Selmer groups over $\Z_p^d$-extensions

0.46 MB·English

by  Ki-Seng Tan

SELMER GROUPS OVER Zd-EXTENSIONS
p
KI-SENGTAN
3
1
0 Abstract. Consider an abelian variety A defined over a global field K and let L/K be a Zd-
p
2 extension,unramifiedoutsideafinitesetofplacesofK,withGal(L/K)=Γ. LetΛ(Γ):=Zp[[Γ]]

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

1
1

]
T 1. Main Results
s:intro
N
Let A be a g-dimensional abelian variety defined over a global field K and let L/K be a Zd-
p
.
h extension, unramified outside a finite set of places of K, with Gal(L/K) = Γ. For each finite
at intermediate extension F/K of L/K, let Selp∞(A/F) denote the p-primary Selmer group (see
m §2.5) and set
[ Selp∞(A/L)=l−i→mSelp∞(A/F).
F
2
v We endow Selp∞(A/L) (resp. Selp∞(A/F)) with the discrete topology and let XL (resp. XF)
denote its Pontryagin dual group. The main aim of this paper is to study how the characteris-
7
0 tic ideal of XL over Λ(Γ) := Zp[[Γ]] (the Iwasawa algebra) varies, when L/K is replaced by an
9 intermediate Ze-extension L′/K. Our result has many applications. In particular, it leads to a
p
3 structure theorem of Z , the Pontryagindual of lim Q /Z ⊗A(F) (see §1.6).
. L −→F p p
5su:ns
0 1.1. Notation. LetS denotethesetofplacesofK ramifiedoverL/K. Foranalgebraicextension
2 F/K and a place w of F, let F denote the w-completion of F. If w is a non-archimedean place,
w
1 letO , m andF (orO , m andF )denotetheringofintegers,themaximalidealandthe
: w w w Fw Fw Fw
v residue field of F . Also, denote q = |F |. We fix an algebraic closure K of K and let Ks ⊂ K
w w w
i denote the separable closure of K, and the same for K .
X v
ForanabeliangroupD,letD (resp. D )denotethep-primary(resp. p-divisible)partofD ,
r p div tor
a the torsion subgroup. For a locally compact group G, let G∨ denote its Pontryagin dual group.
In this paper, we always have G∨ =Hom (G,Q /Z ) as G will be either pro-p or p-primary. If
cont p p
O is the ring of integers of a finite extension Q of Q and G is an O-module, we endow G∨ with
p
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 (G):=rank (G∨).
O O
If G is a Z -module, write OG for O ⊗ G. Then we can identify (OG)∨ with OG∨ by
p Zp
introducing a non-degenerate pairing [ , ] : OG× OG∨ −→ Q /Z as follow. First choose a
p p
generator δ ∈ O of the different of the filed extension Q/Q and set Tr∗(x) =Tr (δ−1·x) for
p Q/Qp
x ∈ Q. If a ∈ Q/O is the residue class of some y ∈ Q modulo O, let T∗(a) ∈ Q /Z denote the
p p
residue class of Tr∗(y) modulo Z . Then Q:O×Q/O−→Q /Z given by Q(x,a):=T∗(xa) is a
p p p
non-degeneratepairing. Let<, >:OG×OG∨ −→Q/ObetheO-pairinggivenby<g,φ>=φ(g),
Acknowledgement: Thisresearchwas supportedinpartbytheNational Science Councilof Taiwan,NSC97-
2115-M-002-006-MY2,NSC99-2115-M-002-002-MY3.
1
2 KI-SENGTAN
for g ∈ G, φ ∈ G∨. Then define [α,β] = T∗(< α,β >). Let e ,...,e be a Z -basis of O. If
1 m p
β = e ⊗φ , φ ∈G∨, satisfies [OG,β]=0, then for all x∈O, g ∈G,
i i i i
P 0=[x⊗g,β]=T∗(x· ei⊗<g,φi >)=Q(x, ei⊗<g,φi >).
i i
X X
Since Q is non-degenerate, e ⊗ <g,φ >=0, and hence <g,φ >=0 for all g. Consequently,
i i i i
φ =0 for every i, whence β =0. Similarly, if α∈OG satisfies [α,OG∨]=0, then α=0.
i
If G is a Γ-module, let ΓPacts 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)∨.
