The structure of Selmer groups

Abstract

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

The results that I will describe here are motivated by a well-known theorem of Iwasawa. Let K be a finite extension of ℚ. Let K_∞/K be the cyclotomic ℤ_p-extension of K, where p is any prime. Thus K_∞ ⊆ K(μ_p^∞) and Γ = Gal(K_∞/K) ≅ ℤ_p, the additive group of p-adic integers. We let Λ = ℤ_p[[Γ]] be the completed group algebra of Γ over ℤ_p, which is isomorphic (noncanonically) to the formal power series ring ℤ_p[[T]]. Let M_∞ denote the maximal abelian pro-p extension of K_∞ unramified outside Σ = {p, ∞}. Let L_∞ denote the maximal abelian pro-p extension of K_∞ unramified at all primes of K_∞. Let X = Gal(M_∞/K_∞) and Y = Gal(L_∞/K_∞). In ref. 1, Iwasawa proves the following important result.

Theorem (Iwasawa):

(i) X and Y are finitely generated Λ-modules.

(ii) Rank_Λ(X) = r₂, where r₂ denotes the number of complex   primes of K.

(iii) Y is a torsion Λ-module.

(iv) X has no nonzero finite Λ-submodules.

We remark also that if K_∞/K is an arbitrary ℤ_p-extension, (i) and (iii) are true (due to Iwasawa). Statement (ii) should conjecturally be true. It is often referred to as the “Weak Leopoldt Conjecture” for K_∞/K and has the following interpretation. Let K_n denote the unique subfield of K such that K_n/K is cyclic of degree pⁿ. Let K̃_n denote the compositum of all ℤ_p-extensions of K_n. Then it is known that

where δ_n ≥ 0. Leopoldt’s Conjecture states that δ_n = 0. The Weak Leopoldt Conjecture states that δ_n is bounded as n → ∞, which is equivalent to the assertion that rank_Λ(X) = r₂. Also if statement (ii) holds, then so does statement (iv). (See proposition 4 of ref. 2.)

Returning to the cyclotomic ℤ_p-extension K_∞/K, we can restate Iwasawa’s theorem in terms of the Pontryagin duals

which are subgroups of H¹(G_K∞, ℚ_p/ℤ_p) = Hom(Gal(K_∞^ab/K_∞), ℚ_p/ℤ_p) defined by imposing certain local conditions. They are examples of what have come to be called “Selmer groups.” Iwasawa’s results then become: (i) Hom(X, ℚ_p/ℤ_p) and Hom(Y, ℚ_p/ℤ_p) are cofinitely generated Λ-modules. (ii) Hom(X, ℚ_p/ℤ_p) has Λ-corank r₂. (iii) Hom(Y, ℚ_p/ℤ_p) is Λ-cotorsion. (iv) Hom(X, ℚ_p/ℤ_p) has no proper Λ-submodules of finite index.

Now consider an abelian variety A defined over K with good, ordinary reductions at the primes of K lying over p. We denote by Sel_A(K_∞)_p the p-primary subgroup of the classical Selmer group for A over K_∞. Over K_n, this Selmer group is defined as follows.

where J_υ(K_n) =   (H¹(K_n,η, A[p^∞])/L_η). Here A[p^∞] denotes the p-power torsion points on A(K̄), υ runs over all primes of K, η over the primes of K_n lying over υ, and L_η denotes the image of the local Kummer homomorphism for A over the η-adic completion K_n,η of K_n. We define J_υ(K_∞) =     J_υ(K_n) (with obvious maps). Then Sel_A(K_∞)_p = Lim_n^→    Sel_A(K_n)_pcan be defined by

where Σ is a finite set of primes of K containing all primes of K where A has bad reduction as well as all primes dividing p or ∞. In the early 1970s, Mazur made the following conjecture, where K_∞/K is assumed to be the cyclotomic ℤ_p-extension.

Conjecture (Mazur): Sel_A(K_∞)_pis Λ-cotorsion.

