Studying the Growth of Mordell-Weil

To Kazuya Kato, with admiration

Barry Mazur and Karl Rubin

Received: December 1, 2002 Revised: February 16, 2003

Abstract. We study the growth of the Mordell-Weil groups E(K_{n}) of an elliptic curve E as K_{n} runs through the intermediate fields of a Z_{p}-extension. We describe those Z_{p}-extensions K_{∞}/K where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms.

2000 Mathematics Subject Classification: Primary 11G05, 11G40; Secondary 11R23, 14G05

1 Introduction

Fix an elliptic curve E over Q of conductor N, and an odd prime number p. Let K be a number field and K_{∞}/K a Z_{p}-extension. For n ≥ 0 denote by K_{n}/K the intermediate extension in K_{∞}/K of degree p^{n}. What is the rate of growth of the Mordell-Weil groups E(K_{n}) as n tends to ∞? The shape of these asymptotics for general number fields K and prime numbers p of good ordinary reduction is given by the following result, conditional on the Shafarevich-Tate conjecture (more specifically, on the finiteness of p-primary components of Shafarevich-Tate groups).

Proposition 1.1. If the Shafarevich-Tate conjecture is true, and E has good, ordinary reduction at p, then there is a non-negative integer r = r(E, K_{∞}/K) such that

rank_{Z}(E(K_{n})) = rp^{n} + O(1)

where the bound O(1) depends on E and K_{∞}/K, but not on n.

We will review the proof of this proposition in §3 below, Proposition 3.2. For convenience, if the above asymptotics hold, we will say that (E, K_{∞}/K) has growth number r = r(E, K_{∞}/K). In particular, if E does not have complex multiplication by a subfield of K, then (E, K_{∞}/K) has growth number zero if and only if E(K_{∞}) is finitely generated. When r(E, K_{∞}/K) > 0 let us say that the Mordell-Weil rank of E has positive growth relative to K_{∞}/K. Recent work has significantly increased our knowledge of questions related to growth number, and has allowed us to compute these growth numbers in two specific instances.

Example 1.2. When K is a real abelian field there is only one Z_{p}-extension K_{∞}/K (the cyclotomic Z_{p}-extension) and Kato’s celebrated work (cf. [K, Sc, Ru]) shows that E(K_{n}) stabilizes for large n. Therefore E(K_{∞}) is a finitely generated group and the growth number vanishes. This had been conjectured in [M1], at least for good, ordinary, primes p.

Example 1.3. Suppose K/Q is a quadratic imaginary field and let K_{∞}/K denote the anti-cyclotomic Z_{p}-extension. This is the unique Z_{p}-extension of K such that K_{∞}/Q is Galois with nonabelian Galois group. It was conjectured in [M2] (for the case of primes p of good, ordinary, reduction for E; but see, e.g., [V] for arbitrary p) that the growth number r(E, K_{∞}/K) is at most 2. Moreover, the conjectures in [M2] assert that

• the growth number r = 2 can only occur in the “exceptional case” where E has complex multiplication by K,

• if we are not in this “exceptional case” then the distinction between growth number 0 and 1 is determined by the root number in the functional equation satisfied by the relevant L-functions.

A good deal of this conjecture, as well as other information about the p-anti-cyclotomic arithmetic of elliptic curves over Q, is now established, thanks to work of Gross & Zagier, Kolyvagin, Perrin-Riou, Bertolini & Darmon, Vatsal, Cornut, Nekovár̆, and Zhang.

The object of this note is to indicate a slightly wider context where “signs of functional equations” would lead one to conjecture positive growth of Mordell-Weil ranks, and to formulate some open problems suggested by the recent work regarding anti-cyclotomic extensions of quadratic imaginary fields. Specifically, we will do two things. Let E be an elliptic curve over Q. For a number field K, call a Z_{p}-extension K_{∞}/K new over K if it is not the base extension of a Z_{p}-extension of a proper subfield of K.

(i) Assuming standard conjectures, we will consider new Z_{p}-extensions K_{∞}/K of number fields K that have the property that (in a sense to be made precise) the root number in the functional equation predicts positive growth for E relative to K_{∞}/K. These Z_{p}-extensions have a very tight structure. As we shall see, in such a situation K has an automorphism of order 2, and if K^{+} ⊂ K is the fixed field of this automorphism then K_{∞}/K^{+} is an infinite dihedral extension. To be sure, the “classical case” where K is a quadratic imaginary field is of this form, but–again as we shall see–there are quite a number of other “nonclassical” instances (in general K need not even be a CM field). An open problem is to prove positive growth in all (or even just one) of these “nonclassical” instances.

(ii) When K is a quadratic imaginary field we shall discuss a certain intriguing p-adic “line” in the Q_{p}-vector space E(K) ⊗ Q_{p} that can be constructed, under certain hypotheses, via universal norms from the anti-cyclotomic Z_{p}-extension of K. This line is stable under the action of complex conjugation and we offer a conjecture, and some beginning numerical evidence, about its “sign.” In the “nonclassical” instances alluded to above, where at present we cannot prove positive growth, if the growth number is 1 we can define a similar “sign” (see Remark 4.12).

We only consider here the growth of Mordell-Weil groups in Z_{p}-power extensions of number fields, i.e., infinite extensions F/K where Gal(F/K) is an abelian p-adic Lie group. There are recent studies (see for example [CH]) by Coates, Howson, and others concerning the growth of Mordell-Weil groups in infinite extensions F/K where Gal(F/K) is a nonabelian p-adic Lie group. We conclude this introduction with an example in which Proposition 1.1, along with the rest of this paper, does not apply.

Example 1.4. Suppose E has complex multiplication by the quadratic imaginary field K, p is a prime of good, supersingular reduction, and K_{∞}/K is the anticyclotomic Z_{p}-extension of K. Then standard conjectures imply (see the remark near the end of §1 of [G1])

rank_{Z}(E(K_{n})) = \begin{cases}\frac{p^{2[n+1/2]}−1}{p+1} + O(1) & if W (E/Q) = +1 \\ \frac{p^{2[n/2]+1}+1}{p+1} + O(1) & if W (E/Q) = −1 \end{cases}

where W (E/Q) is the root number in the functional equation of the L-function L(E/Q, s) (see §2.2) and [ ] is the greatest integer function.

2 Where can we find Mordell-Weil contribution in bulk?

Fix an odd prime p. Let E be an elliptic curve over Q, K/Q a finite Galois extension of degree prime to p, and L the pro-p-abelian extension of K that is the compositum of all Z_{p}-extensions of K in Q̄. Put Δ := Gal(K/Q) and Γ := Gal(L/K) ≅ Z_{p}^{d}, so we have d ≥ r_{2}(K) + 1 with equality if Leopoldt’s conjecture holds, where r_{2}(K) is the number of complex archimedean places of K. The extension L/Q is Galois, and “conjugation” in G := Gal(L/Q) induces a natural action of the group Δ on Γ. By the classical Schur-Zassenhaus Theorem, G is noncanonically isomorphic to the semi-direct product of Δ and Γ, the semi-direct product being formed via this conjugation action.

Example 2.1 (Mordell-Weil growth when K is totally real). Assume that Leopoldt’s conjecture holds. If K is (totally) real, it follows that d = 1 and the unique Z_{p}-extension of K is the base change of the p-cyclotomic Z_{p}-extension over Q. The action of Δ on Γ is therefore trivial. In this case it has been conjectured that E(L) is finitely generated (cf. [M1] when p is of good ordinary reduction for E) and this conjecture has been proved by Kato [K] in the case where K/Q is real abelian.