Letµpm denotethepmthrootofunityandwriteµp∞ = mµpm regardedasadiscretesubgroup
×
of Qp. Let Γ denote the group of all continuous characteSrs from Γ to µp∞ and let Gal(Qp/Qp)
act on it via the action on µp∞. Thus, Γ=Γ∨ as topologicalgroups, while Gal(Qp/Qp) acts non-
triviallyonΓbbuttriviallyonΓ∨. Ifω ∈ΓwiththeimageIm(ω)=µ ,writeO =Z [µ ]⊂Q .
pm ω p pm p
IfO containsO andGisanO-modulebwithacontinuousactionofΓ,write(fortheω-eigenspace)
ω
b G(ω) :=b{g ∈G | γg =ω(γ)·g}.
For a finitely generatedΛ(Γ)-module W, let χ (W) denote its characteristicideal (see §2.1).
Λ(Γ)
Denote Γ′ = Gal(L′/K), Λ(Γ′) = Z [[Γ′]]. Our result also covers the d = 1 case in which Γ′ = 0,
p
Λ(Γ′)=Zp, and we define χΛ(Γ′)(W)=χZp(W), the usual characteristic ideal of Zp-module.
Let A[pm] denote the kernel of the multiplication by pm on A viewed as a sheaf on the flat
topology of K and denote A[p∞]= A[pm]. In particular, A[p∞](K)= A(K) . Let At denote
m p
the dual abelian variety.
su:localcondition S
1.2. A local condition. It is well knownthat if K is a number field, then the following question
has an affirmative answer (see below).
IsX finitelygeneratedover Λ(Γ)?
L
In general, the answer could be obtained via the following local criterion.
p:iwasawa Proposition 1.2.1. The Iwasawa module X is finitely generated over Λ(Γ) if and only if at each
L
place v ∈S, the local cohomology group H1(Γ ,A(L )) is cofinitely generated over Z .
v v p
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 abelianvariety without non-trivial p-torsionpoints, 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 X is finitely generated
L
over Λ(Γ). Also, for simplicity, we assume that char.(K) = p, if K is not a number field. Write
Θ =χ (X ). ExtendthecanonicalmapΓ−→Γ′tothecontinuousZ -algebrahomomorphism
L Λ(Γ) L p
(the specialization map) pL/L′ :Λ(Γ)−→Λ(Γ′). Then the following question arises:
WhatistherelationbetweenpL/L′(ΘL)andΘL′?
To illustrate our answer, some simplification and notation are in order. First, by choosing a
sequence L′ ⊂ L′′ ⊂ ··· ⊂ L(i) ⊂ ··· ⊂ L(d−e) ⊂ L with each Gal(L(i)/K) ≃ Ze−1+i, we can
p
write pL/L′ = pL′′/L′ ◦···◦pL(i+1)/L(i) ◦···◦pL/L(d−e), and hence answer the question for pL/L′
by answeringthat for everyp . Therefore, without loss of generality, we may assume that
L(i+1)/L(i)
e=d−1. WeshallmakesuchassumptionandthenfixatopologicalgeneratorψofΨ:=Gal(L/L′).
SELMER GROUPS OVER Zd-EXTENSIONS 3
p
1.3.1. The global factor. Let K′/K be a Z -extension and let σ be a topological generator of the
p
Galoisgroup. Ifǫ ,...,ǫ areeigenvalues,countedwithmultiplicities, oftheactionofσ onthe Tate
1 l
module T A[p∞](K′). Then the product l (1−ǫ−1σ) ⊂ Z [[Gal(K′/K)]] is nothing but the
p j=1 j p
characteristic ideal of T A[p∞](K′) over Z [[Gal(K′/K)]], and in particular, the ideal
p p
Q
l
wK′/K := (1−ǫ−j1σ)(1−ǫ−j1σ−1)
j=1
Y
is independent of the choice of σ (See Proposition 2.3.5). If K′/K is a Ze-extension with e ≥ 2,
p
set wK′/K =(1).
d:glfactor Definition 1.3.1. Define the Λ(Γ′)-ideal
wL′/K ifd≥2;
̺L/L′ =(|A[p∞](|KA)[pT∞A]([Kp∞)|]2(L)div|2 ifd=1.