One can weaken the assumption that A has good, ordinary reduction at all p dividing p. For each p|p, let h_p denote the height of the formal group associated to the Neron model for A over the integers in any finite extension of K_p where A achieves semistable reduction. Let g = dim(A). Then Mazur’s conjecture should be true if K_∞/K is the cyclotomic ℤ_p-extension and h_p = g for all primes p of K lying over p. Using results of ref. 3, one can show that Sel_A(K_∞)_p has positive Λ-corank if h_p > g for at least one p|p and for any ℤ_p-extension in which p is ramified. On the other hand, we should remark that there may exist noncyclotomic ℤ_p-extensions of K where Sel_A(K_∞)_p fails to be Λ-cotorsion even if A has good, ordinary reduction at all p|p. For example, this can occur if K_∞ is the anticyclotomic ℤ_p-extension of an imaginary quadratic field K. See ref. 4 for a discussion of this issue.

I now will describe various consequences if we assume that K_∞/K is the cyclotomic ℤ_p-extension, A has good, ordinary reduction at all primes of K over p, and Sel_A(K_∞)_p is Λ-cotorsion.

Consequence 1: The Λ-corank of H¹(K_Σ/K_∞, A[p^∞]) can be determined. For i = 0, 1, and 2, the Λ-modules Hⁱ(K_Σ/K_∞, A[p^∞]) are cofinitely generated and their coranks are related by their Euler–Poincaré characteristic

From this one gets the lower bound corank_Λ(H¹(K_Σ/K_∞, A[p^∞])) ≥ [K : ℚ]dim(A), with equality if and only if H²(K_Σ/K_∞, A[p^∞]) is Λ-cotorsion (since H⁰(K_Σ/K_∞, A[p^∞]) is obviously Λ-cotorsion). The calculation of the above global Euler–Poincaré characteristic is a consequence of results of Poitou and Tate for finite Galois modules over number fields. Using their results over local fields one can prove the following fact:

The definition of the Selmer group and the assumption that Sel_A(K_∞)_p is Λ-cotorsion then imply that corank_Λ(H¹(K_Σ/K_∞, A[p^∞])) = [K : ℚ]dim(A).

Consequence 2: The map γ:H¹(K_Σ/K_∞, A[p^∞]) → ⊕_υ∈Ω  J_υ(K_∞) is surjective. It is clear by comparing the Λ-coranks that the cokernel of this map will be Λ-cotorsion. The surjectivity is a consequence of studying the behavior of the corresponding cokernels over the K_n’s. One uses the known fact that A[p^∞]^G_K∞ is finite.

Consequence 3: In addition to the above assumptions, assume that at least one of the following hold: (i) A^t(K) has no p-torsion. (ii) For some υ ∤ p, A[p^∞]^I_υ is finite. (iii) For some p|p, e(p/p) ≤ p − 2. Then Sel_A(K_∞)_p has no proper Λ-submodules of finite index.

The proof of this consequence is discussed in a much more general context in ref. 5. In (ii), I_υ denotes the inertia subgroup of G_Kυ. If A is an elliptic curve, then (ii) is equivalent to A having additive reduction at some υ ∤ p. In (iii), e(p/p) is the ramification index; this assumption clearly holds if p > [K : ℚ] + 1. Assumption (i) also holds if p is sufficiently large, at least for a fixed A and K.

I want to add several remarks about these consequences. Consequence 1 should be true more generally, without the stringent assumptions made above. For any abelian variety defined over K and for any ℤ_p-extension K_∞/K, it is conjecturally true that H¹(K_Σ/K_∞, A[p^∞]) has Λ-corank equal to [K : ℚ]dim(A). This is equivalent to the assertion that H²(K_Σ/K_∞, A[p^∞]) is Λ-cotorsion. I will state later a much more general conjecture which will also include the Weak Leopoldt Conjecture stated earlier.

