On the Galois structure of Selmer groups

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 over extensions of k.


Introduction
Let A be an abelian variety defined over a number field k and write A t for the corresponding dual abelian variety.
Fix a prime number p. For each finite extension field F of k we write Sel p (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, over non-negative integers n, of the Selmer groups associated to the isogenies [p n ] of A over F . It is then 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 Sel p (A/F ). Denote by T (A/F ) the torsion subgroup of 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(A t /F ) p of A t over F , with the associated quotient group canonically isomorphic to Hom Z (A(F ), Z p ). In particular, if X(A/F ) p is finite, then T (A/F ) and X(A/F ) simply identify with X(A t /F ) p and Hom Z A(F ), Z p respectively.
Let now F/k be a Galois extension of group G. In this case we wish to study the structure of X(A/F ) as a G-module. We recall that describing the explicit Krull-Schmidt decomposition of Z p [G]-lattices that occur naturally in arithmetic is known to be a very difficult problem (see, for example, 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 structure of X(A/F ) as a Z p [G]-module. For example, one of our results characterises the projectivity of X(A/F ) as a Z p [G]-module (see Theorem 2.2 and Remark 2.4), whilst another characterises, for certain G, the conditions under which X(A/F ) is a trivial source module over Z p [G] (see Theorem 2.7). More generally, our methods can be used in some cases to give an explicit decomposition of X(A/F ) as a direct sum of indecomposable Z p [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 the intermediate fields of F/k. As a particular example of this, we show here that this information can in some cases be used to tighten the bounds on the growth of ranks of Selmer groups in dihedral towers of numbers fields that were proved by Mazur and Rubin in [12] (see Corollary 2.10). In a similar 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 rank over 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 Z p -extensions that are proved by Greenberg in [7] (for more details in this regard see Remark 2.4).

Statements of the results
In this section we state all of the main results that are to be proved in §3. . We fix a Q c p [Γ]-module V ψ of character ψ. We also write 1 Γ for the trivial character of Γ.
We also note that by a 'Z p [Γ]-lattice' we shall mean a Z p [Γ]-module that is both finitely generated and free over Z p .
For any abelian group M we write M tor for its torsion subgroup. For any prime p we write M [p] for the subgroup {m ∈ M : pm = 0} of M tor and set M p := Z p ⊗ Z M . For any finitely generated Z p -module M and any field extension F of Q p we set F ⊗ M := F ⊗ Zp M and write rk Zp (M ) for the Z p -rank dim Qp (Q p ⊗ M ) of M .
For any Galois extension of fields L/K we write G L/K in place of Gal(L/K). For each nonarchimedean place v of a number field we write κ v for its residue field. We also write F p for the field with 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. (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(κ v )[p] = 0; (e) No place of bad reduction for A /k is ramified in F/k. 2.3. The Galois structure of X(A/F ). We first recall from the introduction that X(A/F ) is defined as the Pontryagin dual of Sel p (A/F ) and that, if the group X(A/F ) p is finite, then the tautological short exact sequence simply identifies with the canonical short exact sequence For each ρ in Ir(P ) we write m ρ , or m F,ρ when we wish to be more precise, for the multiplicity with which ρ occurs in the representation Q c p ⊗ X(A/F ). For each subfield L of F we set srk(A/L) := dim Qp (Q p ⊗ X(A/L)).
For any fields L and L ′ with k ⊆ L ⊆ L ′ ⊆ F we also set where π L ′ L denotes the natural norm map. 2.3.1. Projectivity results. Our first results give an explicit characterisation of the conditions under which X(A/F ) can be a projective Z p [G]-module.