1.3.2. Local factors at unramified places.
d:badunram Definition 1.3.2. For each v, let Π denote the group of the connected components of the closed
v
fiber (over F ) of the N´eron model of A/K and let π denote the Z -ideal (|ΠGal(Fv/Fv)|).
v v v p v
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 F : A¯ −→ A¯ over F are, counted with
v v
multiplicities,
α ,...,α ,q /α ,...,q /α ,
1 g v 1 v g
where α ,...,α are eigenvalues of the (twist) matrix u of the action on the Tate module of A¯[p∞]
1 g
by the Frobenius substitution Frob ∈Gal(F /F ) ([Maz72, Corollary 4.37]).
v v v
d:goodord Definition 1.3.3. Suppose A has good ordinary reduction A¯ at v and L′/K is unramified at v
with the Frobenius element [v]L′/K ∈Γ′. Define
g g
fL′,v := (1−α−i 1·[v]L′/K)× (1−α−i 1·[v]−L′1/K)⊂Λ(Γ′).
i=1 i=1
Y Y
1.3.4. Local factors at split multiplicative places. Suppose A has split multiplicative reduction at
v. This meansthereis arankg latticeΩ ≃Z×···×Zsitting insidethe torusT =(K×)g sothat
v v
T/Ω is isomorphic to the rigid analytic space associated to A (see [Ger72]). In particular,
v
ee::ddeesspplliitt (1) A(K )≃(K×)g/Ω .
v v v
Rg
ConsiderthecompositionΩ −→(K×)g −→v (Γ )g whereR :K× −→Γ isthelocalreciprocity
v v v v v v
map, and extend it Z -linearly to
p
ee::mmccrrvv (2) Rv :Zp⊗ZΩv // (Γv)g.
d:splitmul Definition 1.3.4. Define wv =χZp(coker[Rv]).
su:maint
1.4. The main theorem. Here is our main theorem. Recall that Ψ=Gal(L/L′).
t:compatible Theorem 1. Suppose d≥1 and assume the above notation. Then we have
ΘL′ ·ϑL/L′ =̺L/L′ ·pL/L′(ΘL),
where ϑL/L′ := vϑv with each ϑv an ideal of Λ(Γ′) defined by the following conditions:
(a) Suppose v 6∈S. If Ψ 6=0, then ϑ =π ; otherwise, ϑ =(1).
Q v v v v
4 KI-SENGTAN
(b) Suppose v ∈S and A has good ordinary reduction at v. If v is unramified over L′/K, then
ϑv =fL′,v; otherwise ϑv =(1).
(c) Suppose v ∈S and A has split-multiplicative reduction at v. Then
Λ(Γ′)·w , if Γ′ =0;
v v
ϑ = (σ−1)g, if Ψ ≃Z and Γ′ is topologically generated byσ;
v  v p v
(1), otherwise.
Theproofwillbecompletedin§5.2. Thetools,localandglobal,fortheproofwillbeestablished
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 -extension of K. Then X is
p L
torsion over Λ(Γ).
See [MaRu07] for examples of non-torsion X in the number field case, while examples in
L
characteristic p can be found in [LLTT13, Appendix].
Proof. LetFq denotetheconstantfieldofK andletL0 =KFqp∞ betheconstantZp-extensionover
K with Γ0 = Gal(L /K) topologically generated by the Frobenius substitution Frob : x 7→ xq.
0 q
Thetheoremisalreadyprovedin[OTr09]fortheL=L case. ThismeansΘ 6=0. Byrepeatedly
0 L0
applying Theorem 1 (d−1 times), we deduce
p (Θ )·w =Θ · ϑ .
L/L0 L L0/K L0 v
v
Y
Since Γ0 6= 0, for all v, the factor ϑ equals one of (1), (Frobdeg(v)−1)g, or f . In particular,
v v L0,v
ϑ 6=0, for all v. Therefore, p (Θ )6=0, and hence Θ 6=0. (cid:3)
v L/L0 L L
su:imc
1.5. The Iwasawa main conjecture. Possibly, Theorem 1 could be useful for determining an
explicit generator of χ (X ). Assume that an explicitly given element θ′ ∈ Λ(Γ) is already
Λ(Γ) L L
known to be a generator of the characteristic ideal of a submodule X′ of X , and we want to see
L L
if actually
ee::aaccttuuaallllyy (3) Θ =(θ′ ).
L L
In addition, assume that there exists an intermediate Ze-extension L of L/K such that Θ , the
p 0 L0
characteristic ideal of X over Λ(Gal(L /K)), is explicitly given. Then by applying Theorem 1,
L0 0
we can obtain an explicit expression of p (Θ ) in terms of Θ and other factors. Thus, by
L/L0 L L0
checking the explicit expressions, we would be able to determine if
ee::aabbllee (4) p (Θ )=(p (θ′ )).
