ON THE GALOIS STRUCTURE OF SELMER GROUPSDAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

Abstract. Let A be an abelian variety defined over a number field k and F a finite Galois
extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A
and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F . We
also use the results so obtained to derive new bounds on the growth of the Selmer rank of A overextensions of k.

1. Introduction
Let A be an abelian variety defined over a number field k and write At for the corresponding dual
abelian variety.
Fix a prime number p. For each finite extension field F of k we write Selp (A/F ) for the p-primary
Selmer group of A over F . At the outset we recall that this group can be defined as the direct limit,nover non-negative integers n, of the Selmer groups associated to the isogenies
 [p ] of A over F . It isthen equal to the subgroup of the Galois cohomology group H 1 F, A[p∞ ] given by the usual local
conditions, where A[p∞ ] denotes the Galois module of p-power torsion points on A.
Write X(A/F ) for the Pontryagin dual of Selp (A/F ). Denote by T (A/F ) the torsion subgroupof X(A/F ) and by X(A/F ) the quotient of X(A/F ) by T (A/F ). We recall that X(A/F ) contains
a subgroup canonically isomorphic to the p-primary Tate-Shafarevich group X(At /F )p of At overF , with the associated quotient group canonically isomorphic to HomZ (A(F ), Zp ). In particular, if
X(A/F )p is finite, then T (A/F ) and X(A/F ) simply identify with X(At /F )p and HomZ A(F ), Zp
respectively.
Let now F/k be a Galois extension of group G. In this case we wish to study the structureof X(A/F ) as a G-module. We recall that describing the explicit Krull-Schmidt decomposition of
Zp [G]-lattices that occur naturally in arithmetic is known to be a very difficult problem (see, forexample, the considerable difficulties already encountered by Rzedowski-Calderón et al in [14] when
considering the pro-p completion of the ring of algebraic integers of F ).
Notwithstanding this fact, in this article we will show that under certain not-too-stringent conditions on A and F , there is a very strong interplay between the structures of the G-modules T (A/F )and X(A/F ) and that this interplay can in turn lead to concrete results about the explicit structureof X(A/F ) as a Zp [G]-module. For example, one of our results characterises the projectivity of
X(A/F ) as a Zp [G]-module (see Theorem 2.2 and Remark 2.4), whilst another characterises, forcertain G, the conditions under which X(A/F ) is a trivial source module over Zp [G] (see Theorem 2.7). More generally, our methods can be used in some cases to give an explicit decompositionof X(A/F ) as a direct sum of indecomposable Zp [G]-modules (see the examples in §4).
Whilst such explicit structure results might perhaps be of some intrinsic interest there are also
two ways in which they can have important consequences.
Firstly, they give additional information about the change of rank of X(A/L) as L varies over theintermediate fields of F/k. As a particular example of this, we show here that this information can insome cases be used to tighten the bounds on the growth of ranks of Selmer groups in dihedral towersof numbers fields that were proved by Mazur and Rubin in [12] (see Corollary 2.10). In a similar
1

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DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

way we also show that it sheds new light on Selmer ranks in false Tate curve towers of number fields(see Corollary 2.11).
Secondly, such structure results play an essential role in attempts to understand and investigate
certain equivariant refinements of the Birch and Swinnerton-Dyer conjecture. In this regard, we
note that in [3] the structure results proved here play a key role in our obtaining the first (either
theoretical or numerical) verifications of the p-part of the equivariant Tamagawa number conjecture
for elliptic curves A in the technically most demanding case in which A has strictly positive rankover F and the Galois group G is both non-abelian and of exponent divisible by (an arbitrarily large
power of) p.
Finally we note that some of our structure results concerning projectivity are similar in spirit
to the results on Selmer groups of elliptic curves over cyclotomic Zp -extensions that are proved by
Greenberg in [7] (for more details in this regard see Remark 2.4).
2. Statements of the results
In this section we state all of the main results that are to be proved in §3.
2.1. Basic notations. For any finite group Γ we write Ir(Γ) for the set of irreducible Qcp -valued
characters of Γ. For each ψ in Ir(Γ) we writeψ(1) Xψ(γ −1 ) γ
eψ =
|Γ|
γ∈Γ

c
for the primitive idempotent of the centre ζ Qp [Γ] of the group ring Qcp [Γ]. We fix a Qcp [Γ]-moduleVψ of character ψ. We also write 1Γ for the trivial character of Γ.
We also note that by a ‘Zp [Γ]-lattice’ we shall mean a Zp [Γ]-module that is both finitely generated
and free over Zp .
For any abelian group M we write Mtor for its torsion subgroup. For any prime p we write M [p]for the subgroup {m ∈ M : pm = 0} of Mtor and set Mp := Zp ⊗Z M . For any finitely generated
Zp -module M and any field extension F of Qp we set F ⊗ M := F ⊗Zp M and write rkZp (M ) for
the Zp -rank dimQp (Qp ⊗ M ) of M .
For any Galois extension of fields L/K we write GL/K in place of Gal(L/K). For each nonarchimedean place v of a number field we write κv for its residue field. We also write Fp for the fieldwith p elements.
2.2. The hypotheses. We assume throughout to be given a finite Galois extension of number fields
F/k of group G and an abelian variety A defined over k. We also fix a prime number p and a p-Sylow
subgroup P of G and then set K := F P .
In this section we assume in addition that A and F satisfy the following hypotheses.
(a) A(K)[p] = 0;(b) No Tamagawa number of A/K is divisible by p;(c) A/K has good reduction at all p-adic places;(d) For all p-adic places v that ramify in F/K, the reduction is ordinary and A(κv )[p] = 0;(e) No place of bad reduction for A/k is ramified in F/k.
Remark 2.1. These hypotheses are for the most part motivated by the arguments that are used by
Greenberg in [6] (see, in particular, the proof of Proposition 3.1 below). For a fixed abelian variety A
over k the first three hypotheses are clearly satisfied by all but finitely many primes p. However, thehypothesis (d) excludes the case that is called ‘anomalous’ by Mazur in [10] and, for a given A, there
may be infinitely many primes p for which there are p-adic places v at which A has good ordinary
reduction but A(κv )[p] does not vanish. Nevertheless, it is straightforward to describe examples of

STRUCTURE OF SELMER GROUPS

3

abelian varieties A for which there are only finitely many such anomalous places – see, for example,
the result of Mazur and Rubin in [11, Lemma A.5].
2.3. The Galois structure of X(A/F ). We first recall from the introduction that X(A/F ) isdefined as the Pontryagin dual of Selp (A/F ) and that, if the group X(A/F )p is finite, then thetautological short exact sequence
0 → T (A/F ) → X(A/F ) → X(A/F ) → 0
simply identifies with the canonical short exact sequence
0 → X(At /F )p → X(A/F ) → HomZ (A(F ), Zp ) → 0.
For each ρ in Ir(P ) we write mρ , or mF,ρ when we wish to be more precise, for the multiplicity
with which ρ occurs in the representation Qcp ⊗ X(A/F ). For each subfield L of F we setsrk(A/L) := dimQp (Qp ⊗ X(A/L)).
0

For any fields L and L with k ⊆ L ⊆ L0 ⊆ F we also set


L0
0
πL
T L(A/L) := cok T (A/L0 ) −−
→ T (A/L) ,0

L
where πLdenotes the natural norm map.

