The structure of Selmer groups

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) = r ₂, 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 p ⁿ. 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) = r ₂. 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 H ¹(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 r ₂. (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 _∞)_p is 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 H ¹(K _Σ/K _∞, A[p ^∞]) can be determined. For i = 0, 1, and 2, the Λ-modules H ⁱ(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(H ¹(K _Σ/K _∞, A[p ^∞])) ≥ [K : ℚ]dim(A), with equality if and only if H ²(K _Σ/K _∞, A[p ^∞]) is Λ-cotorsion (since H ⁰(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(H ¹(K _Σ/K _∞, A[p ^∞])) = [K : ℚ]dim(A).

Consequence 2: The map γ:H ¹(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 H ¹(K _Σ/K _∞, A[p ^∞]) has Λ-corank equal to [K : ℚ]dim(A). This is equivalent to the assertion that H ²(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. X ₀(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 X ₀(11) and for K = ℚ, p = 5, the isogenous elliptic curve E = X ₀(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 H ²(K _Σ/K _∞, M) and, as a consequence, H ¹(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 H ¹(K _Σ/K _∞,, M) and H ²(K _Σ/K _∞, M) are cofinitely generated Λ-modules (where Λ = ℤ_p[[Γ]], Γ = Gal(K _∞/K)) and that where δ = r ₂ d + 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 _υ ²(K _∞, M) = Lim_n ^→(⊕_η|υ H ²(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 H ²(K _Σ/K _∞, M) → ⊕_υ∈Σ H _υ ² (K_∞, M) is surjective. The kernel is Λ-cofree.

Our version of the Weak Leopoldt Conjecture is the following.

Conjecture. The map H ²(K _Σ/K _∞, M) → ⊕_υ∈Σ H _υ ² (K_∞, M) is an isomorphism.

One can show that if υ does not split completely in K _∞/K, then H _υ ²(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_p ² 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 _υ ²(K _∞, M) = 0 for υ|∞ and hence conjecturally H ²(K _Σ/K _∞, M) = 0. If p = 2, then H _υ ²(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) = H ⁰(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 _υ ²(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 = r ₂ and the above conjecture states that H ¹(K _Σ/K _∞, M) should have Λ-corank r ₂—i.e., rank_Λ(X) should equal r ₂. 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 _υ ²(K _∞, M) has positive Λ-corank if υ is a nonarchimedean prime which splits completely in K _∞/K. Thus H ²(K _Σ/K, M) can have positive Λ-corank.

Let M = A[p ^∞]. Then H _υ ²(K _∞, M) = 0 for all nonarchimedean υ (and for any ℤ_p-extension K _∞/K). The Weak Leopoldt Conjecture states that H ²(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 R ₂(K _∞,Σ, M)=ker(H ²(K _Σ/K _∞, M)→⊕_υ∈Σ H _υ ²(K _∞, M)).

The Weak Leopoldt Conjecture for M and K _∞/K then asserts that R ₂(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 R ₂(K _∞, Σ, M) and R ₁(K _∞, Σ, M*) have the same Λ-corank. The Weak Leopoldt Conjecture then asserts that R ₁(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 H ¹(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[[G ₀]] in a certain sense, where G ₀ 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 _H Y _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: R ₁(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 H ²(K _Σ/K _∞, M) and hence of H ¹(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 H ²(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 H ¹(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)q ⁿ, 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 _f ¹(ℚ_p,∞, M) = Lim_n ^→ H _f ¹(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 _f ¹(F, M) denotes the image in H ¹(F, M) of H _f ¹(F _υ, V), the ℚ_p-subspace of H ¹(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 _f ¹(ℚ_p,∞, M) = H ¹(ℚ_p,∞, M). Then it would follow that S _M(ℚ_∞) = H ¹(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 X ₀(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(X ₀(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.