L/L0 L L/L0 L
The point is that Equations (3) and (4) are indeed equivalent. To see this, we only need to write
Θ =(θ′ ·θ′′), for someθ′′ ∈Λ(Γ)
L L
and observethat θ′′ is a unit ofΛ(Γ) if and only if its image p (θ′′) is a unit of Λ(Gal(L /K)).
L/L0 0
In the function field case, L could be taken to be the constant Z -extension, since an explicit
0 p
expressionofΘ isalreadygivenin[LLTT13](forsemi-stableA). Wecanalsoapplythetheorem
L0
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 . In [LLTT13] this method is used in the case where
L0
char.(K)=p and A is a constant ordinary abelian variety.
SELMER GROUPS OVER Zd-EXTENSIONS 5
p
su:zero
1.6. The zero set of Θ and the structure of X0. Our theory is useful for determining the
L L
Λ(Γ)-modules structures of
YL :=(l−i→mSelp∞(A/F)div)∨,
F
Z :=(lim(Q /Z )⊗A(F))∨,
L −→ p p
F
and
XL0 :=( (Selp∞(A/L)Gal(L/F))div)∨
F
[
as well. In general, we have the surjections X0 ////Y //// Z due to the maps
L L L
(Qp/Zp)⊗A(F)(cid:31)(cid:127) //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
partoftheTate-ShafarevichgroupofAoverF isfinite. Thus,ifthisholdsforallF thenZ =Y .
L L
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 res is surjective for every F, and hence Y =X0.
F L L
1.6.1. The zero set. The structures of X0, Y and Z are related to the zero set of Θ . For
L L L L
θ ∈Λ(Γ) define the zero set
∆ :={ω ∈Γ | p (θ)=0},
θ ω
where p : O Λ(Γ) −→ O is the O -algebra homomorphism extending ω : Γ −→ O×. Note
ω ω ω ω ω
that for eachω ∈Γ, the eigenspace (OωSelp∞(Ab/L))(ω) is cofinitely generatedoverOω as it is the
Pontryagindual of the finitely generated O -module OX /ker[p ]·X .
ω L ω L
b
d:rs Definition 1.6.1. For each ω ∈Γ, denote s(ω):=corankOω(OωSelp∞(A/L))(ω).
We have the inclusions
b
((OωSelp∞(A/L))(ω))div ⊂((OωSelp∞(A/L)ker[ω])div)(ω) ⊂(Oω (Selp∞(A/L)Gal(L/F))div)(ω),
F
[
where the left term is just the p-divisible part of the term. Hence,
ee::nneewwss (5) s(ω)=corankOω(Oω (Selp∞(A/L)Gal(L/F))div)(ω).
F
[
t:root Theorem 3. A character ω ∈Γ is contained in △ if and only if s(ω)>0.
ΘL
This theorem is proved in §5.3. Let θ ∈ Λ(Γ) be an element vanishing on ∆ . By this, we
b ΘL
meanthatpω(θ)=0foreveryω ∈∆ΘL. SincetheOωΛ(Γ)-structureof(OωSelp∞(A/L))(ω)factors
through OωΛ(Γ) pω //Oω ,wemusthaveθ·(OωSelp∞(A/L))(ω) =pω(θ)·(OωSelp∞(A/L))(ω) =0
for every ω ∈ ∆ΘL. On the other hand, Theorem 3 says if ω 6∈ ∆ΘL then (OωSelp∞(A/L))(ω) is
finite. Thus, θ·(OωSelp∞(A/L))(ω) is always finite for all ω ∈Γ.
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 ∈ Γ(F) := Hom(Γ(F),µp∞)bregard as a finite subgroup of Γ.