Concerning consequence 2, let Ω denote a finite set of primes of K not dividing p or ∞. Define a “nonprimitive” Selmer group Sel_A^Ω(K_∞)_p by

Thus Sel_A(K_∞)_p ⊆ Sel_A^Ω(K_∞)_p. Choose a finite set Σ as before, but also containing Ω. The surjectivity of γ gives an isomorphism

This isomorphism has an interesting interpretation in connection with Mazur’s “Main Conjecture” which asserts that the characteristic ideal of the Λ-module Sel_A(K_∞)_p^∧ is generated by a certain element θ_A ∈ Λ associated to the p-adic L-function for A over K. The existence of this p-adic L-function is known only under very restrictive hypotheses, e.g., if K = ℚ and A is a modular elliptic curve. But if it exists, then it is easy to construct a “nonprimitive” analogue with an interpolation property involving values of the Hasse–Weil L-function for A with the Euler factors for primes in Ω omitted. One could then define an element θ_A^Ω ∈ Λ. It turns out that θ_A^Ω = 𝒫_Ω⋅θ_A, where 𝒫_Ω generates the characteristic ideal of ⊕_υ∈Ω J_υ(K_∞)^∧. Thus the main conjecture is equivalent to a nonprimitive analogue asserting that the characteristic ideal of Sel_A^Ω(K_∞)_p^∧ is generated by θ_A^Ω.

Concerning consequence 3, some restrictive hypotheses are necessary. Here is an example to show that. Let K = ℚ(μ₅) and p = 5. Let E be the elliptic curve/ℚ of conductor 11 such that E(ℚ) is trivial. (The other two elliptic curves of conductor 11 are isogenous to E and contain a ℚ-rational point of order 5.) Now K_∞ = ℚ(μ₅∞) and Gal(K_∞/ℚ) ≅ Δ × Γ, where Δ = Gal(K/ℚ). Let ω denote the Teichmuller character of Δ. Then we can decompose Sel_A(K_∞)_p by the action of Δ:

One can determine the structure as a Λ-module of each factor. The result is that the Pontryagin dual of Sel_A(K_∞)_p^ωi is isomorphic to: Λ/5²Λ if i = 0, 0 if i = 1, the maximal ideal M ⊆ Λ/5²Λ (which has index 5) if i = 2, and ℤ/5ℤ if i = 3. Thus Sel_A(K_∞)_p has a Λ-submodule of index p = 5, the kernel of projecting to the ω³ factor.

It is interesting to note that Iwasawa’s μ-invariant for Sel_A(K_∞)_p is nonzero in the above example. Mazur first gave such examples in ref. 6, e.g. X₀(11) for p = 5, K = ℚ in which case he showed that μ = 1. The behavior of the μ-invariant under isogenies has been studied by Schneider (7) [and in a more general context by Perrin-Riou (8)]. Using their results, the following conjecture would predict the value of μ. Conjecture: μ can be made zero by isogeny. For X₀(11) and for K = ℚ, p = 5, the isogenous elliptic curve E = X₀(11)/μ₅ will have Sel_A(K_∞)_p = 0.

We will now formulate a general version of the Weak Leopoldt Conjecture, which gives a prediction of the Λ-corank of H²(K_Σ/K_∞, M) and, as a consequence, H¹(K_Σ/K_∞, M) for a very general Gal(K_Σ/K)-module M. The previously stated version is the special case M = ℚ_p/ℤ_p, on which Gal(K_Σ/K) acts trivially (and Σ = the set of primes of K lying over p or ∞). Various generalizations and special cases have been considered by Schneider (7), Greenberg (9), Coates and McConnell (10), and Perrin-Riou (11). The form we will give here is inspired by the thesis of McConnell. Let V be a finite dimensional ℚ_p-representation space for Gal(K_Σ/K), where Σ is a finite set of primes of K containing the primes over p and ∞. Let T be a Galois-invariant ℤ_p-lattice in V. Let d = dim_ℚp(V), d_υ^± = dim_ℚp(V^±) for the real primes of K, where V^± denotes the (±1)-eigenspaces for a complex conjugation above υ. Let M = V/T. Let K_∞/K be any ℤ_p-extension. It is known that both H¹(K_Σ/K_∞,, M) and H²(K_Σ/K_∞, M) are cofinitely generated Λ-modules (where Λ = ℤ_p[[Γ]], Γ = Gal(K_∞/K)) and that

where δ = r₂d + Σ_υ real d_υ⁻. (See ref. 9, proposition 3. The Euler–Poincaré characteristic for M over K_∞ is −δ.) For any prime υ of K, we let H_υ²(K_∞, M) = Lim_n^→(⊕_η|υH²(K_n,η, M)), where for each n, η runs over the primes of K_n lying over υ. One can prove the following result.