Example 2.2 (Mordell-Weil growth when K is totally complex). Assume that Leopoldt’s conjecture holds, and that K/Q is (totally) complex. Consider the Q_{p}-vector space of Z_{p}-extensions of K. That is, form the vector space

V (K) := Hom_{cont}(Γ, Q_{p}),

whose one-dimensional Q_{p}-subspaces are in one-to-one correspondence with Z_{p}-extensions of K. The vector space V (K) inherits a natural Q_{p}-linear representation of Δ. An exercise using class field theory gives us that the character of this Δ-representation is

Ind_{⟨σ⟩}^{Δ}χ ⊕ 1,

where 1 is the trivial character on Δ, σ ∈ Δ is a complex conjugation involution,

χ : ⟨σ⟩ → {±1} ⊂ C^{×}

is the nontrivial character on ⟨σ⟩ = {1, σ} ⊂ Δ, and Ind_{⟨σ⟩}^{Δ} refers to induction of characters from the subgroup ⟨σ⟩ to Δ. Here the “1” corresponds to the cyclotomic Z_{p}-extension and the “Ind_{⟨σ⟩}^{Δ}χ” cuts out a hyperplane in V (K) that we call the anti-cyclotomic hyperplane

V (K)^{anti−cyc} ⊂ V (K).

2.1 Representations in Mordell-Weil.

The “big” Mordell-Weil group E(L) of all L-rational points of E is a union of finite-rank G = Gal(L/Q)-stable subgroups, so one way of trying to understanding the growth of Mordell-Weil is to ask, for finite dimensional irreducible complex characters τ of G, whether or not the G-representation space V of τ occurs in E(F) ⊗ C for some intermediate Galois extension F/Q. Mackey’s criterion (cf. [Se] §8.2, Prop. 25) gives a description of all the irreducible characters on G in terms of induced characters. Here, in any event, is how to get quite a few irreducible characters of G. Define an action of Δ on the group of (continuous, hence finite order) characters of Γ as follows. For ψ ∈ Hom_{cont}(Γ, C^{×}) and δ ∈ Δ, define ψ^{δ} by the formula

ψ^{δ}(γ) := ψ(δ^{−1}(γ)) for γ ∈ Γ.

For such a ψ let H_{ψ} ⊂ Δ be the stabilizer of ψ, let K_{ψ} ⊂ K be the fixed field of H_{ψ}, and put G_{ψ} := Gal(L/K_{ψ}) ⊂ G.

Lemma 2.3. With notation as above, every ψ : Γ → C^{×} extends uniquely to a character ψ_{0} : G_{ψ} → C^{×} of p-power order (that is, the restriction of ψ_{0} to Γ is ψ).

Proof. Fix ψ. Since |Δ| is prime to p, G_{ψ} is (noncanonically) a semidirect product of H_{ψ} and Γ. More precisely, there is a subgroup H̃_{ψ} ⊂ G_{ψ} that projects isomorphically to H_{ψ}, and such that G_{ψ} = Γ · H̃_{ψ}. We define ψ_{0} by ψ_{0}(γh) = ψ(γ) for h ∈ H̃_{ψ} and γ ∈ Γ. Then ψ_{0} is a homomorphism because ψ^{h} = ψ for h ∈ H_{ψ}, ψ_{0} clearly has p-power order, and the restriction of ψ_{0} to Γ is ψ. If ψ′_{0} is another such extension of ψ, then ψ′_{0}ψ_{0}^{−1} is trivial on Γ, and hence is the inflation of a character of H_{ψ} of p-power order, which must be trivial.

Definition 2.4. Suppose ψ : Γ → C^{×}. Lemma 2.3 shows that ψ is the restriction to K of a character of a Z_{p}-extension of K_{ψ}, and clearly K_{ψ} is minimal with this property. We will call K_{ψ} the level of ψ, and we say that ψ is new of level K_{ψ}. We will say that ψ is generic if ψ is new of level K, i.e., if H_{ψ} is trivial. Let ψ_{0} : G_{ψ} → C^{×} be the extension of ψ given by Lemma 2.3, and define

ϕ_{ψ} := Ind_{G_{ψ}}^{G}ψ_{0},

the induced character from G_{ψ} to G. By [Se] §8.2, Proposition 25, ϕ_{ψ} is an irreducible character of G. We will also say that ϕ_{ψ} has level K_{ψ}, and that ϕ_{ψ} is generic if ψ is generic.

Proposition 2.5. Suppose ψ : Γ → C^{×}, and let ϕ_{ψ} = Ind_{G_{ψ}}^{G}ψ_{0} be the induced character given by Definition 2.4. Then ϕ_{ψ} is real-valued if and only if there is an element σ ∈ Δ such that ψ^{σ} = ψ^{−1}. Such a σ lies in the normalizer N(H_{ψ}) of H_{ψ} in Δ, and if ψ ≠ 1 then σ has order 2 in N(H_{ψ})/H_{ψ}.

Proof. Let ⟨ , ⟩_{G} denote the usual pairing on characters of a profinite group G. Then using Frobenius reciprocity ([Se] Theorem 13, §7.2) for the second equivalence, and [Se] Proposition 22, §7.3, for the third and fourth,

ϕ_{ψ} is real-valued ⟺ ⟨ϕ_{ψ}, ϕ̄_{ψ}⟩_{G} > 0
⟺ ⟨Res_{G_{ψ}}^{G}ϕ_{ψ}, ψ̄_{0}⟩_{G_{ψ}} > 0
⟺ ⟨Ind_{H_{σ}}^{G_{ψ}}(ψ^{σ})_{0}, ψ̄_{0}⟩_{G_{ψ}} > 0 for some σ ∈ Δ

where H_{σ} = H_{ψ} ∩ H_{ψ^{σ}} and (ψ^{σ})_{0} is the extension of ψ^{σ} to H_{σ} of Lemma 2.3

⟺ ψ^{σ} = ψ̄ = ψ^{−1} for some σ ∈ Δ.

If ψ^{σ} = ψ̄ then we have σH_{ψ}σ^{−1} = H_{ψ^{σ}} = H_{ψ}, so σ ∈ N(H_{ψ}), and ψ^{σ^{2}} = ψ so σ^{2} ∈ H_{ψ}. If further ψ ≠ 1 then (since p is odd) ψ^{σ} ≠ ψ, so σ ∉ H_{ψ}. This completes the proof.

Corollary 2.6. Suppose ψ ≠ 1. Then ϕ_{ψ} is real-valued if and only if there is a subfield K_{ψ}^{+} ⊂ K_{ψ} such that [K_{ψ} : K_{ψ}^{+}] = 2 and L^{ker(ψ_{0})} is Galois over K_{ψ}^{+} with dihedral Galois group. In particular, if ϕ_{ψ} is real-valued then [Δ : H_{ψ}] is even.

Proof. This follows directly from Proposition 2.5. If ϕ_{ψ} is real-valued we take K_{ψ}^{+} to be be the subfield of K_{ψ} fixed by the element σ of Proposition 2.5. Conversely, if there is such a field K_{ψ}^{+}, let σ be a lift to Δ of the nontrivial automorphism of K_{ψ}/K_{ψ}^{+} and apply Proposition 2.5.

2.2 Mordell-Weil and root numbers

Let τ be the character of an irreducible finite dimensional complex Grepresentation V. Appealing to a suitably general version of the conjecture of Birch and Swinnerton-Dyer, and assuming that the relevant L-function L(E, τ, s) has analytic continuation to the entire complex plane, we expect that the representation V occurs in E(F) ⊗ C (for F the fixed field of the G-representation V) if and only if the L-function L(E, τ, s) vanishes at s = 1. We also expect (but do not yet have, in general) a functional equation of the form

L(E, τ, s) = W (E, τ, s)L(E, τ̄, 2 − s),

where W (E, τ, s) is an explicit function involving the exponential function and the Γ-function. Even though the functional equation remains conjectural, there is an explicit definition of W (E, τ, s) (see for example [D, Ro]). If τ is realvalued, then the root number W (E, τ) := W (E, τ, 1) = ±1. Moreover, if W (E, τ, 1) = −1 it would follow that L(E, τ, s) vanishes at s = 1 (to odd order). Then, by the Birch and Swinnerton-Dyer conjecture we would expect that the representation space V occurs in E(L) ⊗ C. It is natural to ask if the really huge contribution to Mordell-Weil (if there is any) will come from this “expected” occurrence of representations in E(L). This leads us to seek out real irreducible characters τ with root number W (E, τ) = −1. Proposition 2.5 provides us with a substantial collection of real characters, and it remains to determine their root numbers.