Consider the elements e := ω(g)−1·g ∈O[Γ(F)], ω ∈Γ(F), which are |Γ(F)|-multiples
ω γ∈Γ(F)
of idempotents. Multiplying any finitbe O[Γ(F)]-module W by e ’s, we can form a homomorphisbm
P ω
b
W(ω) −→W
ω∈MΓb(F)
6 KI-SENGTAN
of finite kernel and cokernel. In particular, by taking W =((OSelp∞(A/L))Gal(L/F))∨ and by the
duality, we have a homomorphism
ee::aammoorrpphhiissmm (6) OSelp∞(A/L)Gal(L/F) −→ (OSelp∞(A/L))(ω)
ω∈MΓb(F)
of finite kernel and cokernel. Then by multiplying both sides of (6) by θ we see that
ee::tthheettaazzeerroo (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 :H1(F,A[p∞])−→H1(L,A[p∞])Gal(L/F)
L/F
denote the restriction map, then
ee::kkeerrrreessllff (8) |ker res |<∞,
L/F
and (cid:2) (cid:3)
ee::ccookkeerrrreessllff (9) |coker res |<∞.
L/F
Then (7) and (8) imply θ·Selp∞(A/F)div =(cid:2)0 as it i(cid:3)s also finite and p-divisible. We have proved:
c:vanishingtheta Corollary 1.6.2. If θ ∈ Λ(Γ) vanishes on ∆ΘL, then θ annihilates (Selp∞(A/L)Gal(L/F))div,
Selp∞(A/F)div and (Qp/Zp)⊗A(F), for all F. Hence θ·XL0 =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 ⊂ Γ is called a Z -flat of codimension k > 0, if there exist γ ,...,γ ∈ Γ
p 1 k
expandable to a Zp-basis of Γ and ζ1,...,ζk ∈µp∞ so that
b
T =T :={ω ∈Γ | ω(γ )=ζ ,i=1,...,k}.
γ1,...,γk;ζ1,...,ζk i i
Theorem 4. (Monsky) If θ ∈Λ(Γ) is non-zero, tbhen ∆θ 6=Γ and is a finite union of Zp-flats.
Note that for a given θ ∈ Λ(Γ), if T ⊂ ∆θ then σT ⊂ ∆bθ for all σ ∈ Gal(Qp/Qp), as ∆θ is
invariant under the action of the Galois group. Also, if T ⊂ ∆ with ζ ∈ O then γ −ζ divides
γ,ζ θ
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−Γp and ζ ∈µp∞ so that
f =f := (γ−σζ).
γ,ζ
σ∈Gal(YQp(ζ)/Qp)
If ζ is of order pn+1 and t = γ − 1, i = 1,...,d, where γ ,...,γ is a Z -basis of Γ, then
i i 1 d p
f is nothing but the polynomial p−1(t +1)ipn that is irreducible in Z [t ]. Hence, a simple
γ1,ζ i=0 1 p 1
element is irreducible in Λ(Γ) = Z [[t ,...,t ]]. Obviously, ∆ = σT . In particular, two
pP1 d fγ,ζ σ γ,ζ
simple elements f and g divide each other if and only if ∆ = ∆ . On the other hand, if T =
f g
S
T , k ≥2,then wecanfindtworelativelyprime simple elements bothvanishing onT,
γ1,...,γk;ζ1,...,ζk
for example, f and f .
γ1,ζ1 γ2,ζ2
SELMER GROUPS OVER Zd-EXTENSIONS 7
p
1.6.3. The structure of X0. If W is a torsion Λ(Γ)-module then θ := χ (W) 6= 0 and there
L Λ(Γ)
exists an pseudo-isomorphism
m
ee::MMssiimm (10) ι:(Λ(Γ)/(f1b1))a1 ⊕···⊕(Λ(Γ)/(flbl))al ⊕ Λ(Γ)/(ξi)−→W.
j=1
M
where a ,...,a , b ,...,b are positive integers, f ,...,f areallthe relatively prime simple factorsof
1 l 1 l 1 l
θ, and ξ ,...,ξ ∈ Λ(Γ) are not divided by any simple element (l = 0 or m = 0 is allowed). The
1 m
product φ=f ·····f vanishes on every codimension one Z -flat of ∆ . By the above argument,
1 l p θ
we can find two products ε=g ·····g and ε′ =g′ ·····g′ , relatively prime to φ and to each
1 m 1 m′
other,ofsimpleelementssothatbothεandε′ vanishoneveryZ -flatof∆ ofcodimensiongrater
p θ
than 1. Then both φε and φε′ vanish on ∆ . Note that ι is actually injective as its domain of
θ
definition contains no non-trivial pseudo-null submodule (see §2.1).