Proposition. The natural map H²(K_Σ/K_∞, M) → ⊕_υ∈ΣH_υ²(K_∞, M) is surjective. The kernel is Λ-cofree.

Our version of the Weak Leopoldt Conjecture is the following.

Conjecture. The map H²(K_Σ/K_∞, M) → ⊕_υ∈Σ H_υ²(K_∞, M) is an isomorphism.

One can show that if υ does not split completely in K_∞/K, then H_υ²(K_∞, M) = 0. However, primes can split completely in a ℤ_p-extension K_∞/K. For example, the archimedean primes of K will split completely. If K is an imaginary quadratic field, then every nonarchimedean prime υ of K not dividing p will split completely in one ℤ_p-extension of K. [This is obvious because Gal(K̃/K) ≅ ℤ_p² and the decomposition subgroup for υ is isomorphic to ℤ_p.] If υ is inert in K/ℚ, then υ splits completely in the anticyclotomic ℤ_p-extension of K. It is conjectured that for any other ℤ_p-extension of K at most one prime of K can split completely. (One can prove that at most two can.)

I discuss several special cases. First assume that K_∞/K is the cyclotomic ℤ_p-extension. Then the above conjecture states that

because nonarchimedean primes of K cannot split completely in K_∞/K. If p is odd, then H_υ²(K_∞, M) = 0 for υ|∞ and hence conjecturally H²(K_Σ/K_∞, M) = 0. If p = 2, then H_υ²(K_∞, M) can be nontrivial. It is (Λ/2Λ)-cofree and its (Λ/2Λ)-corank equals dim_ℤ/2ℤ(M(K_υ)/M(K_υ)_div), where M(K_υ) = H⁰(K_υ, M). In the special case where M = A[p^∞], where A is an abelian variety/K, M(K_υ)/M(K_υ)_div ≅ A(K_υ)/A(K_υ)_con, the group of connected components. This can be nontrivial if K_υ ≅ ℝ.

Let K_∞/K be any ℤ_p-extension. Consider M = ℚ_p/ℤ_p and Σ = {p, ∞}. Then H_υ²(K_∞, M) = 0 for all υ. Also,

where X = Gal(M_∞/K_∞), M_∞ denoting as before the maximal abelian pro-p extension of K_∞ unramified outside Σ. In this case, δ = r₂ and the above conjecture states that H¹(K_Σ/K_∞, M) should have Λ-corank r₂—i.e., rank_Λ(X) should equal r₂. This is the Weak Leopoldt Conjecture for the ℤ_p-extension K_∞/K, as stated earlier.

Let K_∞/K be any ℤ_p-extension. Consider M = μ_p^∞ = ℚ_p(1)/ℤ_p(1). Let Σ be a finite set containing all primes over p and ∞. Then it is not difficult to prove the Weak Leopoldt Conjecture for M and K_∞/K. (This proof is given in ref. 5.) In this case H_υ²(K_∞, M) has positive Λ-corank if υ is a nonarchimedean prime which splits completely in K_∞/K. Thus H²(K_Σ/K, M) can have positive Λ-corank.

