XL is torsion, then we take W = XL to prove (16). In this case, X_L⁰ is pseudo-null if and only if l = 0 (see Theorem 5) which meansP

iδi= 0.

Suppose X_L is non-torsion. Then ∆_ΘL = bΓ, whence by Theorem 3, s(ω) > 0 for all ω. Since (6) is of finite kernel and cokernel, we have

rank_ZpX_L⁰/InX_L⁰ = corank_Zp(Selp^∞(A/L)^Gal(L/Kn))div ≥ p^nd.

By (17), X_L⁰ is non-torsion, and hence not pseudo-null. Let x1, ..., xκ₁ ∈ XL form a basis of the vector space generated by XLover the field of fractions of Λ(Γ). Then we have an exact sequence

0 // P_iΛ(Γ) · x_i // XL // W // 0,

where W is a torsion Λ(Γ)-module. Then we tensor the exact sequence with E_n⊗_ZpΛ(Γ)/I_n.  Remark 1.9.2. By a similar argument, one can prove that: (1) There exists a finite number of Zp-flats T1, ..., Tl so that s(ω) = κ1 for each ω 6∈S

iTi. (2) If XL is torsion, then for each Z^p-flat T ⊂ ∆Θ_L, there is a finite number of proper Zp-flats T₁⁰, ..., T_ν⁰ ⊂ T so that s(ω) is a constant for each ω ∈ T −S

iT_i⁰. (3) There is a bound of s(ω) for all ω ∈ bΓ.

Also, Theorem 9 generalizes [MaRu03, Proposition 1.1], as we have sn = rank A(Kn) if L/K only ramifies at good ordinary places and X(A/Kⁿ)p is finite.

2. preliminary s:pre

In this section, we assume that Γ ' Z^dp, with d ≥ 0, except in Lemma 2.1.1 and Lemma 2.3.1, where we assume d ≥ 1. If d = 0, we set Λ(Γ) = Z^p; otherwise, Gal(L/L⁰) = Ψ ' Z^p is topologically generated by ψ. Let F denote a finite intermediate extension of L/K.

su:char

2.1. The characteristic ideal. Let W be a finitely generated Λ(Γ)-module. Recall that W is pseudo-null if and only if there are relatively prime elements f1, ..., fn, n ≥ 2, in Λ(Γ) so that fi· W = 0 for every i. If W is non-torsion over Λ(Γ), then

χ_Λ(Γ)(W ) = 0.

If W is torsion, then there exist irreducible ξ₁, ..., ξ_m∈ Λ(Γ), and a pseudo-isomorphism φ :Lm

i=1Λ(Γ)/ξ_i^riΛ(Γ) // W, [Bou72, §4.4, Theorem 5]. In this case, the characteristic ideal is

χΛ(Γ)(W ) =

m

Y

k=1

(ξi)^ri 6= 0.

Denote

[W ] =

m

M

i=1

Λ(Γ)/ξ^r_iⁱΛ(Γ).

Since each non-zero element in [W ] cannot be simultaneously annihilated by relatively primes ele- ments in Λ(Γ), there is no non-trivial pseudo-null submodule of [W ], and hence φ is an embedding.

We shall fix a exact sequence (with N pseudo-null):

e:pseu

e:pseu (18) 0 // [W] // W // N // 0.

l:coco Lemma 2.1.1. Suppose M is a Λ(Γ)-module and Γ0 is a closed subgroup of Γ such that the composition Γ0  // Γ // // Γ⁰ is an isomorphism. If M considered as a Λ(Γ₀)-module is finitely generated and torsion, then

χ_Λ(Γ⁰₎(M^Ψ) = χ_Λ(Γ⁰₎(M/(ψ − 1)M ).

Proof. It follows from the exact sequence of Λ(Γ0)-modules:

0 //M^Ψ // M ^ψ−1 // M // M/(ψ − 1)M // 0.