If X is torsion then by taking W =X in (10) we obtain the exact sequence
L L
ee::xxllssiimm (11) 0 // li=1(Λ(Γ)/(fibi))ai ⊕ mj=1Λ(Γ)/(ξi) // XL //N //0,
for some pseudo-nulLl N. Let φε and φε′ beLas above. Let ∼ denote pseudo-isomorphism.
t:x0 Theorem 5. Suppose X is torsion over Λ(Γ) and assume the above notation. Then both φε
L
and φε′ annihilate XL0, YL, ZL and (Selp∞(A/L)Gal(L/F))div, Selp∞(A/F)div, (Qp/Zp)⊗A(F) for
all finite intermediate extension F of L/K. Moreover, X0 is pseudo isomorphic to X /φ·X .
L L L
Namely, if a ,...,a are as in (11), then
1 l
XL0 ∼(Λ(Γ)/(f1))a1 ⊕···⊕(Λ(Γ)/(fl))al.
Proof. The first assertion follows from Corollary 1.6.2. Consequently, φ·X0 is pseudo-null, being
L
annihilated by relatively prime ε and ε′. Thus, X0 ∼ X0/φ·X0. By taking W = X0 in (10) we
L L L L
obtain the exact sequence
0 // li=1(Λ(Γ)/(fi))ci ι // XL0 // M //0,
forsomenon-negativeintegersLc1,...,cl andsomepseudo-nullM. Bycomparingthisexactsequence
with (11) using the fact that Λ(Γ)/(φ,ξ ) is pseudo-null, we see that X /φX ∼X0 if and only if
i L L L
c =a for each i. We shall only show c =a , as the rest can be proved in a similar way.
i i 1 1
Firstchooseaξ ∈Λ(Γ)thatannihilatesM andisrelativelyprime toφ. LetE denotethe field
ω
O Q and via Λ(Γ) pω // O ⊂E we consider the map ι :=E ⊗ ι for an
ω p ω ω ω ω Λ(Γ)
ω ∈∆ −(∆ ∪···∪∆ ∪∆ ∪∆ ∪···∪∆ ).
f1 f2 fl ξ ξ1 ξm
NowtheE -vectorspaceE ⊗ M =0asitisannihilatedbyp (ξ)6=0. Similarly,asp (f )6=0
ω ω Λ(Γ) ω ω i
for i≥2, E ⊗ Λ(Γ)/(f )=0 . On the other hand, as p (f )=0, E ⊗ Λ(Γ)/(f )=E .
ω Λ(Γ) i ω 1 ω Λ(Γ) 1 ω
Also, ker[ιω] = 0 as it is annihilated by pω(ξ). Therefore, ιω is an isomorphism between Eωc1 and
E ⊗ X0. Hence
ω Λ(Γ) L
rank O ⊗ X0 =dim E ⊗ X0 =c .
Oω ω Λ(Γ) L Eω ω Λ(Γ) L 1
Then we deduce s(ω−1)=c by using (5) together with the fact that
1
(Oω (Selp∞(A/L)Gal(L/F))div)(ω−1) =(OωXL0/ker[pω]XL0)∨ ≃(Oω⊗Λ(Γ)XL0)∨.
F
[
Similarly, by tensoring the exact sequence (11) with E , we get a =s(ω−1), whence a =c . (cid:3)
ω 1 1 1
By Theorem 5 there are non-negative integers a′,...,a′, a′′,...,a′′ with a′′ ≤ a′ ≤ a , so that
1 l 1 l i i i
YL ∼(Λ(Γ)/(f1))a′1 ⊕···⊕(Λ(Γ)/(fl))a′l and ZL ∼(Λ(Γ)/(f1))a′1′ ⊕···⊕(Λ(Γ)/(fl))a′l′.
8 KI-SENGTAN
su:algfun
1.7. Algebraic functional equations. Let ♯ : Λ(Γ) −→ Λ(Γ), x 7→ x♯, denote the Z -algebra
p
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 is torsion,
L
the we have the functional equations X0♯ ∼ X0, Y ♯ ∼ Y and Z♯ ∼ Z as well. Taking the
L L L L L L
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 a X Y 0,
L L L
where
a :=lim(X(A/F) /X(A/F) )∨.
L ←− p div
F
Then by using the Cassels-Tate pairing on each X(A/F) ×X(At/F) , one can actually prove
p p
the pseudo-isomorphisms a♯ ∼ a and X♯ ∼ X . The proof is given in [LLTT13], in which the
L L L L
content of Theorem 5 actually plays a key role.