Example 2.7. Suppose first that K is quadratic imaginary. We also suppose, for simplicity, that the discriminant of the field K, the conductor N of the elliptic curve E, and the prime number p are pairwise relatively prime. The unique nontrivial element σ ∈ Δ is complex conjugation, and the nontrivial character χ : ⟨σ⟩ = Δ → C^{×} corresponds in the usual way to the quadratic Dirichlet character χ_{K} attached to the field K. Choose ψ ∈ Hom_{cont}(Γ, C^{×})^{−} (the subgroup of homomorphisms on which σ acts as -1), and put ϕ_{ψ} = Ind_{Γ}^{G}ψ as in Definition 2.4. Then ϕ_{ψ} is real and generic by Proposition 2.5. Viewing det ϕ_{ψ} : G → C^{×} as a Dirichlet character in the usual way we have

W (E, ϕ_{ψ}) = (det ϕ_{ψ})(−N) · W (E)^{dim ϕ_{ψ}}

where W (E) = W (E, 1) is the root number of E (see for example Proposition 10 of [Ro]). But (det ϕ_{ψ}) is simply the inflation of χ, and dim ϕ_{ψ} = 2, so this identity simplifies to

W (E, ϕ_{ψ}) = χ_{K}(−N).  (1)

The surprising consequence of this formula, which we will see repeated even more generally, is that the root number does not depend upon much: it is independent of the choice of generic (real) character ψ. and the elliptic curve E only enters into the formula for the root number via its conductor. This is the case that has seen extraordinary progress recently via the detailed study of the tower of Heegner points. We will return to this in §4 of this article.

We now return to the general case, where K/Q is Galois with group Δ, L is the maximal Z_{p}^{d}-extension of K, and G = Gal(L/Q). From now on we suppose that the discriminant disc(K) of the field K, the conductor N of the elliptic curve E, and the prime p are pairwise relatively prime. If H ⊂ Δ let χ_{Δ/H} : G_{Q} → ±1 be the determinant of Ind_{H}^{Δ}1, the permutation representation of Δ on the set of left cosets Δ/H. (If H = {1} we will write simply χ_{Δ}.) We will view χ_{Δ/H} (and every other one-dimensional character of G_{Q}) as a Dirichlet character in the usual way.

Theorem 2.8. If ψ ≠ 1 and ϕ_{ψ} is real-valued, then W (E, ϕ_{ψ}) = χ_{Δ/H_{ψ}}(−N). In particular characters of the same level have the same root number.

Proof. We have ϕ_{ψ} = Ind_{G_{ψ}}^{G}ψ_{0} from Definition 2.4. By Corollary 2.6, dim(ϕ_{ψ}) = [G : G_{ψ}] = [Δ : H] is even. Hence Proposition 10 of [Ro] (“a special case of a well-known formula”) shows that

W (E, ϕ_{ψ}) = (det ϕ_{ψ})(−N)W (E)^{dim(ϕ_{ψ})} = (det ϕ_{ψ})(−N).

It remains to show that det ϕ_{ψ} = χ_{Δ/H_{ψ}}. We thank the referee for pointing out the following simple argument. Let 𝔭 be a prime of Q̄ above p. Since ψ_{0} has p-power order, ψ_{0} ≡ 1 (mod 𝔭) and so

det ϕ_{ψ} = det(Ind_{G_{ψ}}^{G}ψ_{0}) ≡ det(Ind_{G_{ψ}}^{G}1) = det(Ind_{H_{ψ}}^{Δ}1) = χ_{Δ/H_{ψ}} (mod 𝔭).

Since p is odd and both det ϕ_{ψ} and χ_{Δ/H_{ψ}} take only the values ±1, it follows that det ϕ_{ψ} = χ_{Δ/H_{ψ}}.

Proposition 2.9. The following are equivalent:

(i) χ_{Δ} ≠ 1, (ii) Δ has a nontrivial cyclic 2-Sylow subgroup,

(iii) Δ is the semi-direct product of a (normal) subgroup of odd order with a nontrivial cyclic 2-group.

If these (equivalent) conditions hold then Δ is solvable, K contains a unique quadratic subfield k/Q, and χ_{Δ} = χ_{k}, the quadratic character of k.

Proof. We have

χ_{Δ}(σ) = sign(π_{σ})

where π_{σ} is the permutation of Δ given by left multiplication by σ. If σ ∈ Δ has order d, then the permutation π_{σ} is a product of |Δ|/d d-cycles, so

χ_{Δ}(σ) = sign(π_{σ}) = (−1)^{(d−1)|Δ|/d} = \begin{cases}−1 & if d is even and |Δ|/d is odd, \\ 1 & otherwise. \end{cases}

Thus χ_{Δ}(σ) = −1 if and only if the cyclic subgroup generated by σ contains a nontrivial 2-Sylow subgroup of Δ, and so (i) is equivalent to (ii). If (i) and (ii) hold, then ker(χ) is a (normal) subgroup of index 2 in Δ, which also has a cyclic 2-Sylow subgroup. Proceeding by induction we get a filtration

Δ = Δ_{0} ⊃ Δ_{1} ⊃ ⋯ ⊃ Δ_{k}

where [Δ_{i} : Δ_{i+1}] = 2 and |Δ_{k}| is odd. We claim that Δ_{k} is normal in Δ. For if H is a conjugate of Δ_{k}, then |H| is odd so we see by induction that for every i ≥ 1 the map H → Δ_{i−1}/Δ_{i} is injective, i.e., H ⊂ Δ_{i}. Thus H ⊂ Δ_{k}, so Δ_{k} is normal. Therefore the Schur-Zassenhaus Theorem shows that Δ is the semidirect product of Δ_{k} with a 2-Sylow subgroup. This shows that (ii) implies (iii)^{1}, and it is immediate that (iii) implies (ii). Suppose now that conditions (i)-(iii) hold. It follows from (iii) and the Feit-Thompson theorem that Δ is solvable. By (i), χ_{Δ} = χ_{k} for some quadratic field k ⊂ K, and by (ii), K contains at most one quadratic field. This completes the proof.

Corollary 2.10. Suppose that ϕ_{ψ} is real and generic. Then W (E, ϕ) = −1 if and only if the equivalent conditions of Proposition 2.9 hold and χ_{k}(−N) = −1.

Proof. This is immediate from Theorem 2.8 and Proposition 2.9.

Definition 2.11. Following Theorem 2.8, if F ⊂ K we define W (E/F) to be the common root number of all real characters ϕ_{ψ} of level F. (The proof of Theorem 2.8 shows that this common root number is also the root number in the functional equation of L(E/F, s).)

1 The fact that (ii) implies (iii) was proved by Frobenius in §6, p. 1039 of [F], by the same method we give here. Another proof was given by Burnside in [Bu], Corollary II of §244, p.

327. We thank Persi Diaconis for pointing us toward these references.

Documenta Mathematica · Extra Volume Kato (2003) 585–607

In particular if F = K, then W (E/K) = χ_{k}(−N) if Δ has nontrivial cyclic 2-Sylow subgroup (where k is the quadratic field inside K), and W (E/K) = 1 otherwise.

Suppose now that we are in such a situation where W (E/K) = −1. From the discussion above we expect significant growth of Mordell-Weil rank as we ascend the tower of intermediate finite extensions contained in Z_{p}^{d}-extensions that correspond to the −1-eigenspace in V (K) of some element of order 2 in Δ. We next work out the elementary features of two examples that are more general than the case of K/Q quadratic imaginary, and for which we do not, as yet, have any satisfactory theory of Mordell-Weil growth.

Example 2.12 (K/Q is a complex abelian Galois extension). In this case Leopoldt’s conjecture is known to be true, there is a unique complex conjugation involution, and its −1-eigenspace is the entire anti-cyclotomic hyperplane.