2.3.1. Projectivity results. Our first results give an explicit characterisation of the conditions underwhich X(A/F ) can be a projective Zp [G]-module.
Theorem 2.2. If A and F satisfy hypotheses (a)–(e), then the following conditions are equivalent.
(i) X(A/F ) is a projective Zp [G]-module.
(ii) T F(A/F H ) = 0 for H ≤ P and X(A/F ) spans a free Qp [P ]-module.
(iii) T F(A/F H ) = 0 for H ≤ P and mρ ≥ ρ(1) · srk(A/K) for ρ ∈ Ir(P ).
(iv) For each H ≤ P both T F(A/F H ) = 0 and srk(A/F ) = |H| · srk(A/F H ).
(v) For each H ≤ P with |H| = p both T F (A/F H ) = 0 and srk(A/F ) = p · srk(A/F H ).
Remark 2.3. The equivalence of the conditions (i) and (v) in Theorem 2.2 is somewhat analogous
to a classical theorem of Noether. To see this write OL for the ring of integers of a number field Land κL,p for the direct sum of its residue fields at all p-adic places. Then for H ≤ P with |H| = p
one has rk(OF ) = p · rk(OF H ) (by Galois theory) and Noether’s Theorem [13] asserts that OF,p is a
projective Zp [G]-module if and only if for each such H the cokernel of the trace map κF,p → κF H ,pvanishes.
Remark 2.4. Some aspects of Theorem 2.2 are also reminiscent of, but differ in some important
respects from, results that are proved by Greenberg in [7]. To be more precise, let F∞ and k∞
be the cyclotomic Zp -extensions of F and k, set ∆ := GF∞ /k∞ , let A be an elliptic curve overk that has good ordinary reduction at all p-adic places and for any extension L of k in F∞ set
X(A/L) := HomZp Selp (A/L), Qp /Zp . Then, assuming crucially that X(A/F∞ ) is both torsion
over the relevant Iwasawa algebra and has zero µ-invariant, Greenberg investigates the explicit
structure of X(A/F∞ ) as a Zp [∆]-module. In particular, by combining the same characterisation ofprojectivity in terms of cohomological triviality (that we use in §3.4 to prove Theorem 2.2) together
with certain intricate computations in Galois cohomology, he proves under certain mild additional
conditions on A and F that an ‘imprimitive’ form of X(A/F∞ ) is projective over Zp [∆] and thendeduces families of explicit relations between the multiplicities with which each ψ ∈ Ir(∆) occurs
in Qcp ⊗ X(A/F∞ ). However, even if the natural descent homomorphism Qp ⊗ X(A/F∞ )GF∞ /F →

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DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

Qp ⊗ X(A/F ) is bijective, in order to deduce results about the structure of the Qp [G]-module
Qp ⊗ X(A/F ) one would need to describe Qp ⊗ X(A/F∞ )GF∞ /F as an explicit quotient of the
Qp [∆]-module Qp ⊗ X(A/F∞ ) and, in general, this appears to be difficult.
Regarding Theorem 2.2(v) we note that if P is either cyclic or generalised quaternion (so p = 2),
then it has a unique subgroup of order p.
In another direction, if we assume the vanishing of X(A/K)p , then we are led to the following
simplification of Theorem 2.2. In the subsequent article [3] we use this particular characterisation inorder to obtain important consequences in the context of equivariant refinements of the Birch and
Swinnerton-Dyer conjecture.
Corollary 2.5. If A and F satisfy the hypotheses (a)–(e) and X(A/K)p vanishes, then the following
conditions are equivalent.
(i) X(A/F ) is a projective Zp [G]-module.
(ii) Qp ⊗ X(A/F ) is a free Qp [P ]-module.
(iii) mρ ≥ ρ(1) · srk(A/K) for ρ ∈ Ir(P ).
(iv) For each H ≤ P both T (A/F H ) = 0 and srk(A/F ) = |H| · srk(A/F H ).
(v) For each H ≤ P with |H| = p both T (A/F H ) = 0 and srk(A/F ) = p · srk(A/F H ).
Since obtaining an explicit description (in terms of generators and relations) of any given Zp [G]module is in general very difficult to achieve, the simplicity of the equivalence of the conditions (i)and (ii) in Corollary 2.5 seems striking. In addition, the result of Theorem 2.9 below will show thatthe value of srk(A/F ) that occurs in Theorem 2.2(iv) and Corollary 2.5(iv) is actually the maximal
possible in this context.
2.3.2. General results. In the next result we provide evidence that there is a strong link between the
Galois structures of Mordell-Weil and Tate-Shafarevich groups under much more general hypotheses
than occur in Theorem 2.2 and Corollary 2.5.
For any subgroup H of G we write Zp [G/H] for Zp [G] ⊗Zp [H] Zp , regarded as a left Zp [G]-modulein the obvious way. Given a subgroup J of G, by a ‘trivial source Zp [G]-module with vertices
contained in J’ we shall then mean a direct sum of finitely many (left) Zp [G]-modules, each of which
is isomorphic to a direct summand of Zp [G/H] for some subgroup H of J. In particular, by a ‘trivial
source Zp [G]-module’ we shall mean a trivial source Zp [G]-module with vertices contained in G.
We also note that if G contains a normal subgroup I of p-power index, then by a result of Berman
and Dress (cf. [4, Th. 32.14]) one knows that Zp [G/H] is an indecomposable Zp [G]-module for any
subgroup H of G that contains I. This fact will be useful in the sequel.
Remark 2.6. We wish to point out that the terminology introduced above differs from the oneused in [2], where trivial source modules are instead called ‘permutation modules’ (see §2.2 in loc.
cit.). In particular, under our current terminology and in the context of dihedral extensions of Q,
the relevant assertion of [2, Corollary 5.3 (ii)] would state that the modules A0 (TF ) and Sel0 (TF )tf
defined in loc. cit. are trivial source Zp [G]-modules.
Theorem 2.7. Assume that A and F satisfy (a)–(e) and that P is cyclic of order pn . For each
integer i with 0 ≤ i ≤ n, let Fi denote the unique field with K ⊆ Fi ⊆ F and [F : Fi ] = pn−i . Thenthe following claims are valid.
(i) The Zp [P ]-module X(A/F ) is determined up to isomorphism by the integers srk(A/Fi ) for
each i with 0 ≤ i ≤ n together with the diagram of torsion groups(1)