Theorem 2.2. If A and F satisfy hypotheses (a)-(e), then the following conditions are equivalent.  To be more precise, let F ∞ and k ∞ be the cyclotomic Z p -extensions of F and k, set ∆ := G F∞/k∞ , let A be an elliptic curve over k that has good ordinary reduction at all p-adic places and for any extension L of k in F ∞ set X(A/L) := Hom Zp Sel p (A/L), Q p /Z p . 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 Z p [∆]-module. In particular, by combining the same characterisation of projectivity 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 Z p [∆] and then deduces families of explicit relations between the multiplicities with which each ψ ∈ Ir(∆) occurs in Q c p ⊗ X(A/F ∞ ). However, even if the natural descent homomorphism Q p ⊗ X(A/F ∞ ) G F∞/F → Q p ⊗ X(A/F ) is bijective, in order to deduce results about the structure of the Q p [G]-module Q p ⊗ X(A/F ) one would need to describe Q p ⊗ X(A/F ∞ ) G F∞ /F as an explicit quotient of the Q p [∆]-module Q p ⊗ 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 in order 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. Since obtaining an explicit description (in terms of generators and relations) of any given Z p [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 that the 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.

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. 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 Z p [G/H] is an indecomposable Z p [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 one used 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 A 0 (T F ) and Sel 0 (T F ) tf defined in loc. cit. are trivial source Z p [G]-modules.
Theorem 2.7. Assume that A and F satisfy (a)-(e) and that P is cyclic of order p n . For each integer i with 0 ≤ i ≤ n, let F i denote the unique field with K ⊆ F i ⊆ F and [F : F i ] = p n−i . Then the following claims are valid.
where the upper and lower arrows are induced by the natural corestriction and restriction maps (for more details see §3.2 below).
-module with vertices contained in P . In particular, in this case X(A/F ) is a trivial source Z p [G]-module, and hence also a trivial source Z p [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 Z p [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 Z p [G]-lattices is infinite unless |G| divides p 2 and have only been able to give an explicit classification of all such modules in the special cases |G| = p and |G| = p 2 .
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/F i ) 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 can be applied to Galois extensions F ′ /k ′ with k ⊆ k ′ ⊆ F ′ ⊆ F and for which G F ′ /k ′ 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 Z p [G]-module X(A/F ).
The following result gives an explicit example of this phenomenon (and for more details in this regard 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 Q c p -valued character ρ of G there is a diagram of torsion groups of the form (1) which determines the multiplicity with which ρ occurs in the Q c p [G]-module Q c p ⊗ X(A/F ) up to an error of at most srk(A/K).
In particular, if the group Sel p (A/K) is finite, then the structure of the Q p [G]-module Q p ⊗ 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).
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 L = K and so the hypothesis in claim (ii) is satisfied if T F ker(ρ) (A/K) vanishes and hence, a fortiori, In the remainder of this section we specialise the extension F/k and variety A in order to obtain finer 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 is an 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.
-module with vertices contained in P that has a non-zero projective direct summand. In particular, in this case for all characters ρ in Ir(P ) one has -module and m ρ = 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 natural numbers m and n of fields K m,n d := Q p m √ d, ζ p n where d is a fixed integer, m ≤ n and ζ p n is a choice of primitive p n -th roots of unity in Q c . Such extensions have recently been investigated in the context of non-commutative Iwasawa theory.
Since K m,n d is a bicyclic p-extension of Q(ζ p ) there are several ways in which the results stated above can be used to restrict the structure of X(A/K m,n d ).
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 order p over Q(ζ p ), that the reduction of A at p has no point of order p over F p and that no Tamagawa Let n be any natural number for which X A/Q(ζ p n ) p vanishes and set Write P n and G n for the Galois groups of K n d ,n d over Q(ζ p n ) and over the subextension of Q(ζ p n ) of degree p n−1 over Q respectively. Then the following claims are valid.
is at most srk A/Q(ζ p n ) . Further if ρ and ρ ′ are any non-linear characters in Ir(G n ), then m ρ ≤ m ρ ′ if and only if ker(ρ) ⊆ ker(ρ ′ ). (iii) For any complementary subgroup H to P n in G n and any φ in

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 N G (J) for the normaliser of a subgroup J of G.