Question 2.13. Can one use towers of Heegner points in Shimura curves over totally real fields to account for (at least some of) the expected Mordell-Weil growth as one ascends the finite intermediate extensions of the anti-cyclotomic hyperplane?

Example 2.14 (K/Q is a complex S_{3}-extension). To guard against complacency, we wish to mention this relatively simple case, where no analogue of the detailed results known in the quadratic imaginary case is currently available. Suppose Δ ≅ S_{3}, the symmetric group on three letters, and suppose that W (E/K) = −1. Let k be the quadratic field contained in K. Leopoldt’s conjecture holds for K (for “easy” reasons: if Leopoldt’s conjecture failed the natural homomorphism from global units to local units would be trivial, which it is not). We may decompose the four-dimensional Δ-representation space V (K) into the sum of two Δ-stable planes

V (K) = V (K)_{new} ⊕ V (k)

where the new plane V (K)_{new} ⊂ V (K)^{anti−cyc} is of codimension one in the anti-cyclotomic hyperplane. There are three involutions σ, σ′, σ′′ in Δ, none of which act as scalars either on V (K)_{new} or on V (k). Let L_{σ}/K denote the unique Z_{p}^{2}-extension of K on whose Galois group σ acts as −1, and similarly for σ′ and σ′′. We have, then, three Z_{p}^{2}-extensions, L_{σ}/K, L_{σ′}/K and L_{σ′′}/K, sub-extensions of the anti-cyclotomic Z_{p}^{3}-extension of K. Each of these Z_{p}^{2}- extensions is the compositum of the “old” anti-cyclotomic Z_{p}-extension of the quadratic subfield k ⊂ K and the unique Z_{p}-extension of K that corresponds to a line in V (K)_{new} that is a −1-eigenspace for one of the three involutions in Δ.

Challenge 2.15. As one ascends the intermediate fields of finite degree in each of these Z_{p}^{2}-extensions we expect significant growth in the ranks of the corresponding Mordell-Weil groups. Find this Mordell-Weil contribution.

Remark 2.16. Theorem 2.10 fails in the situation of Example 1.4 because in that case E has complex multiplication by K, so the conductor of E is not relatively prime to the discriminant of K.

3 Where is the support of the Selmer module?

Keep the notation of the previous section. In particular we continue to assume that the conductor of E, the discriminant of K, and the prime p are pairwise relatively prime.

3.1 Selmer modules

If F is an abelian extension of K we put Λ_{F} = Z_{p}[[Gal(F/K)]]; if [F : K] is finite this is just the group ring Z_{p}[Gal(F/K)], and if Gal(F/K) ≅ Z_{p}^{d} then Λ_{F} is noncanonically isomorphic to a power series ring Z_{p}[[T_{1}, …, T_{d}]]. If F′ ⊂ F then Λ_{F′} is naturally a quotient of Λ_{F}. For every number field F let S_{p}(E/F) ⊂ H^{1}(F, E[p^{∞}]) be the classical Selmer group, which sits in the center of the exact sequence

0 ⟶ E(F) ⊗ Q_{p}/Z_{p} ⟶ S_{p}(E/F) ⟶ � (E/F)[p^{∞}] ⟶ 0  (2)

where � (E/F)[p^{∞}] is the p-part of the Shafarevich-Tate group of E over F. We extend this definition to possibly infinite algebraic extensions of Q by passing to a direct limit

S_{p}(E/F) := \underrightarrow{lim}_{0} F′ ⊂ F S_{p}(E/F′)

where F′ ranges through the finite extensions of Q in F. If F/K is Galois then S_{p}(E/F) has a natural Z_{p}[[Gal(F/K)]]-module structure. Specifically, S_{p}(E/L) is a Λ_{L}-module. We will refer to S_{p}(E/L) as the discrete Selmer module attached to L/K. If K_{∞} ⊂ L is a Z_{p}-extension of K we will write K_{n} ⊂ K_{∞} for the subfield of degree p^{n} over K, and we have the Λ_{K_{∞}}-module

S_{p}(E/K_{∞}) = \underrightarrow{lim}_{0} nS_{p}(E/K_{n}).

The following Control Theorem is a strengthening by Greenberg ([G2] Proposition 3.1) of a theorem of the first author [M1].

Theorem 3.1 (Control Theorem). Suppose that K_{∞} is a Z_{p}-extension of K, and E has good ordinary reduction at p. Then the restriction maps H^{1}(K_{n}, E[p^{∞}]) → H^{1}(K_{∞}, E[p^{∞}]) induce maps

S_{p}(E/K_{n}) ⟶ S_{p}(E/K_{∞})^{Gal(K_{∞}/K_{n})}

whose kernels and cokernels are finite and bounded independently of n.

For every F we form the corresponding compact Selmer group

X(E/F) := Hom(S_{p}(E/F), Q_{p}/Z_{p}).

Then X(E/L) is a finitely generated Λ_{L}-module, and if K_{∞} is a Z_{p}-extension of K then X(E/K_{∞}) is finitely generated over Λ_{K_{∞}}.

Proposition 3.2. Suppose K_{∞} is a Z_{p}-extension of K. If E has good ordinary reduction at p, and � (E/K_{n})[p^{∞}] is finite for every n, then the growth number r(E, K_{∞}/K) defined in Proposition 1.1 is equal to the rank of the compact Selmer Λ_{K_{∞}}-module X(E/K_{∞}).

Proof. Write Λ_{∞} = Λ_{K_{∞}} and Λ_{n} = Λ_{K_{n}} = Z_{p}[Gal(K_{n}/K)]. The Control Theorem (Theorem 3.1) shows that there is a map with finite kernel and cokernel

X(E/K_{∞}) ⊗_{Λ_{∞}} Λ_{n} → X(E/K_{n}).

If � (E/K_{n})[p^{∞}] is finite then (2) shows that

rank_{Z}(E(K_{n})) = rank_{Z_{p}}(X(E/K_{n})) = rank_{Z_{p}}(X(E/K_{∞}) ⊗_{Λ_{∞}} Λ_{n}),

and the structure theory of Λ_{K_{∞}}-modules shows that

rank_{Z_{p}}(X(E/K_{∞}) ⊗_{Λ_{∞}} Λ_{n}) = p^{n}rank_{Λ_{∞}}(X(E/K_{∞})) + O(1).

3.2 The support of the Selmer module

Definition 3.3. Recall that Γ = Gal(L/K) ≅ Z_{p}^{d}, so Λ_{L} = Z_{p}[[Γ]]. The group Δ acts naturally and Z_{p}-linearly on Γ. If F ⊂ K let L_{F} denote the compositum of all Z_{p}-extensions of F. The group Aut(F) of automorphisms of F acts on Γ_{F} := Gal(L_{F} /F). If σ ∈ Aut(F) has order 2, we let Γ^{σ,−} denote the maximal quotient of Γ on which σ acts as −1, and L_{F}^{σ,−} ⊂ L_{F} the extension of F. The minus locus in Spec Λ_{L} is the closed subscheme of Spec Λ_{L}

⋃_{{F ⊂K \atop W (E/F)=−1}} ⋃_{{σ∈Aut(F) \atop σ^{2}=1,σ≠1}} Spec Λ_{KL_{F}^{σ,−}},

where we view Spec Λ_{KL_{F}^{σ,−}} ⊂ Spec Λ_{L} via the projection map Λ_{L} ↠ Λ_{KL_{F}^{σ,−}}.

Using the root number considerations above, the following conjecture follows from a suitably general version of the Birch and Swinnerton-Dyer conjecture.

Conjecture 3.4. The support of the compact Selmer module X(E/L) in Spec Λ_{L} contains the minus locus.

Example 3.5. For each quadratic imaginary field F ⊂ K such that χ_{F} (−N) = −1, the minus locus contains a Z_{p}-line corresponding to the anticyclotomic Z_{p}- extension of F, and Heegner points should provide the corresponding part of the Selmer group. On the other hand, if χ_{F} (−N) = +1 for every quadratic field F contained in K, then Theorem 2.8 shows that W (E/F) = +1 for every subfield F of K, and the minus locus is empty. In Example 2.14, the minus locus consists of the union of the 3 Z_{p}-planes Spec Λ_{L^{σ}}, Spec Λ_{L^{σ′}}, and Spec Λ_{L^{σ′′}}.