T F(A/F0 )  T F(A/F1 )  · · ·  T F(A/Fn−1 )STRUCTURE OF SELMER GROUPS

5

where the upper and lower arrows are induced by the natural corestriction and restriction
maps (for more details see §3.2 below).
(ii) Moreover, if T F(A/Fi ) vanishes for each i with 0 ≤ i < n, then X(A/F ) is a trivial source
Zp [G]-module with vertices contained in P . In particular, in this case X(A/F ) is a trivial
source Zp [G]-module, and hence also a trivial source Zp [P ]-module.
In the context of Theorem 2.7 it is worth bearing in mind that even if G is cyclic of p-power order(and so G = P ) the category of Zp [G]-lattices is in general very complicated. For example, Heller
and Reiner [9] have shown that in this case the number of isomorphism classes of indecomposable
Zp [G]-lattices is infinite unless |G| divides p2 and have only been able to give an explicit classificationof all such modules in the special cases |G| = p and |G| = p2 .
Despite these difficulties, in §4 we will show that in certain cases the classification results of Heller
and Reiner can be combined with our approach to make much more explicit the interplay between
X(A/F ) and the groups T F(A/Fi ) that is described in Theorem 2.7.
In a more general direction, we note that if A and F satisfy the hypotheses (a)–(e) then, irrespective of the abstract structure of P , the result of Lemma 3.4 below shows that Theorem 2.7 canbe applied to Galois extensions F 0 /k 0 with k ⊆ k 0 ⊆ F 0 ⊆ F and for which GF 0 /k0 has cyclic Sylow
p-subgroups. This shows that in all cases the behaviour of the natural restriction and corestriction
maps on T (A/L) as L varies over all intermediate fields of F/k imposes a strong restriction on the
explicit structure of the Zp [G]-module X(A/F ).
The following result gives an explicit example of this phenomenon (and for more details in thisregard see Remark 3.6 below). We recall that a finite group Γ is defined to be supersolvable if it has
a normal series in which all subgroups are normal in Γ and all factors are cyclic groups.
Corollary 2.8. Assume that A/k has good reduction at all p-adic places and ordinary reduction at
all p-adic places that ramify in F/k. Assume also that A and F satisfy the hypotheses (a)–(e), that
P is both abelian and normal and that the quotient G/P is supersolvable.
Then for each irreducible Qcp -valued character ρ of G there is a diagram of torsion groups of theform (1) which determines the multiplicity with which ρ occurs in the Qcp [G]-module Qcp ⊗ X(A/F )up to an error of at most srk(A/K).
In particular, if the group Selp (A/K) is finite, then the structure of the Qp [G]-module Qp ⊗X(A/F ) is uniquely determined by the torsion groups T (A/L) as L ranges over intermediate fields
of F/k together with the natural restriction and corestriction maps between them.
2.4. Some consequences for ranks. In this section we describe several explicit consequences
concerning the ranks of Selmer modules which follow from the structure results stated above.
In the first such result we do not restrict the abstract structure of G.
Theorem 2.9. Assume that A and F satisfy the hypotheses (a)–(e).
(i) Fix a chain of fields K = F0 ⊂ F1 ⊂ · · · ⊂ Fn = F with each extension Fi /Fi−1 Galois ofdegree p and for each index i define an integer di by the equality pdi = |T Fi (A/Fi−1 )|. Then
one hassrk(A/K) + (p − 1)i=nXdi ≤ srk(A/F ) ≤ [F : K] · srk(A/K) + (p − 1)i=1(ii) Fix ρ ∈ Ir(P ). If Tρ(1) · srk(A/K).

i=nX

pn−i di .

i=1

F ker(ρ)

H(A/F ) vanishes for all H with ker(ρ) < H ≤ P , then mρ ≤

6

DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH(iii) If T F (A/F H ) vanishes for all H ≤ P , then srk(A/F ) ≤ [F : K] · srk(A/K).
In certain cases the additional hypotheses in Theorem 2.9 can be simplified. For example, if P
is abelian of exponent p, then for any non-trivial ρ in Ir(P ) the inclusion K ⊆ L ( F ker(ρ) implies
ker(ρ)
L = K and so the hypothesis in claim (ii) is satisfied if T F(A/K) vanishes and hence, a fortiori,
if X(A/K)p vanishes. Similarly, if |P | = p then the hypothesis in claim (iii) is satisfied if X(A/K)pvanishes.
In the remainder of this section we specialise the extension F/k and variety A in order to obtainfiner bounds on Selmer rank than are given in Theorem 2.9.
2.4.1. Dihedral extensions. In the next result we focus on the case that the Sylow p-subgroup P isan abelian (normal) subgroup of G of index two and that the conjugation action of any lift to G of
the generator of G/P inverts elements of P . Using the terminology of Mazur and Rubin [12], we
therefore say that the pair (G, P ) is ‘dihedral’.
This result tightens the bounds on ranks of Selmer modules in dihedral towers of numbers fields
that are proved by Mazur and Rubin in [12].
Corollary 2.10. Assume that p is odd, that A is an elliptic curve for which A and F satisfy the
hypotheses (a)–(e) and that the pair (G, P ) is dihedral (in the above sense). Assume also that all
p-adic places of k split in the quadratic extension K/k. Then the following claims are valid.
(i) If srk(A/K) is odd, P is cyclic and T F (A/F H ) vanishes for all subgroups H of P , then
X(A/F ) is a trivial source Zp [G]-module with vertices contained in P that has a non-zeroprojective direct summand. In particular, in this case for all characters ρ in Ir(P ) one has
1 ≤ mρ ≤ srk(A/K).
(ii) If srk(A/K) = 1 and X(A/K)p vanishes, then X(A/F H )p vanishes for all subgroups H of
P , X(A/F ) is a projective Zp [G]-module and mρ = 1 for all characters ρ in Ir(P ).
2.4.2. False Tate curve towers. A ‘false Tate
√ curve tower’ is obtained as the union over naturalmnumbers m and n of fields Kdm,n := Q p d, ζpn where d is a fixed integer, m ≤ n and ζpn is achoice of primitive pn -th roots of unity in Qc . Such extensions have recently been investigated in
the context of non-commutative Iwasawa theory.
Since Kdm,n is a bicyclic p-extension of Q(ζp ) there are several ways in which the results statedabove can be used to restrict the structure of X(A/Kdm,n ).
In the following result we give an example of this phenomenon.
Corollary 2.11. Fix a natural number d. Let A be an abelian variety over Q that has good reduction
at all prime divisors of d and good ordinary reduction at p. Assume that A has no point of orderp over Q(ζp ), that the reduction of A at p has no point of order p over Fp and that no Tamagawa
number of A over Q(ζp ) is divisible by p.
Let n be any natural number for which X A/Q(ζpn ) p vanishes and set(
least integer m with both 0 ≤ m ≤ n and X(A/Kdm,n )p 6= 0,
nd :=n if no such integer m exists.
Write Pn and Gn for the Galois groups of Kdnd ,n over Q(ζpn ) and over the subextension of Q(ζpn )of degree pn−1 over Q respectively. Then the following claims are valid.
(i) X(A/Kdnd ,n ) is a trivial source Zp [Gn ]-module with vertices contained in Pn .

STRUCTURE OF SELMER GROUPS

7(ii) The multiplicity mρ with which each ρ in Ir(Gn ) occurs in the module Qcp ⊗ X(A/Kdnd ,n ) is
at most srk A/Q(ζpn ) . Further if ρ and ρ0 are any non-linear characters in Ir(Gn ), thenmρ ≤ mρ0 if and only if ker(ρ) ⊆ ker(ρ0 ).
(iii) For any complementary subgroup H to Pn in Gn and any φ in Ir(H) one has rkZp (eφ ·X(A/Kdnd ,n )) ≤ pnd · rkZp eφ · X(A/Q(ζpn )) .
3. Proofs
In this section we prove all of the results that are stated above. More precisely, after making some
important general observations in §3.1 we prove Theorem 2.7 and Corollary 2.8 in §3.2, Theorem
2.9 in §3.3, Theorem 2.2 in §3.4, Corollary 2.5 in §3.5 and then finally Corollaries 2.10 and 2.11 in§3.6.
3.1. General observations. In the following result we write NG (J) for the normaliser of a subgroup
J of G.
Proposition 3.1. Assume that A and F satisfy the hypotheses (a)–(e). Then for each subgroup J ofF
J
P the natural
norm
map X(A/F )J → X(A/F J ) is bijective and the group
 T (A/F ) is isomorphic
−1
as a Zp NG (J) -module to the Tate cohomology group Ĥ
J, X(A/F ) .
Proof. We first note that, if we strengthen condition (d) to impose that the reduction is good ordinary
and non-anomalous at all places above p, then our conditions coincide precisely with the hypotheses
J:of Proposition 5.6 used by Greenberg in [6]. In this case the restriction homomorphism resF
F
Selp (A/F J ) → Selp (A/F )J is therefore bijective. However, the proof of Proposition 5.6 and the result
J
in Proposition 4.3 in [6] show that resF
F is also still bijective if there are places above p which do not
ramify in F/K and for which the reduction is good but not necessarily ordinary or non-anomalousand so the restriction map Selp (A/F J ) → Selp (A/F )J is bijective under the given hypotheses. By
taking Pontryagin duals this implies that the norm map X(A/F )J → X(A/F J ) is bijective and hence
restricts on torsion subgroups to give an isomorphism X(A/F )J,tor ∼= X(A/F J )tor = T (A/F J ).
Upon taking J-coinvariants of the tautological exact sequence
0 → T (A/F ) → X(A/F ) → X(A/F ) → 0
and then passing to torsion subgroups in the resulting exact sequence we therefore obtain an exact
sequence of Zp [NG (J)]-modules
πF J