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 of Proposition 5.6 used by Greenberg in [6]. In this case the restriction homomorphism res F J F : However, the proof of Proposition 5.6 and the result in Proposition 4.3 in [6] show that res F J 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-anomalous and so the restriction map Sel Given this exact sequence and the definition of T F (A/F J ) as the cokernel of π F F J , the claimed isomorphism is a direct consequence of the fact that X(A/F ) J,tor is equal toĤ −1 J, X(A/F ) . Indeed, sinceĤ −1 J, X(A/F ) is finite and X(A/F ) is Z p -free, the latter equality follows immediately from the tautological exact sequence where the third arrow is induced by the action of the element g∈J g of Z p [G].  (2) to imply that and in particular that srk(A/F J ) = rk Zp X(A/F ) J . In fact, these statements at the rational level hold true in full generality and do not depend on the arguments or hypotheses required to prove the integral assertions of Proposition 3.1. For convenience and 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).
(i) Assume that A /k and F satisfy the hypotheses (a), (b), (e) and the second part of (d) (with respect 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 /k ′ and F ′ satisfy the hypotheses (a)-(e) with respect to K ′ . (ii) If A /k and F satisfy the hypotheses (a)-(e) (with respect to K) and K ⊆ K ′ , then A /k ′ and F ′ satisfy the hypotheses (a)-(e) with respect to K ′ .
Proof. We note first that the validity of hypotheses (a) and (e) for A /k and F implies their validity for A /k ′ and F ′ . This is clear for (e) and for (a) follows from the fact that F/K is a Galois extension of p-power degree and so the vanishing of A(K)[p] implies the vanishing of A(F )[p] and hence also that of A(K ′ )[p].
To consider hypothesis (b) we fix a place v ′ of bad reduction for A /K ′ and a place w of F above v ′ and write v for the place of K which lies beneath w. Then hypothesis (e) implies that v and v ′ are unramified in F/K and F/K ′ 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 also that of A /K ′ at v ′ .
In a similar way one can show that the second part of hypothesis (d) for A /K and F implies the analogous assertion for A /K ′ and F ′ . Indeed, if v ′ is any p-adic place of K ′ which ramifies in F ′ and w any place of F above v ′ , then the restriction v of w to K is ramified in F/K and hence (by assumption) A(κ v )[p] vanishes. This implies that A(κ w )[p] vanishes (since F/K is a Galois extension of p-power degree) and hence also that A(κ v ′ )[p] vanishes, as required.
In particular, since the validity of hypothesis (c) and the first part of hypothesis (d) for A /K ′ and F ′ 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 ′ by its Galois closure over K (if necessary) in order to assume, without loss of generality, that F ′ /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 /k ′ and F ′ . In addition, since F ′ /K is a Galois p-extension, the same arguments as used to prove claim (i) show that the hypotheses (a), (b) and the second part of (d) will also hold for A /K ′ and F ′ . 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 P i := G F/Fi , N i := N G (P i ) and For each such integer i, one can check that the natural restrictionĤ −1 P i , X →Ĥ −1 P i+1 , X and corestriction mapsĤ −1 P i+1 , X →Ĥ −1 P i , X on Tate cohomology correspond under Proposition 3.1 to the homomorphisms X i → X i+1 and X i+1 → X i 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 the isomorphism class of the Z p [G]-module X up to addition of trivial source Z p [G]-modules with vertices contained in P . To be more precise, we recall that [16, Theorem 2.4 and Lemma 5.2] combine to give the following result:  By simply applying the final assertion of Lemma 3.5 to the module M = X we complete the proof of Theorem 2.7 (ii).
At this stage, to complete the proof of Theorem 2.7 (i) it suffices to assume that there is a decomposition of Z p [G]-modules X = M 1 ⊕ M 2 where M 1 is uniquely determined up to isomorphism and M 2 is a trivial source Z p [G]-module, and then to show that M 2 is determined up to isomorphism as a Z p [P ]-module by the value of srk(A/F i ) for each i with 0 ≤ i ≤ n. But, it is easy to see that any trivial source Z p [P ]-module W is determined uniquely up to isomorphism by the integers rk Zp (W Pi ) for each i with 0 ≤ i ≤ n, and so the required fact follows directly from the equalities where the second equality follows directly from (3). This completes the proof of Theorem 2.7. The key point here is that the given structure of G implies that for any character ρ in Ir(G) there exists a subgroup G ρ of G which contains P and a linear character ρ ′ of G ρ such that ρ = Ind G Gρ (ρ ′ ) (for a proof of this fact see [15,, Exercice] and the argument of [15,).