Let

MW_{p}(E/L) := Hom(E(L), Z_{p}) = Hom(E(L) ⊗ Q_{p}/Z_{p}, Q_{p}/Z_{p}),

a quotient of X(E/L). We will say that L/K has unexpected large Mordell-Weil if there exists a subscheme of Krull dimension greater than 1 in Spec Λ_{L} that is in the support of MW_{p}(E/L), but in the complement of the minus locus.

Question 3.6. Are there Galois extensions K/Q such that L/K has unexpected large Mordell-Weil?

We will call an abelian extension F/K a Z_{p}-power extension of K if Gal(F/K) = Z_{p}^{k} for some k ≥ 0.

Question 3.7. Are there Z_{p}-power extensions L_{1}, …, L_{k} of K, and a finite subset Ψ ⊂ Hom(Γ, C^{×}), such that

{ψ : Γ → C^{×} : ψ occurs in E(L) ⊗ C} = ∪_{i} Hom(Gal(L_{i}/K), C^{×}) ∪ Ψ?

Remark 3.8. Question 3.7 is motivated by work of Monsky, who was the first to study what the support of a Selmer module (in his case, an ideal class group module) over a Z_{p}^{d}-extension should look like when d ≥ 2. Iwasawa theory suggests that there should be an ideal A ⊂ Λ such that ψ occurs in E(L)⊗C if and only if ψ(A) = 0 (where we fix an embedding Q̄_{p} ↪ C so that we can view ψ as a character into Q̄_{p}). Monsky (Theorem 2.6 of [Mo]) showed that there is a set {L_{1}, …, L_{k}} of Z_{p}-power extensions of K, and ψ_{i}, …, ψ_{k} ∈ Hom(Γ, C^{×}) such that

{ψ : ψ(A) = 0} = ∪_{i=1}^{k} ψ_{i} · Hom(Gal(L_{i}/K), C^{×}).

Combining this with the root number calculation of Theorem 2.8 leads to Question 3.7. One can also ask the following variant of Question 3.7.

Question 3.9. Is there a collection L_{1}, …, L_{k} of Z_{p}-power extensions of K (not necessarily distinct) such that for every finite extension F of K in L we have

rank_{Z}(E(F)) = ∑_{i=1}^{k}[F ∩ L_{i} : K] + O(1)?

Here the bound O(1) should depend only on E and K, not on F.

4 Where are the universal norms in Mordell-Weil?

Fix for this section a Z_{p}-extension K_{∞}/K, and let Γ := Gal(K_{∞}/K) and Λ := Λ_{K_{∞}} = Z_{p}[[Γ]]. As before, K_{n} will denote the extension of K of degree p^{n} in K_{∞}, and Λ_{n} = Λ_{K_{n}} = Z_{p}[Gal(K_{n}/K)].

4.1 Universal norms

Definition 4.1. For m ≥ n consider the (Z_{p}-linear) norm maps (or, perhaps, “trace maps” since one usually writes Mordell-Weil groups additively)

N_{m,n} : E(K_{m}) ⊗ Z_{p} ⟶ E(K_{n}) ⊗ Z_{p}.

Define

U(E, K_{∞}/K_{n}) := ⋂_{m≥n} N_{m,n}(E(K_{m}) ⊗ Z_{p}) ⊂ E(K_{n}) ⊗ Z_{p}.

Passing to the projective limits we define a Gal(K/Q)-semi-linear Λ-module

U(E/K_{∞}) := \underleftarrow{lim}_{0} n E(K_{n}) ⊗ Z_{p} = \underleftarrow{lim}_{0} n U(E, K_{∞}/K_{n}),

and U(E, K_{∞}/K_{n}) is the image of the projection U(E/K_{∞}) → E(K_{n}) ⊗ Z_{p}.

Theorem 4.2. Suppose E has good ordinary reduction at p, and � (E/K_{n})[p^{∞}] is finite for every n. Let r = rank_{Λ}(X(E/K_{∞})). Then

(i) U(E/K_{∞}) has finite index in a free Λ-module of rank r,

(ii) U(E, K_{∞}/K_{n}) ⊗ Q_{p} is a free Λ_{n} ⊗ Q_{p}-submodule of E(K_{n}) ⊗ Q_{p} of rank r.

Proof. Write simply X for the finitely generated Λ-module X(E/K_{∞}), and X_{n} = X ⊗ Λ_{n}. It follows from the Control Theorem (Theorem 3.1) that there is an injection with finite cokernel bounded independently of n

Hom(X(E/K_{n}), Z_{p}) ⟶ Hom(X_{n}, Z_{p}).

Further, it follows from (2) that if � (E/K_{n})[p^{∞}] is finite then

Hom(X(E/K_{n}), Z_{p}) ≅ E(K_{n}) ⊗ Z_{p}.

Taking inverse limits we get an injective map

U(E/K_{∞}) = \underleftarrow{lim}(E(K_{n}) ⊗ Z_{p}) ⟶ \underleftarrow{lim}(Hom(X_{n}, Z_{p})) =: U(X)

with finite cokernel, where U(X) is the abstract universal norm Λ-module studied in Appendix A.2. Since U(X) is free of rank r (Proposition A.20), this proves (i).

Consider the diagram

\xymatrix{U(E/K_{∞}) ⊗ Q_{p} \ar[r] \ar[d]^{≅} & U(E/K_{∞}) ⊗ Λ_{n} ⊗ Q_{p} \ar[r] \ar[d]^{≅} & E(K_{n}) ⊗ Q_{p} \ar[d] \\ U(X) ⊗ Q_{p} \ar@{->>}[r] & U(X) ⊗ Λ_{n} ⊗ Q_{p} \ar@{^{(}->}[r] & Hom(X_{n}, Q_{p})}

where the horizontal maps are the natural projections. The lower right-hand map is injective by Proposition A.21, so the upper right-hand map is injective, which proves (ii).

4.2 Anticyclotomic Z_{p}-extensions of quadratic imaginary fields

From now on we assume that K is quadratic imaginary, and K_{∞} is the anticyclotomic Z_{p}-extension of K (the unique Z_{p}-extension, Galois over Q, on which the nontrivial element of Gal(K/Q) acts as −1). Let Λ = Z_{p}[[Gal(K_{∞}/K)]]. We assume in addition that E has good ordinary reduction at p, that � (E/K_{n})[p^{∞}] is finite for every n, and that every prime dividing the conductor N of E splits in K. The last assumption guarantees that W (E/K) = χ_{K}(−N) = −1.

Theorem 4.3 (Vatsal [V], Cornut [Co]). With assumptions as above, rank_{Λ}(X(E/K_{∞})) = 1.

Let E(K)^{±} ⊂ E(K) denote the +1 and −1 eigenspaces for the action of Δ = Gal(K/Q) on E(K). Thus E(K)^{+} = E(Q), and E(K)^{−} ≅ E^{(K)}(Q) where E^{(K)} is the quadratic twist of E by K/Q. Also write r_{±} = rank_{Z}(E(K)^{±}), and let W (E/Q) = ±1 denote the root number in the functional equation of L(E/Q, s).

Corollary 4.4. The universal norm subgroup U(E, K_{∞}/K) ⊗ Q_{p} is a onedimensional Q_{p}-subspace of E(K) ⊗ Q_{p}, contained either in E(K)^{+} ⊗ Q_{p} or E(K)^{−} ⊗ Q_{p}.

Proof. By Theorems 4.2(ii) and 4.3, dim_{Q_{p}}(U(E, K_{∞}/K)⊗Q_{p}) = 1. It is clear from the definition that U(E, K_{∞}/K) ⊗ Q_{p} is stable under Δ, so it must lie in E(K)^{±} ⊗ Q_{p}.