T (A/F ) −−F−→ T (A/F J ) → X(A/F )J,tor → 0.
Given this exact sequence and the definition of T F (A/F J ) as the cokernel of πFF J , the claimed
isomorphism is a direct consequence of the fact that X(A/F )J,tor is equal to Ĥ −1 J, X(A/F ) . In
deed, since Ĥ −1 J, X(A/F ) is finite and X(A/F ) is Zp -free, the latter equality follows immediatelyfrom the tautological exact sequence
 ⊆

(2)
0 → Ĥ −1 J, X(A/F ) −→ X(A/F )J → X(A/F )J → Ĥ 0 J, X(A/F ) → 0
P
where the third arrow is induced by the action of the element g∈J g of Zp [G].
Remark 3.2. If in the setting of Proposition 3.1 one has srk(A/F ) = srk(A/F J ), then X(A/F ) is
srk(A/F )isomorphic as a Zp [J]-module to Zp
and so the group Ĥ −1 J, X(A/F ) vanishes. Proposition

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DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

3.1 therefore implies that in this case the group T F (A/F J ) also vanishes and hence, a fortiori,that the order of T (A/F ) is divisible by the order of T (A/F J ). This shows that, under the statedhypotheses, Proposition 3.1 provides a useful quantitative measure of the general principle that any
decrease in the order of Tate-Shafarevich groups must be accompanied by a corresponding increasein the rank of Mordell-Weil groups. For details of some explicit applications of this result see §4.
Remark 3.3. Since the Tate cohomology groups of X(A/F ) are finite, for any subgroup J of P
the isomorphism X(A/F )J ∼= X(A/F J ) in Proposition 3.1 combines with the exact sequence (2) toimply that(3)
Qp ⊗ X(A/F J ) ∼= Qp ⊗ X(A/F )Jand in particular that
srk(A/F J ) = rkZp X(A/F )J .
In fact, these statements at the rational level hold true in full generality and do not depend on thearguments or hypotheses required to prove the integral assertions of Proposition 3.1. For convenienceand brevity, we shall often in the sequel use these facts without further explicit comment.
To end this section we prove a useful technical result concerning the hypotheses (a)-(e).
Lemma 3.4. Let k 0 , K 0 and F 0 be fields with k ⊆ k 0 ⊆ K 0 ⊆ F 0 ⊆ F and such that F 0 /k 0 is Galois
and GF 0 /K 0 is a Sylow p-subgroup of GF 0 /k0 .
(i) Assume that A/k and F satisfy the hypotheses (a), (b), (e) and the second part of (d) (withrespect to K) and that A/k has good reduction at all p-adic places and ordinary reduction at
all p-adic places that ramify in F/k. Then A/k0 and F 0 satisfy the hypotheses (a)-(e) withrespect to K 0 .
(ii) If A/k and F satisfy the hypotheses (a)–(e) (with respect to K) and K ⊆ K 0 , then A/k0 andF 0 satisfy the hypotheses (a)–(e) with respect to K 0 .
Proof. We note first that the validity of hypotheses (a) and (e) for A/k and F implies their validity
for A/k0 and F 0 . This is clear for (e) and for (a) follows from the fact that F/K is a Galois extensionof p-power degree and so the vanishing of A(K)[p] implies the vanishing of A(F )[p] and hence alsothat of A(K 0 )[p].
To consider hypothesis (b) we fix a place v 0 of bad reduction for A/K 0 and a place w of F above v 0
and write v for the place of K which lies beneath w. Then hypothesis (e) implies that v and v 0 are
unramified in F/K and F/K 0 respectively. Thus, if the Tamagawa number of A/K at v is coprime
to p, then so is that of A/F at w since F/K is a Galois extension of p-power degree, and hence alsothat of A/K 0 at v 0 .
In a similar way one can show that the second part of hypothesis (d) for A/K and F implies theanalogous assertion for A/K 0 and F 0 . Indeed, if v 0 is any p-adic place of K 0 which ramifies in F 0
and w any place of F above v 0 , then the restriction v of w to K is ramified in F/K and hence (byassumption) A(κv )[p] vanishes. This implies that A(κw )[p] vanishes (since F/K is a Galois extensionof p-power degree) and hence also that A(κv0 )[p] vanishes, as required.
In particular, since the validity of hypothesis (c) and the first part of hypothesis (d) for A/K 0
and F 0 follow directly from the reduction assumptions that are made in claim (i), that result is now
clear.
To discuss claim (ii) we assume that A/k and F satisfy the hypotheses (a)–(e). We also replace F 0
by its Galois closure over K (if necessary) in order to assume, without loss of generality, that F 0 /K
is a Galois extension. Then it is easy to see that the hypotheses (c), (e) and the first part of (d)will hold for A/k0 and F 0 . In addition, since F 0 /K is a Galois p-extension, the same arguments as

STRUCTURE OF SELMER GROUPS

9

used to prove claim (i) show that the hypotheses (a), (b) and the second part of (d) will also holdfor A/K 0 and F 0 . This proves claim (ii).
3.2. The proofs of Theorem 2.7 and Corollary 2.8. In this section we make precise and prove
all of the claims in Theorem 2.7 and Corollary 2.8.
3.2.1. The proof of Theorem 2.7. We use the notation and hypotheses of Theorem 2.7. We set
X := X(A/F ) and for each integer i with 0 ≤ i ≤ n also Pi := GF/Fi , Ni := NG (Pi ) and
Xi := T F (A/Fi ).
For each such integer i, one can check that the natural restriction Ĥ −1 Pi , X → Ĥ −1 Pi+1 , Xand corestriction maps Ĥ −1 Pi+1 , X → Ĥ −1 Pi , X on Tate cohomology correspond under Proposition 3.1 to the homomorphisms Xi → Xi+1 and Xi+1 → Xi that are induced by the natural
restriction and corestriction homomorphisms on Galois cohomology.
Given this, the main result of Yakovlev in [16] implies that the diagram (1) determines theisomorphism class of the Zp [G]-module X up to addition of trivial source Zp [G]-modules with vertices
contained in P . To be more precise, we recall that [16, Theorem 2.4 and Lemma 5.2] combine togive the following result:Lemma 3.5. Assume that M and M 0 are any finitely generated Zp [G]-modules for which, for each
i with 0 ≤ i < n, there are isomorphisms of Zp [Ni ]-modules κi : Ĥ −1 (Pi , M ) → Ĥ −1 (Pi , M 0 ) thatlie in commutative diagramsĤ −1 (Pi , M )κiĤ −1 (Pi , M 0 )/ Ĥ −1 (Pi+1 , M ) Ĥ −1 (Pi , M ) o

κi+1

Ĥ −1 (Pi+1 , M )κi/ Ĥ −1 (Pi+1 , M 0 ) Ĥ −1 (Pi , M 0 ) o



κi+1

Ĥ −1 (Pi+1 , M 0 )where the horizontal arrows are the natural restriction and corestriction homomorphisms. Then
there are direct sum decompositions of Zp [G]-modules(4)

M = M1 ⊕ M2 ,andM 0 = M10 ⊕ M20

where M1 and M10 are isomorphic and M2 and M20 are both trivial source Zp [G]-modules with vertices
contained in P . In particular, if Ĥ −1 (Pi , M ) = 0 for each i, then M is itself a trivial source Zp [G]module with vertices contained in P .
By simply applying the final assertion of Lemma 3.5 to the module M = X we complete the proofof Theorem 2.7 (ii).
At this stage, to complete the proof of Theorem 2.7 (i) it suffices to assume that there is adecomposition of Zp [G]-modules X = M1 ⊕ M2 where M1 is uniquely determined up to isomorphism
and M2 is a trivial source Zp [G]-module, and then to show that M2 is determined up to isomorphism
as a Zp [P ]-module by the value of srk(A/Fi ) for each i with 0 ≤ i ≤ n. But, it is easy to see that any
trivial source Zp [P ]-module W is determined uniquely up to isomorphism by the integers rkZp (W Pi )for each i with 0 ≤ i ≤ n, and so the required fact follows directly from the equalities


 P




i
rkZp M2Pi = rkZp X− rkZp M1Pi = srk(A/Fi ) − rkZp M1Pi ,where the second equality follows directly from (3).
This completes the proof of Theorem 2.7.