su:hs 

2.2. The Hochschild-Serre spectral sequence. In this subsection, let K be any field. Let S be a sheaf of abelian groups on the flat topology of K and let F /K be a finite Galois extension with G = Gal(F /K). Then there is the Hochschild-Serre spectral sequence

e:spect

e:spect (19) E₂^p,q = H^p(G, H^q(F , S)) =⇒ H^p+q(K, S), and in particular (see [Mil80, p.105]), the exact sequence

e:infres

e:infres (20) 0 // H¹(G, S(F ))

inf_{F /K}¹

// H¹(K, S)

res¹_{F /K}

// H¹(F , S)^G

d^0,1_F

ssgggggggggggggggggggggggggg H²(G, S(F ))

inf_{F /K}² // kerhres²_{F /K}i

// H¹(G, H¹(F , S))

d^1,1_F // H³(G, S(F )), where for each i, inf_{F /K}ⁱ : Hⁱ(G, S(F )) −→ Hⁱ(K, S) and resⁱ_{F /K}: Hⁱ(K, S) −→ Hⁱ(F , S) denote the inflation and the restriction maps.

l:spc Lemma 2.2.1. Suppose K ⊂ E ⊂ F and E /K is Galois with Gal(F /E ) = H. Then we have commutative diagrams:

e:spcoh1

e:spcoh1 (21) H¹(K, S)

res¹_E/K

// H¹(E , S)^{G/H d}

0,1 E //

res^0,1_{F /E}



H²(G/H, S(E ))

inf_{F /E}²



H¹(K, S)

res¹_{F /K}

// H¹(F , S)^G

d^0,1_F

// H²(G, S(F )), and

e:spcoh3

e:spcoh3 (22) kerh

res²_E/Ki

//



H¹(G/H, H¹(E , S))

d^1,1_E

//

res^1,1_{F /E}



H³(G/H, S(E ))

inf_{F /E}³



kerh

res²_{F /K}i

// H¹(G, H¹(F , S)) ^d

1,1

F //// H³(G, S(F )), where

res^i,j_{F /E} : Hⁱ(G/H, H^j(E , S)) −→ Hⁱ(G, H^j(F , S))

is induced by the restriction map H^j(E , S) −→ H^j(F , S) that respects the actions of G/H (on the left-hand side ) and G (on the right-hand side ).

Proof. Recall that if the complex C is an injective resolution of S (in the category of sheaves on the flat site of Spec K) and the bi-complex I is a fully injective (Cartan-Eilenberg) resolution of C(F ) (in the category of G-modules), then the spectral sequence (19) is obtained from the bi-complex I^G (with (I^G)^pq = (I^pq)^G, the part of I^pq fixed by G). Let the bi-complex J be a fully injective resolution of C(E ) (in the category of G/H-modules) that gives rise to the spectral sequence

E₂^p,q= H^p(G/H, H^q(E , S)) =⇒ H^p+q(K, S).

Since an injective G/H-module is also injective as G-module, we have a G-morphism J −→ I. The commutative diagrams are obtained from the induced morphism J^G/H −→ I^G.  We are mostly interested in the case where G is cyclic. Denote g = |G|, h = |H|. We fix a generator of G and choose its H-coset as a generator of G/H. Then we have the commutative diagram (see [Ser79, VIII.4]):

H²(G/H, Z)

inf²



Hom(G/H, Q/Z) Z/_h^gZ



H²(G, Z) Hom(G, Q/Z) Z/gZ,

where the right down-arrow is induced by the homomorphism Z −→ Z, 1 7→ h. Let δG∈ H²(G, Z) be the class corresponding to 1 (mod gZ) in the above diagram. Then we have the induced commutative diagram:

e:inf2

e:inf2 (23) S(K)/ N_G/H(S(E ))



Hb⁰(G/H, S(E )) // H²(G/H, S(E ))

inf_{F /E}²



S(K)/ NG(S(F )) Hb⁰(G, S(F )) // H²(G, S(F )),

where the upper and lower right-arrows are, respectively, cup-product with δG/H and δG, and the left down-arrow is induced by the multiplication by h on S(K). Similarly, we have the commutative diagram

e:inf3

e:inf3 (24) H¹(G/H, S(E )) //



H³(G/H, S(E ))

inf_{F /E}³



H¹(G, S(F )) // H³(G, S(F )), where the left down-arrow is the h multiple of inf_{F /E}¹ .

su:torsion

2.3. Cohomology groups of A[p^∞]. In this section let D be a discrete p-primary abelian group cofinitely generated over Z^p.

2.3.1. Assume that Γ acts continuously on D.

l:cofito Lemma 2.3.1. Suppose D^Γ is finite. Write C1 = H¹(Ψ, D), C2 = D^Ψ. Then C₁^∨ is finitely generated over Z^p and

χ_Λ(Γ⁰₎(C1∨) =

(χΛ(Γ⁰)(C2∨), if d ≥ 2;

χ_Λ(Γ0)(C2/C2∩ Ddiv), if d = 1.