Let M = A[p^∞]. Then H_υ²(K_∞, M) = 0 for all nonarchimedean υ (and for any ℤ_p-extension K_∞/K). The Weak Leopoldt Conjecture states that H²(K_Σ/K_∞, M) = 0 if p is any odd prime. There are some known cases. For example, if A is an elliptic curve/ℚ, K_∞/K is the cyclotomic ℤ_p-extension, and K/ℚ is abelian, then the conjecture is settled if A has complex multiplication and good, ordinary reduction at p [Rubin (12), where he proves Mazur’s conjecture in this case], if A has complex multiplication and good, supersingular reduction at p (McConnell), and, more generally if E is modular and has good reduction at p (Kato). All of these results use a nonvanishing theorem of Rohrlich for the Hasse–Weil L-function.

Let R₂(K_∞,Σ, M)=ker(H²(K_Σ/K_∞, M)→⊕_υ∈Σ H_υ²(K_∞, M)).

The Weak Leopoldt Conjecture for M and K_∞/K then asserts that R₂(K_∞, Σ, M) = 0. We want to state an equivalent version (inspired by McConnell). Let V* = Hom_ℚp(V, ℚ_p(1)) and T* = Hom_ℤp(T, ℤ_p(1)). Let M* = V*/T*. Define

Then, as a consequence of Tate’s global duality theorem, one can show that R₂(K_∞, Σ, M) and R₁(K_∞, Σ, M*) have the same Λ-corank. The Weak Leopoldt Conjecture then asserts that R₁(K_∞, Σ, M*) is Λ-cotorsion.

Let F_∞ denote the fixed field for the kernel of the action of G_K∞ on M*. Let H = Gal(F_∞/K_∞). Thus the action of G_K∞ on M* factors through H. Let L_F∞ denote the maximal abelian pro-p extension of F_∞, which is unramified at all primes of F_∞. Then G = Gal(F_∞/K) acts on Y_F∞ = Gal(L_F∞/F_∞). Here G is a p-adic Lie group, H is a closed subgroup, and one has an exact sequence 1 → H → G → Γ → 1. One also has the restriction map

The kernel of ρ is a subgroup of H¹(H, M*), which is Λ-cotorsion. We assume now that K_∞/K is the cyclotomic ℤ_p-extension. Then the cokernel of ρ is also Λ-cotorsion. Thus the Weak Leopoldt Conjecture would then be equivalent to asserting that Hom_H(Y_F∞, M*) is Λ-cotorsion. A theorem of Harris (13) states that Y_F∞ is a torsion-module over ℤ_p[[G₀]] in a certain sense, where G₀ is a suitable open subgroup of G. If we replace K by a finite extension contained in F_∞ (so that G_K acts trivially on M*[p]), then H is a pro-p group. Assume that μ(K_∞/K) = 0, which of course is a well-known conjecture of Iwasawa. This means that Y_K∞ = Gal(L_K∞/K_∞) is a finitely generated ℤ_p-module, where L_K∞ is the maximal abelian pro-p extension of K_∞ unramified everywhere (denoted by L_∞ earlier). By studying the map Y_F∞/I_HY_F∞ → Y_K∞, where I_H is the augmentation ideal of ℤ_p[[H]], and by using a version of Nakayama’s lemma, one finds that Y_F∞ must be a finitely generated ℤ_p[[H]]-module. But the Weak Leopoldt Conjecture for M (and for the cyclotomic ℤ_p-extension K_∞/K) would then follow because Hom_H(Y_F∞, M*) would consequently be cofinitely generated as a ℤ_p-module and therefore Λ-cotorsion.

Continuing to assume that K_∞/K is the cyclotomic ℤ_p-extension, let M*(t) denote the tth Tate twist, where t ∈ ℤ. Assume that μ_p ⊆ K (or μ₄ ⊆ K if p = 2). Then another equivalent form of the Weak Leopoldt Conjecture for M and K_∞/K is the following statement: R₁(K, Σ, M*(t)) is finite for all but finitely many t ∈ ℤ. Here

which has finite ℤ_p-corank for all t. This formulation illustrates the “Deformation” point of view since M*(t) = V*(t)/T*(t) and T*(t), t ∈ ℤ, are specializations of a representation Gal(K_Σ/K) → GL_d(Λ), which is a deformation of T* (the “cyclotomic” deformation as defined in ref. 14).