10

DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

Remark 3.6. If A and F satisfy the hypotheses (a)–(e), then Lemma 3.4 implies that the resultof Theorem 2.7 can be applied to a variety of Galois extensions F 0 /k 0 , with k ⊆ k 0 ⊆ F 0 ⊆ F ,for which GF 0 /k0 has cyclic Sylow p-subgroups. The rank of X(A/L) and the behaviour of thenatural restriction and corestriction maps on T (A/L) as L varies over intermediate fields of F/K
has therefore a strong influence on the explicit structure of the Zp [G]-module X(A/F ). Nevertheless,
it is unlikely that this data alone will suffice to determine the structure of X(A/F ) as a Zp [G]-module
unless P is cyclic. For example, if G = P is non-cyclic of order p2 , then there exist (non-isomorphic)
abstract Zp [G]-lattices M1 and M2 such that for all subgroups J 0 and J with J 0 ≤ J ≤ G both00
rkZp (M1J ) = rkZp (M2J ) and the groups Ĥ −1 (J/J 0 , M1J ) and Ĥ −1 (J/J 0 , M2J ) vanish and yet only
M1 is a trivial source Zp [G]-module. (We are grateful to Griff Elder [5] for pointing out to us theexistence of such examples).
3.2.2. The proof of Corollary 2.8. The key point here is that the given structure of G implies thatfor any character ρ in Ir(G) there exists a subgroup Gρ of G which contains P and a linear character0ρ0 of Gρ such that ρ = IndG
Gρ (ρ ) (for a proof of this fact see [15, II-22, Exercice] and the argumentof [15, II-18]).
0
We set kρ := F Gρ , Fρ := F ker(ρ ) and Kρ := F J where J contains ker(ρ0 ) and is such thatJ/ ker(ρ0 ) is the Sylow p-subgroup of Gρ / ker(ρ0 ). We also set G0ρ := Gρ / ker(ρ0 ) and Pρ0 := J/ ker(ρ0 ).
Then kρ ⊆ F P = K and hence also Kρ ⊆ K since the degree of Kρ /kρ is prime to p, the extension
Fρ /kρ is cyclic and the multiplicity with which ρ occurs in the Qcp [G]-module Qcp ⊗ X(A/F ) is equal
to the multiplicity mρ0 with which ρ0 occurs in the Qcp [G0ρ ]-module Qcp ⊗ X(A/Fρ ).
In addition, the result of Lemma 3.4(i) combines with the given hypotheses on A and F to imply
that the pair A/kρ and Fρ satisfy the hypotheses (a)–(e). From the proof of Theorem 2.7 we therefore
know that knowledge of the diagram (1) with F/k replaced by Fρ /kρ determines the isomorphismclass of the Zp [G0ρ ]-module X(A/Fρ ) up to decompositions of the form (4). Such decompositions,with M = X(A/Fρ ) say, determine mρ0 up to the multiplicity m0ρ0 with which ρ0 occurs in the scalar
extension Qcp ⊗ M2 of the trivial source Zp [G0ρ ]-module M2 . We also write m0ρ00 for the multiplicity
with which the character ρ00 ∈ Ir(Pρ0 ) induced by the restriction of ρ0 to J occurs in the Qcp [Pρ0 ]module Qcp ⊗ M2 . Then, since M2 is also a trivial source Zp [Pρ0 ]-module and Pρ0 is a p-group, theexplicit structure of such modules (and in particular the fact that Zp [Pρ0 /H] is indecomposable forany subgroup H of Pρ0 ) implies that m0ρ0 ≤ m0ρ00 ≤ m01P 0 . Sinceρ0P0
Qp ⊗ M2 ρ ⊆ Qp ⊗ X(A/Fρ )Pρ ∼= Qp ⊗ X(A/Kρ ) ⊆ Qp ⊗ X(A/K)(with the isomorphism a consequence of (3)), one therefore hasP0
m0ρ0 ≤ m01P 0 = dimQp Qp ⊗ M2 ρ ≤ dimQp Qp ⊗ X(A/K) = srk(A/K).
ρThis proves the first assertion of Corollary 2.8.
We next note that if Selp (A/K) is finite, then srk(A/K) = 0 and so the above argument showsthat the multiplicity mρ with which any ρ in Ir(G) occurs in the Qcp [G]-module Qcp ⊗ X(A/F ) is
uniquely determined by a suitable diagram of the form (1). This implies the second assertion in
Corollary 2.8 because the Qp [G]-module structure of Qp ⊗ X(A/F ) is determined up to isomorphismby the multiplicities mρ for all ρ in Ir(G). This completes the proof of Corollary 2.8.
3.3. The proof of Theorem 2.9. For each ρ in Ir(P ) we set Fρ := F ker(ρ) . Then for each subgroupF
H with ker(ρ) < H ≤ P one has πFF H = πF ρH ◦ πFFρ and so the module T Fρ (A/F H ) is a quotient ofT F (A/F H ). Thus if T F (A/F H ) vanishes for all H ≤ P (as assumed in claim (iii)), then T Fρ (A/F H )vanishes for any given ρ and H as above (as assumed in claim (ii)).