Remark 3.6. If A and F satisfy the hypotheses (a)-(e), then Lemma 3.4 implies that the result of Theorem 2.7 can be applied to a variety of Galois extensions
We set k ρ := F Gρ , F ρ := F ker(ρ ′ ) and K ρ := F J where J contains ker(ρ ′ ) and is such that J/ ker(ρ ′ ) is the Sylow p-subgroup of G ρ / ker(ρ ′ ). We also set G ′ ρ := G ρ / ker(ρ ′ ) and P ′ ρ := J/ ker(ρ ′ ). Then k ρ ⊆ F P = K and hence also K ρ ⊆ K since the degree of K ρ /k ρ is prime to p, the extension F ρ /k ρ is cyclic and the multiplicity with which ρ occurs in the Q c 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 F ρ 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 F ρ /k ρ determines the isomorphism class of the Z p [G ′ ρ ]-module X(A/F ρ ) up to decompositions of the form (4). Such decompositions, with M = X(A/F ρ ) say, determine m ρ ′ up to the multiplicity m ′ ρ ′ with which ρ ′ occurs in the scalar extension Q c p ⊗ M 2 of the trivial source Z p [G ′ ρ ]-module M 2 . We also write m ′ ρ ′′ for the multiplicity with which the character ρ ′′ ∈ Ir(P ′ ρ ) induced by the restriction of ρ ′ to J occurs in the Q c p [P ′ ρ ]module Q c p ⊗ M 2 . Then, since M 2 is also a trivial source Z p [P ′ ρ ]-module and P ′ ρ is a p-group, the explicit structure of such modules (and in particular the fact that Z p [P ′ ρ /H] is indecomposable for (3)), one therefore has This proves the first assertion of Corollary 2.8. We next note that if Sel p (A/K) is finite, then srk(A/K) = 0 and so the above argument shows that the multiplicity m ρ with which any ρ in Ir(G) occurs in the Q c p [G]-module Q c p ⊗ 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 Q p [G]-module structure of Q p ⊗ X(A/F ) is determined up to isomorphism by the multiplicities m ρ 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 F ρ := F ker(ρ) . Then for each subgroup H with ker(ρ) < H ≤ P one has π F F H = π Since the Q c p [P ]-module Q c p ⊗ X(A/F ) is isomorphic to ρ∈Ir(P ) V mρ ρ the assertion of claim (iii) is therefore an immediate consequence of claim (ii). Indeed, if one has m ρ ≤ ρ(1) · srk(A/K) for all ρ ∈ Ir(P ) (as would follow from claim (ii) under the hypotheses of claim (iii)), then 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 To Turning to the proof of claim (ii) we write ρ for the character of P/ ker(ρ) that inflates to give ρ. Then, since m F ker(ρ) ,ρ = m F,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 subgroup J of P , a cyclic quotient Q = J/J ′ of J and a character ψ in Ir(Q) with ρ = Ind P J • Inf J Q (ψ). We set F ′ := F J ′ and K ′ := F J . Then m F,ρ = m F ′ ,ψ and Lemma 3.4(ii) implies that A /k and F ′ satisfy hypotheses (a)-(e) with respect to K ′ . In addition, for each L with K ′ ⊆ L F ′ the module T F ′ (A/K ′ ) is a quotient of T F (A/K ′ ) and so vanishes under our present hypotheses. Since Q = G F ′ /K ′ is cyclic we may therefore apply Theorem 2.7 (ii) to F ′ /K ′ in order to deduce that X(A/F ′ ) is a trivial source Z p [Q]-module. From the explicit structure of a trivial source Z p [Q]-module it is then clear that m F,ρ = m F ′ ,ψ ≤ m F ′ ,1Q = srk(A/K ′ ). Since ρ(1) is equal to [P : J] = [K ′ : K] it thus suffices to prove that srk(A/K ′ ) ≤ [K ′ : 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 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( To prove this implication we note that the Z p [G]-module X(A/F ) is both finitely generated and Z p -free and hence that [1, Chapter VI, (8.7), (8.8) and (8.10)] combine to imply that it is a projective Z p [G]-module if it is a cohomologically trivial Z p [P ]-module. We next claim that it is a cohomologically trivial Z p [P ]-module if it is a free Z p [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 is normal in J and has order p and hence also a Hochschild-Serre spectral sequence in Tate  To prove this implication it suffices to prove that, under the hypotheses of Corollary 2.5, if Q p ⊗X(A/F ) is a free Q p [P ]-module, then T (A/F ) vanishes and X(A/F ) is a cohomologically trivial Z p [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 thatĤ −1 C, X(A/F ) = 0. Since Q p ⊗ X(A/F ) is by assumption a free Q p [C]-module, a Herbrand quotient argument then implies thatĤ 0 C, X(A/F ) = 0 and hence that X(A/F ) is a cohomologically trivial Z p [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 Z p [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 proof of Corollary 2.5.