The Weak Leopoldt Conjecture for M and for an arbitrary ℤ_p-extension K_∞/K has two consequences, which are analogues of parts of Iwasawa’s theorem stated earlier. The first is the obvious consequence that one could then determine the Λ-corank of H²(K_Σ/K_∞, M) and hence of H¹(K_Σ/K_∞, M), in terms of the Euler–Poincaré characteristic δ for M and the ℤ_p-corank of the local Galois cohomology groups H²(K_υ, M) for those υ ∈ Σ which split completely in K_∞/K. The second consequence is the following result.

Proposition: Assume that the Weak Leopoldt Conjecture holds for M and K_∞/K. Then H¹(K_Σ/K_∞, M) has no proper Λ-submodule of finite index.

I would like to now discuss briefly Selmer groups associated to modular forms. To illustrate, consider Δ = Σ_n=1^∞ τ(n)qⁿ, where τ is Ramanujan’s tau-function. We let V denote V_p(Δ), the p-adic representation associated to Δ. Let M = V/T, where T = T_p(Δ) is a G_ℚ-invariant ℤ_p-lattice. Let Σ = {p, ∞}. Assume p is odd. Then the Selmer group for M over the cyclotomic ℤ_p-extension of ℚ has the following definition.

where L_p = H_f¹(ℚ_p,∞, M) = Lim_n^→ H_f¹(ℚ_p,n, M). Here ℚ_p,∞  = ∪_nℚ_p,n is the cyclotomic ℤ_p-extension of ℚ_p, ℚ_p,n is the nth layer. For any finite extension F/ℚ_p, H_f¹(F, M) denotes the image in H¹(F, M) of H_f¹(F_υ, V), the ℚ_p-subspace of H¹(F_υ, V) defined by Bloch and Kato. In the so-called ordinary case [which means p ∤ τ(p)], one can describe L_p as follows. It is known that there exists a one-dimensional ℚ_p-subspace W of V which is G_ℚp-invariant and such that V/W is unramified for the action of G_ℚp. Let N denote the image of W under the map V → M. Then it turns out that

In contrast, if p|τ(p), then it seems likely that H_f¹(ℚ_p,∞, M) = H¹(ℚ_p,∞, M). Then it would follow that S_M(ℚ_∞) = H¹(ℚ_Σ/ℚ_∞, M). This has been proven by Perrin-Riou if p∥τ(p).

If p ∤ τ(p), then S_M(ℚ_∞) is Λ-cotorsion (proved by Kato). We consider two ordinary primes: p = 11, p = 23. In ref. 15, I have calculated the structure of S_M(ℚ_∞) for these primes (even as a Λ-module for p = 11). As groups, S_M(ℚ_∞) ≅ ℚ_p/ℤ_p in both cases. The idea behind the calculation is to use certain congruences between modular forms: Δ ≡ f_E(mod 11), where f_E is the modular form of weight 2 associated to X₀(11), and Δ ≡ f_ρ(mod 23), where f_ρ is the weight 1 modular form associated to a certain dihedral two-dimensional Artin character. One can use an easily verified fact that S_M(ℚ_∞)[p]= S_M[p](ℚ_∞), where one defines the Selmer group for the finite Galois module M[p] in a way analogous to the definition of S_M(ℚ_∞), using the subgroup N[p] of M[p]. (One needs mild hypotheses on M to verify this fact.) One can calculate the Selmer group over ℚ_∞ for V_p(X₀(11)) and for V_p(ρ) (modulo ℤ_p-lattices). This allows one to show that in both cases S_M(ℚ_∞)has order p. One concludes that S_M(ℚ_∞) ≅ ℚ_p/ℤ_p by using the result that S_M(ℚ_∞) has no proper Λ-submodule of finite index [and hence S_M(ℚ_∞) cannot be finite]. A very general result of this nature is proved in ref. 9 under rather restrictive hypotheses, and much more generally in ref. 5. However, as indicated earlier, there are cases where such a result fails to be true.