STRUCTURE OF SELMER GROUPS

11

L
m
Since the Qcp [P ]-module Qcp ⊗ X(A/F ) is isomorphic to ρ∈Ir(P ) Vρ ρ the assertion of claim (iii)
is therefore an immediate consequence of claim (ii). Indeed, if one has mρ ≤ ρ(1) · srk(A/K) for allρ ∈ Ir(P ) (as would follow from claim (ii) under the hypotheses of claim (iii)), thenX
X
srk(A/F ) =mρ · ρ(1) ≤ρ(1)2 · srk(A/K) = |P | · srk(A/K),ρ∈Ir(P )ρ∈Ir(P )as required.
It therefore suffices to prove claims (i) and (ii) of Theorem 2.9 and to do this we argue by induction
on |P |, using the result of Lemma 3.4(ii).
To prove claim (i) by induction on n, it suffices to show that
srk(A/Fn−1 ) + (p − 1) dimFp T Fn (A/Fn−1 ) ≤ srk(A/Fn )≤ p · srk(A/Fn−1 ) + (p − 1) dimFp T Fn (A/Fn−1 ) .
To do this we set H := GFn /Fn−1 and note that (as per the discussion in §4.1 below) any Zp [H]lattice is isomorphic to a direct sum of the form N := Zap ⊕ Zp [H]b ⊕ Rc for suitable non-negativePH
integers a, b and c and with R := Zp [H]/h∈H h . This implies that rk(N ) = a + b andrk(N ) = a + pb + (p − 1)c so rk(N H ) + (p − 1)c ≤ rk(N ), with equality if and only if b = 0, andrk(N ) ≤ p · rk(N H ) + (p − 1)c, with equality if and only if a = 0. In addition, an easy computationshows that Ĥ −1 (H, R) ∼= Fp and so the group Ĥ −1 (H, N ) ∼= Ĥ −1 (H, R)c is an Fp -space of dimension
c. Given these observations for N = X(A/F ) and the result of Lemma 3.4(ii), the above displayed
inequalities follow by applying Proposition 3.1 with J = H.
Turning to the proof of claim (ii) we write ρ for the character of P/ ker(ρ) that inflates to give ρ.
Then, since mF ker(ρ) ,ρ = mF,ρ , Lemma 3.4 allows us to replace F by F ker(ρ) and thus assume that
T F (A/L) = 0 for all L with K ⊆ L ( F . This is what we do in the rest of this argument.
We next note that P is monomial and hence that for each ρ in Ir(P ) there is a non-trivial subgroupJ
J of P , a cyclic quotient Q = J/J 0 of J and a character ψ in Ir(Q) with ρ = IndPJ ◦ Inf Q (ψ). We
0
set F 0 := F J and K 0 := F J . Then mF,ρ = mF 0 ,ψ and Lemma 3.4(ii) implies that A/k andF 0 satisfy hypotheses (a)–(e) with respect to K 0 . In addition, for each L with K 0 ⊆ L ( F 00
the module T F (A/K 0 ) is a quotient of T F (A/K 0 ) and so vanishes under our present hypotheses.
Since Q = GF 0 /K 0 is cyclic we may therefore apply Theorem 2.7 (ii) to F 0 /K 0 in order to deducethat X(A/F 0 ) is a trivial source Zp [Q]-module. From the explicit structure of a trivial source
Zp [Q]-module it is then clear that mF,ρ = mF 0 ,ψ ≤ mF 0 ,1Q = srk(A/K 0 ). Since ρ(1) is equal to[P : J] = [K 0 : K] it thus suffices to prove that srk(A/K 0 ) ≤ [K 0 : K] · srk(A/K).
To prove this we use the fact that, as P is a p-group, there exists a finite chain of subgroups
J = J0 E J1 E · · · E Jn = P(5)

in which |Ji+1 /Ji | < |P | for all i with 0 ≤ i < n. For each such i we set F i := F Ji . Then
our hypotheses combine with Lemma 3.4(ii) to imply that, for each i, A/F i+1 and F i satisfy the
hypotheses of Theorem 2.9(iii) and hence, by induction, that srk(A/F i ) ≤ |Ji+1 /Ji | · srk(A/F i+1 ) =[F i : F i+1 ] · srk(A/F i+1 ). It follows thatsrk(A/K 0 ) = srk(A/F 0 ) ≤

n−1
Y


|Ji+1 /Ji | · srk(A/F n )i=0= [Jn : J0 ] · srk(A/K) = [K 0 : K] · srk(A/K)
as required. This completes the proof of Theorem 2.9.

12

DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

3.4. The proof of Theorem 2.2. From (the proof of) Theorem 2.9(iii) we know that the conditionsin Theorem 2.2(ii) and (iii) are equivalent. Next we note that if X(A/F ) is a projective Zp [G]-module,
then it is a free Zp [P ]-module (by Swan’s Theorem [4, (32.1)]), and hence also a free Zp [H]-module
for each H ≤ P . It follows that Qp ⊗X(A/F ) is a free Qp [P ]-module and hence that both srk(A/F ) =|H| · srk(A/F H ) for each H ≤ P and mρ = ρ(1) · srk(A/K) for each ρ ∈ Ir(P ). In addition, theprojectivity of X(A/F ) implies that it is also cohomologically trivial, and so Proposition 3.1 implies
that T F (A/F H ) = 0 for all H ≤ P . At this stage we have verified the implications (i) =⇒ (ii) ⇐⇒(iii) =⇒ (iv) =⇒ (v) in Theorem 2.2 and so it suffices to prove (v) =⇒ (i).
To prove this implication we note that the Zp [G]-module X(A/F ) is both finitely generated
and Zp -free and hence that [1, Chapter VI, (8.7), (8.8) and (8.10)] combine to imply that it is a
projective Zp [G]-module if it is a cohomologically trivial Zp [P ]-module. We next claim that it isa cohomologically trivial Zp [P ]-module if it is a free Zp [C]-module for each C ≤ P with |C| = p.
The point here is that for any non-trivial subgroup J of P there exists a subgroup C of P which isnormal in J and has orderp and hence also a Hochschild-Serre
spectral sequence in Tate cohomologyĤ a J/C, Ĥ b (C, X(A/F )) =⇒ Ĥ a+b J, X(A/F ) : thus if X(A/F ) is a free Zp [C]-module, thenĤ m J, X(A/F ) = 0 for all integers m, as required. It therefore suffices to prove that the hypotheses
in claim (v) imply that X(A/F ) is a free Zp [C]-module for each subgroup C of P of order p. Now,
under the given conditions, the hypotheses of Theorem 2.7 (ii) are satisfied by the data A/F C , F/F C
and so we know that X(A/F ) is a trivial source Zp [C]-module, and hence of the form Zp [C]m ⊕ Znp
for suitable non-negative integers m and n. It follows that srk(A/F ) = p m + n and, using (3), thatsrk(A/F C ) = m + n and so the equality srk(A/F ) = p · srk(A/F C ) in claim (v) implies n = 0, andhence that X(A/F ) = Zp [C]m is a free Zp [C]-module, as required.
This completes the proof of Theorem 2.2.
3.5. The proof of Corollary 2.5. We first note that, under the assumed validity of hypotheses (a)–(e), Proposition 3.1 shows that for every subgroup J of P the natural homomorphism X(A/F )J →X(A/F J ) is bijective.
In particular, if we now assume that X(A/F ) is a projective Zp [G]-module, then there is a direct
sum decomposition of Zp [G]-modules X(A/F ) ∼= T (A/F ) ⊕ X(A/F ) and X(A/F )J is Zp -free andso the above isomorphism induces an isomorphism T (A/F )J ∼= T (A/F J ). Now, under the assumed∼
vanishing of X(A/K)p the module T (A/K) = T (A/F )P vanishes and so Nakayama’s Lemma implies
T (A/F ), and hence also T (A/F J ) for each J ≤ P , vanishes. Taken in conjunction with the proofof Theorem 2.2 given above, this verifies the implications (iv) ⇐⇒ (v) ⇐⇒ (i) =⇒ (ii) ⇐⇒ (iii) in
Corollary 2.5 and hence reduces us to proving that (ii) =⇒ (v) under the stated hypotheses.
To prove this implication it suffices to prove that, under the hypotheses of Corollary 2.5, if
Qp ⊗X(A/F ) is a free Qp [P ]-module, then T (A/F ) vanishes and X(A/F ) is a cohomologically trivial
Zp [C]-module for each subgroup C of P of order p. To prove this we use induction on |P |. We thus fix
a subgroup C of P of order p and then choose a chain of subgroups as in (5) but with J replaced by C.
Then Lemma 3.4 shows that each set of data A/F i+1 and F i satisfies (a)–(e) and hence, by induction,
we can deduce that T (A/F C ) = 0. By Proposition 3.1, we have that Ĥ −1 C, X(A/F ) = 0. Since
Qp ⊗ X(A/F ) is by assumption a free Qp [C]-module, a Herbrand quotient argument then impliesthat Ĥ 0 C, X(A/F ) = 0 and hence that X(A/F ) is a cohomologically trivial Zp [C]-module (since,for example, the Tate cohomology of C is periodic of order two). By [1, Chapter VI, (8.7)] we may
now deduce that X(A/F ) is a projective Zp [C]-module and then the same argument as used at the
beginning of this section shows that T (A/F C ) = 0 implies T (A/F ) = 0. This completes the proofof Corollary 2.5.