Definition 4.5. Following Corollary 4.4, if U(E, K_{∞}/K)⊗Q_{p} ⊂ E(K)^{+} ⊗Q_{p} (resp., E(K)^{−} ⊗ Q_{p}) we say that the sign of the p-anti-cyclotomic norms in E(K) is +1 (resp., −1). In the language of Appendix A.1, the sign of the p-anti-cyclotomic norms is sign(U(E/K_{∞})), the sign of the Δ-semi-linear Λ-module U(E/K_{∞}).

Theorem 4.6 (Nekovár̆ [N]). Under the hypotheses above,

(−1)^{r_{±}} = ±W (E/Q) and rank_{Z}(E(K)) is odd.

Proof. Using (1), the root numbers in the functional equations of L(E/Q, s), L(E^{(K)}/Q, s), and L(E/K, s) are, respectively, W (E/Q), χ_{K}(−N)W (E/Q), and χ_{K}(−N). By Theorem 2.10 and our assumption that all primes dividing N split in K (the “Heegner hypothesis”), χ_{K}(−N) = χ_{K}(−1) = −1. Thus the conclusions of the theorem follow from the Parity Conjecture (which predicts, for every number field F, that rank_{Z}(E(F) ≡ ord_{s=1}L(E, s) (mod 2)), which under our hypotheses was proved by Nekovár̆.

Definition 4.7. By Theorem 4.6, rank_{Z}(E(K)^{+}) ≠ rank_{Z}(E(K)^{−}). We define

ϵ(E/K) = \begin{cases}+1 & if rank_{Z}(E(K)^{+}) > rank_{Z}(E(K)^{−}), \\ −1 & if rank_{Z}(E(K)^{−}) > rank_{Z}(E(K)^{+}). \end{cases}

Conjecture 4.8 (Sign Conjecture). We have

sign(U(E/K_{∞})) = ϵ(E/K).

In other words U(E, K_{∞}/K) ⊗ Q_{p} lies in the larger of E(K)^{+} ⊗ Q_{p} and E(K)^{−} ⊗ Q_{p}, and in particular sign(U(E/K_{∞})) is independent of p.

Remark 4.9. Here is a brief heuristic argument that supports the Sign Conjecture. It is straightforward to prove that U(E, K_{∞}/K) is contained in the nullspace of the (symmetric, bilinear) p-anti-cyclotomic height pairing (cf. [MT])

⟨ , ⟩_{anti−cycl} : E(K) ⊗ Z_{p} × E(K) ⊗ Z_{p} ⟶ Gal(K_{∞}/K) ⊗_{Z_{p}} Q_{p}.

Moreover, this height pairing is compatible with the action of Gal(K/Q) in the sense that if τ is the nontrivial automorphism of K, then

⟨τ(v), τ(w)⟩_{anti−cycl} = −⟨v, w⟩_{anti−cycl}.

It follows that E(K)^{+} ⊗ Z_{p} and E(K)^{−} ⊗ Z_{p} are each isotropic under the anticyclotomic height pairing, and the null-space of ⟨ , ⟩_{anti−cycl} is the sum of the left and right nullspaces of the restriction of ⟨ , ⟩_{anti−cycl} to

E(K)^{+} ⊗ Z_{p} × E(K)^{−} ⊗ Z_{p} ⟶ Gal(K_{∞}/K) ⊗_{Z_{p}} Q_{p}.  (3)

It seems natural to conjecture that the pairing (3) is as nondegenerate as possible. (For a precise statement of this conjecture, and further discussion, see §3 of [BD1]). This “maximal nondegeneracy” would imply that (3) is either left or right nondegenerate, depending upon which of the two spaces E(K)^{−} or E(K)^{+} is larger, and this in turn would imply the Sign Conjecture.

4.3 Heegner points and the Sign Conjecture

We are grateful to Christophe Cornut who communicated to us Proposition 4.10 below and its proof. Keep the assumptions of the previous section, and fix a modular parametrization X_{0}(N) ↠ E and an embedding K_{∞} ↪ C. Let H_{n} ⊂ E(K_{n}) be the module of Heegner points as defined, for example, in [BD2] §2.5 or [M2] §19, and

H_{∞} = \underleftarrow{lim}(H_{n} ⊗ Z_{p}) ⊂ U(E/K_{∞}).

Then H_{∞} is free of rank one over Λ, and stable under the action of Δ = Gal(K/Q). In particular it is a Δ-semi-linear Λ-module in the sense of Appendix A.1, and sign(H) ∈ {1, −1} describes the action of Δ on H ⊗_{Λ} Z_{p} (Definition A.16). Since U(E/K_{∞}) is a torsion-free, rank-one Λ-modules (Theorem 4.2(i)), we can fix L ∈ Λ so that LU(E/K_{∞}) ⊂ H_{∞} and [H_{∞} : LU(E/K_{∞})] is finite (take L to be a generator of the characteristic ideal of U(E/K_{∞})/H_{∞}). Let I denote the augmentation ideal of Λ, the kernel of the natural map Λ ↠ Z_{p}, and let ord_{I}(L) denote the largest power of I that contains L. If we fix an isomorphism Λ ≅ Z_{p}[[T]], then ord_{I}(L) = ord_{T =0}(L).

Proposition 4.10. (i) sign(H_{∞}) = −W (E/Q).

(ii) sign(U(E/K_{∞})) = −W (E/Q)(−1)^{ord_{I}(L)}.

Proof. The Heegner module H_{∞} has a generator h_{∞} with the property that

τh_{∞} = −W (E/Q)σh_{∞}

for some σ ∈ Gal(K_{∞}/K), where τ is complex conjugation (see [Gr] §§1.4-1.5). Thus τh_{∞} ≡ −W (E/Q)h_{∞} in H_{∞}/IH_{∞}, which proves (i). The second assertion then follows by Proposition A.17(ii).

Conjecture 3.10 of Bertolini and Darmon in [BD1] asserts (in our notation) that

ord_{I}(L) = max{r_{+}, r_{−}} − 1.  (4)

Corollary 4.11. Conjecture 3.10 of Bertolini and Darmon [BD1] implies the Sign Conjecture.

Proof. Let ϵ = ϵ(E/K) as given by Definition 4.7, so r_{ϵ} = max{r_{+}, r_{−}}. Combining Proposition 4.10(ii), the conjecture (4), and Theorem 4.6 gives

sign(U(E/K_{∞})) = (−1)^{r_{ϵ}}W (E/Q) = ϵ,

which is the Sign Conjecture.

Remark 4.12. Suppose now that K is arbitrary (possibly even non-Galois), and K_{∞}/K is a Z_{p}-extension that is “new”, in the sense that it is not the base-change of a Z_{p}-extension of a proper subfield of K. Suppose further that the compact Selmer module X(E/K_{∞}) has rank 1 over Λ = Z_{p}[[Gal(K_{∞}/K)]]. Then by Proposition 3.2 (assuming the Shafarevich-Tate conjecture) the growth number of (E, K_{∞}/K) is 1. In the spirit of the questions from §3, we expect this can only happen if the characters ϕ_{ψ} are real-valued for every ψ ∈ Hom_{cont}(Gal(K_{∞}/K), C^{×}), and have root number W (E, ϕ_{ψ}) = W (E/K) = −1. In this case Corollary 2.6 shows that K has an automorphism of order 2, with fixed field K^{+}, such that K_{∞}/K^{+} is Galois with dihedral Galois group. Thus in this setting, exactly as when K is quadratic imaginary, the universal norm module U(E, K_{∞}) is a Gal(K/K^{+})-semi-linear Λ-module that is torsionfree of rank one over Λ, and we get a one-dimensional universal norm subspace U(E, K_{∞}/K) ⊂ E(K) ⊗ Q_{p}. Again U(E, K_{∞}/K) ⊗ Q_{p} ⊂ E(K)^{±} ⊗ Q_{p}, where E(K)^{±} means the plus and minus spaces for the action of Gal(K/K^{+}), rank_{Z}(E(K)^{+}) and rank_{Z}(E(K)^{−}) (conjecturally) have opposite parity, and we can conjecture that U(E, K_{∞}/K) ⊗ Q_{p} lies in the larger of E(K)^{+} ⊗ Q_{p} and E(K)^{−} ⊗ Q_{p}. As in Remark 4.9, such a conjecture would follow from the “maximal nondegeneracy” of the p-adic height attached to (E, K_{∞}/K).