1.7.1. a is torsion. Thefollowingisprovedin[LLTT13]byusingTheorem3,whileifeveryv ∈S
L
is a good ordinary place, then it can be proved by the control theorem.
t:a1 Theorem 6. The module a is finitely generated and torsion over Λ(Γ).
L
su:when
1.8. When is X torsion? For convenience, call an intermediate extension M of L/K simple,
M
if Gal(M/K)≃Zc, for some c. For such M, by repeatedly applying Theorem 1, we deduce
p
ee::dd--cc (12) p (Θ )·̺=Θ ·ϑ,
L/M L M
where ̺6=0 and ϑ is a product of local factors obtained from those ϑ ’s in Theorem 1. It is easy
v
to see that ϑ6=0 unless M is fixed by the decomposition subgroupΓ of some split-multiplicative
v
place v ∈S. Thus, the following theorem is proved by taking
T={LΓv | v ∈S is a split-multiplicative place}.
t:nontor Theorem 7. Suppose X is non-torsion. If L/K only ramifies at good ordinary places, then X
L 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 X is non-torsion unless
M
M ⊂M for some M ∈T.
j j
By (12), if Θ = 0, then p (Θ ) = 0. Put T := {ω ∈ Γ | ω(γ) = 1, for all γ ∈
M L/M L M
Gal(L/M)}, the Z -flat of codimension d−c determined by M. Then p (Θ ) = 0 if and only
p L/M L
if T ⊂ ∆ , or equivalently, T ⊂ T for some j if ∆ = ν bT . Let M be the maximal
M ΘL M j ΘL j=1 j j
simple intermediate extension of L/K, so that T ⊂ T . Then the following theorem is proved
Mj j S
by setting
T={M | j =1,...,ν}.
j
Theorem 8. Suppose X is torsion. Then there is a finite set T consisting of proper simple
L
intermediate extensions of L/K, such that for each simple intermediate extension M outside T,
X is torsion.
M
Hence, if d = 2 and X is non-torsion (resp. torsion), then X is non-torsion (resp. torsion)
L M
for almost all intermediate Z -extensions M.
p
SELMER GROUPS OVER Zd-EXTENSIONS 9
p
su:mw
1.9. The growth of s . Let K denote the nth layer of L/K and write I for the kernel of
n n n
Λ(Γ) ////Zp[Γ(Kn)]. Then Selp∞(A/L)Gal(L/Kn) is the Pontryagin dual of XL/InXL, whence
cofinitely generated over Z . Let s denote its corank. Theorem 9 below gives an asymptotic
p n
formula of s . The following lemma as well as its proof is by I. Longhi. Denote E = Q (µ ),
n n p pn
E =pn(d−2), for d≥2; E =1, for d=1. Let J be an ideal of Λ(Γ).
n n
Lemma 1.9.1. (Longhi) If J =(fm ) for some positive integer m and δ :=[Q (ζ):Q ] then
l:longhi γ,ζ ζ p p
ee::aassssyyttoottiiccff (13) rankZpΛ/(In+J)=δζ ·pn(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
ee::aassssyyttoottiiccffgg (14) rankZpΛ/(In+J)=O(En).
Proof. Write Γ =Γ(K ) and V :=E ⊗ (Λ/I ). Then
n n n n Zp n
rank Λ/(I +J)=dim E ⊗ Λ/(I +J) =dim (V /JV ).
Zp n En n Zp n En n n
One has a decomposition of E -vector spaces (cid:0) (cid:1)
n
V =⊕ V(ω).
n ω∈Γbn n
Moreover dim V(ω) =1 because V ≃E [Γ ] is the regular representation. Obviously
En n n n n
0 if p (J)=0
JV(ω) = ω
n (Vn(ω) if pω(J)6=0.
Denote ∆ ={ω ∈Γ | p (J)=0}. Then
J ω
dim (V /JV )=|{ω ∈Γ :p (J)=0}|=|∆ ∩Γ |=|∆ [pn]|
bEn n n n ω J n J
(the last equality comes from Γ = Γ[pn]). Here G[pn] denotes the pn torsion subgroup of G.
n b b
Monsky’stheoremyields∆ = T ,wheretheT ’sareZ -flats. Besides,by[Mon81,Lemma1.6],
J j j p
b b
S |T [pn]|=pn(d−kj) for n≫0,
j
where k denotes the codimension of T . If J =(fm ), then every T is of codimension 1. Hence
j j γ,ζ i
dim (V /(fm )V )=|∆ [pn]|=δ ·pn(d−1) for n≫0.