STRUCTURE OF SELMER GROUPS

13

3.6. The proofs of Corollaries 2.10 and 2.11. To complete the proof of all of the results statedin §2 it now only remains to prove Corollaries 2.10 and 2.11.
3.6.1. The proof of Corollary 2.10. The given hypotheses imply that the main results of Mazur andRubin in [12] are valid. Their Theorem B thus shows that the Qp [P ]-module Qp ⊗ X(A/F ) has adirect summand that is isomorphic to Qp [P ]. Given this fact, claim (ii) follows immediately from theequivalence of claims (i) and (iii) in Corollary 2.5, whilst claim (i) is a straightforward consequenceof Theorem 2.7 (ii) and the fact that if P is cyclic then any trivial source Zp [P ]-module M for which
the associated Qp [P ]-module Qp ⊗ M has a free rank one direct summand must itself have a direct
summand that is isomorphic to Zp [P ].
3.6.2. The proof of Corollary 2.11. The first thing to note is that, since the extension Q(ζpn )/Q(ζp )is a p-extension, the given hypotheses imply (via Lemma 3.4) that the abelian variety A satisfies
the hypotheses (a)–(e) with F = Kdnd ,n , K = Q(ζpn ) and k equal to the subextension of Q(ζpn ) ofdegree pn−1 over Q (so that G = Gn and P = Pn ).
Given this, claim (i) follows directly from Theorem 2.7 (ii). In order to prove claim (ii), we firstnote that any non-linear character ρ of G is of the form IndG
P (ψ) for a linear character ψ of P . Fromthe explicit structure of a trivial source module over Zp [P ] it is then clear that mρ = mψ ≤ m1P =00
srk A/Q(ζpn ) . It is also clear that, if ρ0 = IndG
P (ψ ) for a linear character ψ of P , then mψ ≤ mψ 0
0
if and only if ker(ψ) ⊆ ker(ψ ).
Similarly, by using the fact that X(A/Kdnd ,n ) is a trivial source module in the setting of claim(iii), one finds that mρ ≤ ρ(1) mφ for any ρ in Ir(G) whose restriction to H has φ occurring with
non-zero multiplicity. In this inequality mφ denotes the multiplicity with which φ occurs in themodule Qcp ⊗ X A/Q(ζpn ) = Qcp ⊗ X(A/Kdnd ,n )P . By arguing as in the proof of Theorem 2.9(iii)it then follows that rkZp eφ · X(A/Kdnd ,n ) ≤ pnd · rkZp eφ · X(A/Q(ζpn )) , as required to complete
the proof of claim (iii).
4. Examples
To end the article we discuss several concrete examples in which the classification results of Heller
and Reiner in [9] can be combined with the results of §3.1 to make the result of Theorem 2.7 much
more explicit.
To do this we assume to be given a (possibly finite) pro-cyclic pro-p extension of number fields
F/k and an abelian variety A over k which satisfies the hypotheses (a)–(e) with respect to the field
K = k. We set G := GF/k and for each non-negative integer i we write Fi for the unique field withk ⊆ Fi ⊆ F and [Fi : k] equal to pi unless pi > |G| in which case we set Fi = F . For each such i wethen set ri := srk(A/Fi ) and define an integer ti by the equality pti := |T F(A/Fi )|. We also write
ι(G) for the number of isomorphism classes of indecomposable Zp [G]-lattices.
For each non-negative integer i we write Ri for the quotient Zp [Z/(pi )]/(T ) where T denotes
the sum in Zp [Z/(pi )] of all elements of (pi−1 )/(pi ) ⊂ Z/(pi ). For any integer i such that G
has a quotient Gi of order pi we regard Ri as a Zp [G]-module by means of the homomorphism
Zp [G] → Zp [Gi ] ∼= Zp [Z/(pi )] → Ri where the isomorphism is induced by choosing a generator ofGi (the precise choice of which will not matter in what follows) and the unlabelled arrows are thenatural projection maps.
4.1. |G| = p. In this case ι(G) = 3 (by [9, Theorem 2.6]) with representative modules Zp , Zp [G] andR1 . In addition, the groups Ĥ −1 (G, Zp ) and Ĥ −1 (G, Zp [G]) vanish and Ĥ −1 (G, R1 ) has order p. In
particular, if one sets δ := (r0 − r1 )/(p − 1), then a comparison of ranks and of Tate cohomology

14

DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

shows (via Proposition 3.1) that t1 ≤ δ and that there is an isomorphism of Zp [G]-modules
X(A/F ) ∼= Zrp1 +t1 −δ ⊕ Zp [G]δ−t1 ⊕ R1t1 .
4.2. |G| = p2 . In this case ι(G) = 4p + 1 (by [9, §4]) but one finds that the group Ĥ −1 (G, M )only vanishes for modules M in an explicit subset Υ of p + 2 of these isomorphism classes (see, forexample, Table 2 in [14]). Hence, if X(A/k)p vanishes, then t2 = 0 and Proposition 3.1 implies
X(A/F ) is a direct sum of modules from Υ. If one further assumes for example that r1 = r2 , then acomparison of ranks and Tate cohomology groups (over the subgroup of G of order p) of the modulesin Υ shows the existence of an integer s1 with both (p − 1)s1 = t1 and s1 ≤ r2 and such that there
is an isomorphism of Zp [G]-modules
X(A/F ) ∼= Zpr2 −s1 ⊕ (R2 , Zp , 1)s1
where the indecomposable module (R2 , Zp , 1) is an extension of R2 by Zp which corresponds to theimage of 1 in Ext1Zp [G] (R2 , Zp ) ∼= Z/p.
4.3. |G| > p2 . In this case ι(G) is infinite (as proved in [8]) but Proposition 3.1 still imposes verystrong restrictions on those indecomposable modules that can occur in the decomposition of themodules X(A/Fi ). For example, if Selp (A/k) vanishes, then the isomorphism X(A/Fi )Gi ∼= X(A/k)coming from Proposition 3.1 combines with Nakayama’s Lemma to imply that Selp (A/Fi ) vanishes
for all i ≥ 0.
In the following result we describe the next simplest case.
Proposition 4.1. Assume that the abelian variety A and the pro-cyclic extension F/k satisfy all of
the above hypotheses. Assume in addition that Selp (A/k) is both finite and non-trivial. Then either(i) X(A/Fi )p is non-trivial for all i ≥ 0, or(ii) there exists a natural number n, a strictly positive integer mn and for each integer i with1 ≤ i < n a non-negative integer mi such that the following conditions are satisfied.
(a) X(A/Fn )p is trivial;
Ln−1 mn−i(b) There is a decreasing filtration of X(A/F ) with associated graded object i=0 Rn−i.
Pia
In particular, for each integer i with 1 ≤ i ≤ n one has srk(A/Fi ) = a=1 ma (p −pa−1 );Ln−1(c) There is a decreasing filtration of X(A/k) Q
with associated graded object i=0 (Z/p)mn−i .
nIn particular, the order of Selp (A/k) is a=1 pma and its exponent divides pe with eequal to the number of integers ma that are strictly positive.
Proof. It is enough to assume the existence of a natural number m for which X(A/Fm )p , and hence
also T (A/Fm ), is trivial and then use this to prove all assertions in claim (ii). We write n for the
least such m and note that, since X(A/Fn )p is trivial, for each integer i with 0 ≤ i < n the group
X(A/Fi )p is finite and hence equal to T (A/Fi ).
We may also assume henceforth that F = Fn so that G = Gn . We then set X := X(A/F ) =X(A/F ) and for each integer i with 0 ≤ i ≤ n we write Gi for the subgroup of G of order pi and Gi
for the quotient of G of order pi (so that Gi ∼= G/Gn−i ).
Then for each integer i with 0 ≤ i < n there exists a non-negative integer mn−i and a short exact
sequence of Zp [G]-modules of the form
(6)

0→X

Gi+1

→X

Gi

m

n−i
→ Rn−i→ 0.