4.4 Computational examples

The Sign Conjecture is trivially true if either E(K)^{+} or E(K)^{−} is finite. In the first nontrivial case, i.e., if min{rank(E(K)^{±})} = 1, the Sign Conjecture would follow (using Remark 4.9) if the anti-cyclotomic height pairing is not identically zero. Let us focus on this “first nontrivial case”. Fix an elliptic curve E with rank_{Z}E(Q) = 2, and let R, S ∈ E(Q) generate a subgroup of finite index. If K is a quadratic field, let E^{(K)} be the twist of E by the quadratic character attached to K. Fix a prime number p where E has good ordinary reduction. Let K run through the set K_{E} of all quadratic imaginary fields such that E and K satisfy our running hypotheses, and such that the rank of the Mordell-Weil group E^{(K)}(Q) is 1; for each of these fields, choose a rational point P_{K} ∈ E^{(K)}(Q) ⊂ E(K) of infinite order. If the anti-cyclotomic height pairing is not identically zero, then the Sign Conjecture is true, so the p-anti-cyclotomic norm line for K ∈ K_{E} (call it ℓ_{K}) will lie in the Q_{p} vector space E(Q) ⊗ Q_{p}. The slope of ℓ_{K} ⊂ E(Q) ⊗ Q_{p} with respect to the basis {R, S} is given by

slope(ℓ_{K}) = −\frac{h(S + P_{K})}{h(R + P_{K})}

where h = h_{K_{∞}/K} is the quadratic function h(P) = ⟨P, P ⟩_{anti−cycl} on E(K).

Some numerical experiments. We are grateful to William Stein for providing us with the following data. Stein works with the prime p = 3 and performs two types of numerical experiment. The first experiment is to test whether or not the 3-anti-cyclotomic height pairing is not identically zero (and hence whether or not the Sign Conjecture is true) for optimal curves E of Mordell-Weil rank 2 of small conductor, and for the quadratic imaginary field K of smallest discriminant in K_{E}. This Stein did for the elliptic curves labelled

389A, 433A,446D, 571B, 643A, 655A, 664A, 707A, 718B, 794A, 817A

in [Cr]. In all these instances, the height pairing is indeed not identically zero. We also expect that the lines ℓ_{K} (for K ∈ K_{E}) generate the Q_{p}-vector space E(Q) ⊗ Q_{p}, and further that it is never the case that ℓ_{K} = ℓ_{K′} for different fields K, K′. To test this, Stein computed the slopes of ℓ_{K} (mod 9) (and mod 27, when necessary) for certain selected curves E and various K ∈ K_{E}. In particular

• for E = 389A the first 6 K ∈ K_{E} have the property that the ℓ_{K} are distinct, with slopes (mod 9) equal to 4, 4, 4, 0, 7, 4,

• for E = 433A the first 4 K ∈ K_{E} have the property that the ℓ_{K} are distinct, with slopes (mod 9) equal to 2, 1, 6, ∞,

• for E = 571B the first 4 K ∈ K_{E} have the property that the ℓ_{K} are distinct, with slopes (mod 9) all equal to 6.

Remark 4.13. If the anti-cyclotomic height pairing is not identically zero, then computing it modulo some sufficiently high power of p will detect its nontriviality. Unfortunately, at present, we know of no recipe for an a priori upper bound on how high a power of p one must check, before being able to conclude with certainty that the height pairing is trivial. In other words, if the Sign Conjecture were to fail in some example, it seems difficult to formulate a numerical experiment that would prove this failure.

Appendix

A.1 Semi-linear Λ-modules

Suppose G is a profinite group with a normal subgroup of finite index Γ ≅ Z_{p}^{d} for some natural number d. This is the situation discussed earlier in the paper, with G = Gal(L/Q) and Γ = Gal(L/K). We let Δ = G/Γ and we assume further that the order of Δ is prime to p. Let Λ = Z_{p}[[Γ]]. Then Δ acts on both Γ and Λ, and G is noncanonically isomorphic to the the semidirect product of Δ and Γ. We fix such a semidirect product decomposition.

Definition A.14. A Z_{p}[[G]]-module M is a Λ module with a semi-linear action of Δ in the following sense:

δ(λ · m) = λ^{δ} · δ(m)

for all δ ∈ Δ, λ ∈ Λ, m ∈ M. We will call such a module a Δ-semi-linear Λ-module, or simply a semi-linear Λ-module if the group Δ and its action on Λ are understood. Given a semi-linear Λ-module M define M_{Γ} := M ⊗_{Λ} Z_{p}, and note that the semi-linear Δ-action on M induces a Z_{p}-linear action of Δ on M_{Γ}. We can also produce semi-linear Λ-modules by “inverting” this procedure. Namely, if N is a Z_{p}[Δ]-module, then M = N ⊗_{Z_{p}} Λ (with the natural “diagonal” action of Δ) is a semi-linear Λ-module and M_{Γ} = N. We say that a semi-linear Λ-module M is split if M ≅ N ⊗_{Z_{p}} Λ for some Z_{p}[Δ]-module N. Clearly a split semi-linear Λ-module M is determined, up to isomorphism, by the Z_{p}[Δ]-module M_{Γ}.

Lemma A.15. If M is a semi-linear Λ-module that is free of finite rank over Λ, then M is split.

Proof. For every subgroup H of finite index in Γ that is stable under Δ, the projection M ⊗_{Λ} Z_{p}[Γ/H] ↠ M_{Γ} of finite-type Z_{p}[Δ]-modules admits a section because Z_{p}[Δ] is semisimple. It follows by a compactness argument that M ↠ M_{Γ} admits a Z_{p}[Δ]-section ι : M_{Γ} ↪ M, and it is straightforward to verify that M = ι(M_{Γ}) ⊗ Λ.

Definition A.16. Suppose that M is a semi-linear Λ-module. If M ⊗ Q_{p} is free of rank one over Λ ⊗ Q_{p}, we define the sign sign(M) to be the character

χ_{M} : Δ → Aut(M_{Γ} ⊗ Q_{p}) ≅ Q_{p}^{×}.

By Lemma A.15, if M is free of rank one over Λ then sign(M) determines M up to isomorphism. If Δ has order two (which is the case of most interest in this paper), and τ is the nontrivial element of Δ, we will also call χ_{M}(τ) = ±1 the sign of M.

Let I denote the augmentation ideal of Λ, the kernel of the natural map Λ ↠ Z_{p}.

Proposition A.17. Suppose that Γ ≅ Z_{p}, and let ψ : Δ → Aut(Γ) ≅ Z_{p}^{×} be the character giving the action of Δ on Γ. Suppose N ⊂ M are semi-linear Λ-modules, and N ⊗ Q_{p}, M ⊗ Q_{p} are free of rank one over Λ ⊗ Q_{p}. Then there is an L ∈ Λ such that N ⊗ Q_{p} = LM ⊗ Q_{p}, and for every such L

(i) sign(N) = ψ^{λ(L)}sign(M), where λ(L) = dim_{Q_{p}}(M/N ⊗ Q_{p}) is the λinvariant of L,

(ii) sign(N) = ψ^{r}sign(M), where r = ord_{I}(L) is maximal such that L ∈ I^{r}.