3.6. The proofs of Corollaries 2.10 and 2.11. To complete the proof of all of the results stated in §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 and Rubin in [12] are valid. Their Theorem B thus shows that the Q p [P ]-module Q p ⊗ X(A/F ) has a direct summand that is isomorphic to Q p [P ]. Given this fact, claim (ii) follows immediately from the equivalence of claims (i) and (iii) in Corollary 2.5, whilst claim (i) is a straightforward consequence of Theorem 2.7 (ii) and the fact that if P is cyclic then any trivial source Z p [P ]-module M for which the associated Q p [P ]-module Q p ⊗ M has a free rank one direct summand must itself have a direct summand that is isomorphic to Z p [P ].
3.6.2. The proof of Corollary 2.11. The first thing to note is that, since the extension Q(ζ p n )/Q(ζ p ) is a p-extension, the given hypotheses imply (via Lemma 3.4) that the abelian variety A satisfies the hypotheses (a)-(e) with F = K n d ,n d , K = Q(ζ p n ) and k equal to the subextension of Q(ζ p n ) of degree p n−1 over Q (so that G = G n and P = P n ).
Given this, claim (i) follows directly from Theorem 2.7 (ii). In order to prove claim (ii), we first note that any non-linear character ρ of G is of the form Ind G P (ψ) for a linear character ψ of P . From the explicit structure of a trivial source module over Z p [P ] it is then clear that m ρ = m ψ ≤ m 1 P = srk A/Q(ζ p n ) . It is also clear that, if ρ ′ = Ind G P (ψ ′ ) for a linear character ψ ′ of P , then m ψ ≤ m ψ ′ if and only if ker(ψ) ⊆ ker(ψ ′ ).
Similarly, by using the fact that X(A/K n d ,n d ) is a trivial source module in the setting of claim (iii), one finds that m ρ ≤ ρ(1) m φ for any ρ in Ir(G) whose restriction to H has φ occurring with non-zero multiplicity. In this inequality m φ denotes the multiplicity with which φ occurs in the module Q c p ⊗ X A/Q(ζ p n ) = Q c p ⊗ X(A/K n d ,n d ) P . By arguing as in the proof of Theorem 2.9(iii) it then follows that rk Zp e φ · X(A/K n d ,n d ) ≤ p n d · rk Zp e φ · X(A/Q(ζ p n )) , as required to complete the proof of claim (iii).

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 := G F/k and for each non-negative integer i we write F i for the unique field with k ⊆ F i ⊆ F and [F i : k] equal to p i unless p i > |G| in which case we set F i = F . For each such i we then set r i := srk(A/F i ) and define an integer t i by the equality p ti := |T F (A/F i )|. We also write ι(G) for the number of isomorphism classes of indecomposable Z p [G]-lattices.