STRUCTURE OF SELMER GROUPS

15

Gn

In particular, since the finiteness of Selp (A/k) implies that X
vanishes, the Zp [G]-module X has
a decreasing filtration{0} = X n ⊆ X n−1 ⊆ · · · ⊆ X 0 = X
Gi

m

n−i
. Thiswith each X i := X a Zp [Gn−i ]-module and each quotient X i /X i+1 isomorphic to Rn−igives the filtration on X = X(A/F ) described in claim (ii)(b) and also implies that the Qp [G]-module
Li=nQp ⊗ X is isomorphic to i=1 (Qp ⊗ Ri )mi .
Now for each integer a with 1 ≤ a ≤ n and each subgroup H of G the module of invariants RaH
is equal to Ra if H ⊆ Gn−a and otherwise vanishes. For each integer i with 1 ≤ i ≤ n, by using (3),one therefore has
a=i
a=i

 X
X
n−i
srk(A/Fi ) = dimQp (Qp ⊗ X)G=
ma · dimQp (Qp ⊗ Ra ) =ma · (pa − pa−1 )

a=1

a=1

where the last equality follows from the fact that for each a there is a short exact sequence ofZp [G]-modules
0 → Zp [Ga−1 ] → Zp [Ga ] → Ra → 0(7)and hence that dimQp (Qp ⊗ Ra ) = |Ga | − |Ga−1 | = pa − pa−1 . This proves claim (ii)(b).
To prove claim (ii)(c) the key point is that Proposition 3.1 combines with the vanishing of T (A/F )and the finiteness of Selp (A/k) to give an isomorphism of finite abelian groups(8)Ĥ −1 (G, X) ∼= T F (A/k) = T (A/k) = X(A/k)

whilst the exact sequences (6) and (7) provide a means of explicitly computing Ĥ −1 (G, X). MoreG
G
precisely, since the module of invariants X vanishes, as does the module Rn−afor each integer awith 0 ≤ a < n, and since Tate cohomology with respect to G is periodic of order two, the exact
sequences (6) give rise to associated short exact sequences
0 → Ĥ −1 (G, X

Ga+1) → Ĥ −1 (G, X

Ga) → Ĥ −1 (G, Rn−a )mn−a → 0

whilst the exact sequence (7) gives rise to an isomorphismĤ −1 (G, Ra ) = Ĥ −1 (Ga , Ra ) ∼= Ĥ 0 Ga , Zp [Ga−1 ]∼= Z/p.
= Ĥ 0 Gn−a+1 /Gn−a , Zp ∼By combining the above exact sequences with the isomorphism (8), we thereby obtain a decreasing
filtration of X(A/k) of the sort described in claim (ii)(c).
Given this filtration, it is then easy to deduce the assertions in claim (ii)(c) concerning the orderand exponent of Selp (A/k).
At this stage it only remains to prove that the integer mn is strictly positive. However, if thisis not the case, then the sequence (6) with i = 0 implies that G1 acts trivially on the module X
and hence that the group Ĥ −1 (G1 , X) vanishes. Moreover, Lemma 3.4 implies that the data A
and F/Fn−1 satisfy the hypotheses (a)–(e) and hence, by applying Proposition 3.1 to this data, wededuce that the group T F (A/Fn−1 ) also vanishes. In particular, since T (A/F ) vanishes the group
T (A/Fn−1 ) = X(A/Fn−1 )p must also vanish and this contradicts the minimality of n.

Example 4.2. We give some simple examples showing that in certain situations the result of Proposition 4.1 becomes very explicit. To do this we assume the notation and hypotheses of Proposition
4.1. We also assume that Selp (A/k) has order p2 and that T (A/Fm ) vanishes for some integer m.
In this case Proposition 4.1(ii)(c) implies that the least such m is greater than or equal to two and16

DAVID BURNS, DANIEL MACIAS CASTILLO AND CHRISTIAN WUTHRICH

we assume, for simplicity, that this integer is equal to two. We then also assume that F = F2 andhence that G = G2 .
(i) If Selp (A/k) is cyclic of order p2 , then Proposition 4.1(ii)(c) implies that m1 = m2 = 1 andhence that X(A/F2 ) is an extension of R2 by R1 . Moreover, the Heller-Reiner classification of
Zp [G]-lattices shows (via Table 2 in [14]) that, up to isomorphism, there is a unique such extension
M for which Ĥ −1 (G, M ) is isomorphic to Z/p2 (rather than having exponent dividing p). Putting
everything together one finds that in this case X(A/F2 ) is isomorphic as a Zp [G]-module to theindecomposable module (R2 , R1 , 1) which corresponds to the image of 1 in Ext1Zp [G] (R2 , R1 ) ∼= Z/p.
Using the isomorphism X(A/F1 ) ∼= X(A/F2 )G1 ∼= (R2 , R1 , 1)G1 one then computes that the Zp [G]modules T (A/F1 ) and X(A/F1 ) are isomorphic to Z/p and R1 respectively.
(ii) If Selp (A/k) has exponent p and order p2 , then Proposition 4.1(ii)(c) implies that either m2 = 2
and m1 = 0 or that m2 = m1 = 1. In the former case X(A/F2 ) is isomorphic to R22 and hence
X(A/F1 ) ∼= X(A/F2 )G1 is isomorphic to (R22 )G1 ∼= (R2,G1 )2 ∼= (Z/p)2p . In the latter case the HellerReiner classification implies that X(A/F2 ) is isomorphic to either R1 ⊕ R2 or, for some integer iwith 0 < i < p, to the indecomposable Zp [G]-module (R2 , R1 , i) which corresponds to the image ofthe polynomial (1 − x)i in the group Ext1Zp [G] (R2 , R1 ) ∼= R1 /p ∼= (Z/p)[x]/(1 − x)p−1 .
∼If X(A/F2 ) = R1 ⊕ R2 , then one computes that X(A/F1 ) ∼= X(A/F2 )G1 is isomorphic as a
Zp [G]-module to (Z/p)p ⊕ R1 . In a similar way, if X(A/F2 ) ∼= (R2 , R1 , i) with0 < i < p, then
one computes that the abelian group T (A/F1 ) = T F (A/F1 ) ∼= Ĥ −1 G1 , X(A/F2 ) is isomorphic to
i+1
(Z/p)
and that the Zp [G]-module X(A/F1 ) is isomorphic to R1 .
Acknowledgements. It is a pleasure for us to thank Werner Bley and Griff Elder for helpful
discussions and correspondence concerning this work, and the anonymous referees for making several
useful suggestions and for correcting an error in one of our arguments.
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STRUCTURE OF SELMER GROUPS

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Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), no. 2, 206–221, Representation theory of groups, algebras,and orders (Constanţa, 1995).
King’s College London, Department of Mathematics, London WC2R 2LS, U.K.
E-mail address: david.burns@kcl.ac.uk
Instituto de Ciencias Matemáticas (ICMAT), 28049 Madrid, Spain.
E-mail address: daniel.macias@icmat.es
School of Math. Sciences, University of Nottingham, Nottingham NG7 2RD, U.K.
E-mail address: christian.wuthrich@nottingham.ac.uk