Proof. Denote Ψ⁽ⁿ⁾ = Ψ^pn and Ψn = Ψ/Ψ⁽ⁿ⁾. Every x ∈ D is fixed by Ψ⁽ⁿ⁾, for some n. If x is of order p^m, then, for l ≥ n + m, the norm NΨ_l(x) = 0, and hence x determines a class in H¹(Ψ_l, D^Ψ(l)). Therefore, C₁= D/(ψ − 1)D and consequently, C₁^∨ = (D^∨)^Ψ. On the other hand, C₂^∨= D^∨/(ψ − 1)D^∨. If d ≥ 2, then the lemma follows from Lemma 2.1.1, as D^∨, being finitely generated over Zp, is torsion over Λ(Γ₀) for any choice of Γ₀. If d = 1, then Ψ = Γ and D^Ψ is finite. It follows that D_div −→ D_div, x 7→ (ψ − 1)x, is surjective. Thus, the snake lemma for the multiplication of (ψ − 1) on the exact sequence

0 −→ Ddiv −→ D −→ ¯D −→ 0 gives rise to the exact sequence

0 −→ D^Ψ_div −→ C2−→ ¯D^Ψ−→ 0 as well as the isomorphism

C1' ¯D/(ψ − 1) ¯D.

Since ¯D is torsion over Λ(Γ0) = Z^p, for Γ0= 0 = Γ⁰, Lemma 2.1.1 and the above exact sequences imply χ_Zp(C₁) = χ_Zp( ¯D^Ψ) = χ_Zp(C₂/C₂∩ D_div). Now, C₁^∨' C₁, as C₁is finite.  c:chicoh1 Corollary 2.3.2. If d ≥ 3, then H¹(Ψ, A[p^∞](L))^∨ is pseudo-null over Λ(Γ⁰). In general,

χ_Λ(Γ0)(H¹(Ψ, A[p^∞](L))^∨) =

(χ_Λ(Γ0)(A[p^∞](L⁰)^∨), if d ≥ 2;

(_|A[p∞](K)∩A[p^{|A[p∞](K)|∞}](L)_div|), if d = 1.

Proof. The second assertion is immediately from Lemma 2.3.1. Since A[p^∞](L⁰)^∨is finitely gener- ated over Zp, it is pseudo-null over Λ(Γ⁰), if d ≥ 3, whence the first assertion follows.  c:cofito Corollary 2.3.3. Suppose Γ ' Z^p and D is finite. Then

| H¹(Γ, D)| = |D^Γ|.

2.3.2. Next, consider the case where a topological group C ' Zp acts continuously on D. We are going to discuss some associated modules over Λ(C) := Zp[[C]] as well as their characteristic ideals.

Denote C^(m) = C^pm and Dm = D^C(m) for m = 0, 1, ..., ∞. Also, denote G_m^m0 = C^(m)/C^(m0) for m⁰= m, ...., ∞. Define M := lim←−mDm, with the limit taken over norm maps N_Gm0

m : Dm⁰ −→ Dm. Define N := D^∨, the Pontryagin dual of D, and T := lim←−nD[pⁿ], the Tate-module of D.

Let c be a topological generator of C. Let ₁, ..., _m, counted with multiplicities, be the eigen- values of the action of c on T. Note that ₁· · · _m= det(c) is a p-adic unit, and hence so is each

_i. Also, _j= 1 for some j if and only if D₀= D^C is infinite.

p:torcomp Lemma 2.3.4. We have χ_Zp((D0∩ Ddiv)^∨) =Qm

j=1(1 − j) =Qm

j=1(1 − ⁻¹_j ).

Proof. Let N⁰ denote the Z^p free part of N. Then (D0∩ Ddiv)^∨ = N⁰/(c − 1)N⁰ whose p-adic valuation is the same as that of the determinant (of 1 − c acting on Q^p⊗_ZpN⁰)

det(1 − c) =

m

Y

j=1

(1 − ⁻¹_j ) = ±

m

Y

j=1

⁻¹_j (1 − j).

 p:torcomp1 Proposition 2.3.5. Suppose Dmis finite for each m. Then M ∼ T as Λ(C)-modules, and

χ_Λ(C)(M) = χ_Λ(C)(T) = (

m

Y

j=1

(1 − ⁻¹_j c)),