Proof. The existence of such an L ∈ Λ is clear. Further, replacing M and N by their double duals Hom(Hom(M, Λ), Λ) and Hom(Hom(N, Λ), Λ), we may assume that M and N are both free of rank one over Λ and that N = LM. The natural isomorphism Γ ⟶^{∼} I/I^{2} by γ ↦ γ − 1 is Δ-equivariant, so Δ acts on I/I^{2} via ψ. Hence sign(IM), the character giving the action of Δ on (IM)_{Γ} = (IM) ⊗_{Λ} Z_{p} = M_{Γ} ⊗ I/I^{2}, is ψ · sign(M) and by induction

sign(I^{r}M) = ψ^{r}sign(M)  (5)

for every r ≥ 0. Identify Λ with a power series ring Z_{p}[[T]], so that I = T Z_{p}[[T]]. The Weierstrass Preparation Theorem shows that L = p^{μ}T^{r}g(T), where g is a distinguished polynomial of degree λ(L)−r and g(0) ≠ 0. Since sign(M) = sign(pM), we may suppose that μ = 0. Write M̄ = M/pM and N̄ = N/pN. Then we have N̄ = I^{λ(L)}M̄. Using (5) we see that sign(N) ≡ ψ^{λ(L)}sign(M) (mod p). Since sign(N) is a character of order prime to p, this proves (i). To prove (ii), thanks to (5) it is enough to consider the case r = 0, i.e., N = g(T)M with g(0) ≠ 0. In that case

N_{Γ} = N ⊗_{Λ} Z_{p} = g(0)(M ⊗_{Λ} Z_{p}) = g(0)M_{Γ}

so sign(N) = sign(M) as desired.

A.2 Universal norms and growth of ranks in Iwasawa modules.

In this appendix Γ will denote a topological pro-p-group written multiplicatively and isomorphic to Z_{p}, and Γ_{n} is the unique open subgroup in Γ of index p^{n}. If γ ∈ Γ is a topological generator, then γ^{p^{n}} is a topological generator of Γ_{n}. Put Λ := Z_{p}[[Γ]] and for each n ≥ 0, Λ_{n} := Z_{p}[Γ/Γ_{n}] = Z_{p}[γ]/(γ^{p^{n}} − 1). We have Λ_{n} = Λ/(γ^{p^{n}} − 1)Λ, and Λ = \underleftarrow{lim} Λ_{n}. IF n ≥ M there are natural Λ-module homomorphisms

• the canonical surjection π_{n,m} : Λ_{n} → Λ_{m},

• an injection ν_{n,m} : Λ_{m} → Λ_{n}, given by multiplication by

\frac{γ^{p^{n}} − 1}{γ^{p^{m}} − 1} = ∑_{i=0}^{p^{n−m}−1} γ^{ip^{m}} ∈ Λ

which takes cosets of (γ^{p^{m}} − 1)Λ to cosets of (γ^{p^{n}} − 1)Λ,

and π_{n,m} ∘ ν_{n,m} = p^{n−m}. Suppose M is a module of finite type over Λ. Put

M_{n} := M ⊗_{Λ} Λ_{n} = M/(γ^{p^{n}} − 1)M,

and let rank_{Z_{p}}(M_{n}) = dim_{Q_{p}}(M_{n} ⊗_{Z_{p}} Q_{p}), the the Z_{p}-rank of M_{n}. Then the structure theory of Λ-modules shows that

rank_{Z_{p}}(M_{n}) = rp^{n} + ϵ_{n}

where the error term ϵ_{n} is non-negative, monotone non-decreasing with n, and bounded (i.e., there is an n_{0} ∈ Z^{+} such that if n ≥ n_{0} then ϵ_{n} = ϵ_{n_{0}}).

Definition A.18. Put

M_{n}^{∗} := Hom_{Z_{p}}(M_{n}, Z_{p}).

If n ≥ m the map 1 ⊗ ν_{m,n} : M_{m} → M_{n} induces a map ν_{m,n}^{∗} : M_{n}^{∗} → M_{m}^{∗}. The universal norm Λ-module attached to M is the projective limit

U(M) := \underleftarrow{lim}_{0} nM_{n}^{∗}

with respect to the maps ν_{n,m}^{∗}, with its natural Λ-module structure.

Lemma A.19. There are canonical horizontal isomorphisms making the following diagram commute

M_{n}^{∗} ⟶^{∼} Hom_{Λ_{n}}(M_{n}, Λ_{n})
ν_{m,n}^{∗}↓  ↓
M_{m}^{∗} ⟶^{∼} Hom_{Λ_{m}}(M_{m}, Λ_{m})

where the right-hand map is induced by the isomorphism M_{m} = M_{n} ⊗_{Λ} Λ_{m}.

Proof. Define p_{1} : Λ_{n} → Z_{p} by p_{1}(∑_{i=0}^{p^{n}−1} a_{i}γ^{i}) = a_{0}. It is straightforward to check that composition with p_{1} induces a Λ_{n}-module isomorphism Hom_{Λ_{n}}(M_{n}, Λ_{n}) ⟶^{∼} Hom(M_{n}, Z_{p}) (see for example [Br] Proposition VI.3.4), and that with these isomorphisms the diagram of the lemma commutes.

The Λ-dual of M is M^{•} := Hom_{Λ}(M, Λ), which is a free Λ-module (of finite rank). The Λ-rank of M is

rank_{Λ}(M) := dim_{K}(M ⊗ K)

where K is the field of fractions of Λ. Equivalently, the Λ-rank of M is the rank of the free module M^{•}.

Proposition A.20. Suppose M is a finitely generated Λ-module. Then U(M) ≅ M^{•}, so in particular U(M) is free over Λ of rank rank_{Λ}(M).

Proof. Clearly M^{•} ≅ \underleftarrow{lim} Hom_{Λ_{n}}(M_{n}, Λ_{n}). Thus by Lemma A.19 we have an isomorphism

U(M) ⟶^{∼} M^{•}

and the proposition follows.

Proposition A.21. Suppose M is a finitely generated Λ-module. Then the projection map U(M) → M_{n}^{∗} factors through an injective map U(M) ⊗ Λ_{n} ↪ M_{n}^{∗}. In particular ∩_{m>n}ν_{m,n}^{∗}(M_{m}^{∗}) is a free Λ_{n}-submodule of M_{n}^{∗} of rank equal to rank_{Λ}(M).

Proof. Using the identification U(M) ≅ M^{•} of Proposition A.20, and the identification M_{n}^{∗} ≅ Hom_{Λ}(M_{n}, Λ_{n}) = Hom_{Λ}(M, Λ_{n}) of Lemma A.19, the projection map U(M) → M_{n}^{∗} is identified with the natural composition

Hom_{Λ}(M, Λ) ⟶ Hom_{Λ}(M, Λ) ⊗ Λ_{n} ⟶^{β_{n}} Hom_{Λ}(M, Λ_{n}).  (6)

We need to show that the map β_{n} is injective. Let r = rank_{Λ}(M). By Proposition A.20, Hom_{Λ}(M, Λ)⊗Λ_{n} is a free Z_{p}-module of rank rp^{n}, so to prove the injectivity of β_{n} it will suffice to show that the image of the composition (6) contains a Z_{p}-module of rank rp^{n}. Fix a free Λ-module N of rank r and a map M → N with finite cokernel (for example, we can take N to be the double dual M^{••}, with the canonical map). We have a commutative diagram

\xymatrix{Hom_{Λ}(M, Λ) \ar[r] & Hom_{Λ}(M, Λ_{n}) \\ Hom_{Λ}(N, Λ) \ar@{->>}[r] \ar@{^{(}->}[u] & Hom_{Λ}(N, Λ_{n}) \ar@{^{(}->}[u]}

The vertical maps are injective because Λ and Λ_{n} have no p-torsion, and the bottom map is surjective because N is free. Since Hom_{Λ}(N, Λ_{n}) is free of rank rp^{n} over Z_{p}, this proves that β_{n} is injective. By definition the image of U(M) ⊗ Λ_{n} ↪ M_{n}^{∗} is ∩_{m>n}ν_{m,n}^{∗}(M_{m}^{∗}), and since U(M) is free of rank r over Λ, this module is free of rank r over Λ_{n}

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Barry Mazur Department of Mathematics Harvard University Cambridge MA 02138 USA mazur@math.harvard.edu Karl Rubin Department of Mathematics Stanford University Stanford CA 94305 USA rubin@math.stanford.edu 608

Documenta Mathematica · Extra Volume Kato (2003)