En n γ,ζ n (fγm,ζ) ζ
To show the second assertion we observe that every T should be of codimension greater than 1,
i
because if some T =T , then ∆ ⊂∆ , whence f divides all elements of J. Thus,
j γ;ζ fγ,ζ J γ,ζ
dim (V /JV )=|∆ [pn]|=O(E ).
En n n J n
(cid:3)
t:mw Theorem 9. There exists a non-negative integer κ such that
1
(15) s =κ pnd+O(pn(d−1)).
n 1
X is torsion if and only if κ =0, in this case there exists a non-negative integer κ such that
L 1 2
ee::kkaappppaa22 (16) s =κ pn(d−1)+O(E ).
n 2 n
X0 is pseudo-null if and only if s =O(E ).
L n n
10 KI-SENGTAN
Proof. Suppose W is a finitely generatedΛ(Γ)-module. Then (10) givesrise to the exactsequence
0 // li=1(Λ(Γ)/(fibi))a1 ⊕ mj=1Λ(Γι)/(ξi) // W // M //0
for some pseudo-nulLl M. Then we tensor tLhe exact sequence with E ⊗ Λ(Γ)/I . Since M is
n Zp n
annihilatedbysomerelativelyprimef andg,bothM/I M andTor (Λ(Γ)/I ,M)arequotients
n Λ(Γ) n
of some direct sums of finite copies of V /(f,g)V . Hence, formulae (13) and (14) imply
n n
ee::aassssyyttoottiiccww (17) rankZpW/InW =dimEnEn⊗Zp W/InW = δipn(d−1)+O(En),
i
X
where δ = δ if f = f . If X is torsion, then we take W = X to prove (16). In this case,
i ζi i γi,ζi L L
X0 is pseudo-null if and only if l =0 (see Theorem 5) which means δ =0.
L i i
Suppose X is non-torsion. Then ∆ = Γ, whence by Theorem 3, s(ω) > 0 for all ω. Since
L ΘL P
(6) is of finite kernel and cokernel, we have
b
rankZpXL0/InXL0 =corankZp(Selp∞(A/L)Gal(L/Kn))div ≥pnd.
By (17), X0 is non-torsion, and hence not pseudo-null. Let x ,...,x ∈ X form a basis of the
L 1 κ1 L
vector space generatedby X over the field of fractions of Λ(Γ). Then we have an exact sequence
L
// // // //
0 Λ(Γ)·x X W 0,
i i L
where W is a torsion Λ(Γ)-moduPle. Then we tensor the exact sequence with En⊗ZpΛ(Γ)/In. (cid:3)
Remark 1.9.2. By a similar argument, one can prove that: (1) There exists a finite number of
Z -flats T ,...,T so that s(ω)=κ for each ω 6∈ T . (2) If X is torsion, then for each Z -flat
p 1 l 1 i i L p
T ⊂ ∆ , there is a finite number of proper Z -flats T′,...,T′ ⊂ T so that s(ω) is a constant for
ΘL p S 1 ν
each ω ∈T − T′. (3) There is a bound of s(ω) for all ω ∈Γ.
i i
Also, Theorem 9 generalizes [MaRu03, Proposition 1.1], as we have s = rankA(K ) if L/K
n n
only ramifies aSt good ordinary places andX(A/K ) is finiteb.
n p
2. preliminary
s:pre
In this section, we assume that Γ ≃ Zd, with d ≥ 0, except in Lemma 2.1.1 and Lemma
p
2.3.1, where we assume d ≥ 1. If d = 0, we set Λ(Γ) = Z ; otherwise, Gal(L/L′) = Ψ ≃ Z is
p p
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 f ,...,f , n ≥ 2, in Λ(Γ) so that
1 n
f ·W =0 for every i. If W is non-torsion over Λ(Γ), then
i
χ (W)=0.
Λ(Γ)
If W is torsion, then there exist irreducible ξ ,...,ξ ∈Λ(Γ), and a pseudo-isomorphism
1 m
φ: m Λ(Γ)/ξriΛ(Γ) // W,
i=1 i
[Bou72, §4.4, Theorem 5]. In this caLse, the characteristic ideal is
m
χ (W)= (ξ )ri 6=0.
Λ(Γ) i
k=1
Y
Denote
m
[W]= Λ(Γ)/ξriΛ(Γ).
i
i=1
M