For each non-negative integer i we write R i for the quotient Z p [Z/(p i )]/(T ) where T denotes the sum in Z p [Z/(p i )] of all elements of (p i−1 )/(p i ) ⊂ Z/(p i ). For any integer i such that G has a quotient G i of order p i we regard R i as a Z p [G]-module by means of the homomorphism where the isomorphism is induced by choosing a generator of G i (the precise choice of which will not matter in what follows) and the unlabelled arrows are the natural projection maps. 4.1. |G| = p. In this case ι(G) = 3 (by [9, Theorem 2.6]) with representative modules Z p , Z p [G] and R 1 . In addition, the groupsĤ −1 (G, Z p ) andĤ −1 (G, Z p [G]) vanish andĤ −1 (G, R 1 ) has order p. In particular, if one sets δ := (r 0 − r 1 )/(p − 1), then a comparison of ranks and of Tate cohomology shows (via Proposition 3.1) that t 1 ≤ δ and that there is an isomorphism of Z p [G]-modules 4.2. |G| = p 2 . In this case ι(G) = 4p + 1 (by [9, §4]) but one finds that the groupĤ −1 (G, M ) only vanishes for modules M in an explicit subset Υ of p + 2 of these isomorphism classes (see, for example, Table 2 in [14]). Hence, if X(A/k) p vanishes, then t 2 = 0 and Proposition 3.1 implies X(A/F ) is a direct sum of modules from Υ. If one further assumes for example that r 1 = r 2 , then a comparison of ranks and Tate cohomology groups (over the subgroup of G of order p) of the modules in Υ shows the existence of an integer s 1 with both (p − 1)s 1 = t 1 and s 1 ≤ r 2 and such that there is an isomorphism of Z p [G]-modules where the indecomposable module (R 2 , Z p , 1) is an extension of R 2 by Z p which corresponds to the image of 1 in Ext  Proof. It is enough to assume the existence of a natural number m for which X(A/F m ) p , and hence also T (A/F m ), 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/F n ) p is trivial, for each integer i with 0 ≤ i < n the group X(A/F i ) p is finite and hence equal to T (A/F i ).
We may also assume henceforth that F = F n so that G = G n . We then set X := X(A/F ) = X(A/F ) and for each integer i with 0 ≤ i ≤ n we write G i for the subgroup of G of order p i and G i for the quotient of G of order p i (so that G i ∼ = G/G n−i ).
Then for each integer i with 0 ≤ i < n there exists a non-negative integer m n−i and a short exact sequence of Z p [G]-modules of the form In particular, since the finiteness of Sel p (A/k) implies that X G n vanishes, the Z p [G]-module X has a decreasing filtration {0} = X n ⊆ X n−1 ⊆ · · · ⊆ X 0 = X with each X i := X G i a Z p [G n−i ]-module and each quotient X i /X i+1 isomorphic to R mn−i n−i . This gives the filtration on X = X(A/F ) described in claim (ii)(b) and also implies that the Q p [G]-module Q p ⊗ X is isomorphic to i=n i=1 (Q p ⊗ R i ) mi . Now for each integer a with 1 ≤ a ≤ n and each subgroup H of G the module of invariants R H a is equal to R a if H ⊆ G n−a and otherwise vanishes. For each integer i with 1 ≤ i ≤ n, by using (3), one therefore has where the last equality follows from the fact that for each a there is a short exact sequence of Z p [G]-modules and hence that dim Qp (Q p ⊗ R a ) = |G a | − |G a−1 | = p a − p a−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 Sel p (A/k) to give an isomorphism of finite abelian groups  (6) and (7) provide a means of explicitly computingĤ −1 (G, X). More precisely, since the module of invariants X G vanishes, as does the module R G n−a for each integer a with 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 ∼ =Ĥ 0 G n−a+1 /G n−a , Z p ∼ = Z/p. 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 order and exponent of Sel p (A/k).
At this stage it only remains to prove that the integer m n is strictly positive. However, if this is not the case, then the sequence (6) with i = 0 implies that G 1 acts trivially on the module X and hence that the groupĤ −1 (G 1 , X) vanishes. Moreover, Lemma 3.4 implies that the data A and F/F n−1 satisfy the hypotheses (a)-(e) and hence, by applying Proposition 3.1 to this data, we deduce that the group T F (A/F n−1 ) also vanishes. In particular, since T (A/F ) vanishes the group T (A/F n−1 ) = X(A/F n−1 ) p must also vanish and this contradicts the minimality of n. To do this we assume the notation and hypotheses of Proposition 4.1. We also assume that Sel p (A/k) has order p 2 and that T (A/F m ) 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