RANDOM MAXIMAL ISOTROPIC SUBSPACES AND SELMER GROUPS

BJORN POONEN AND ERIC RAINS

Abstract. Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over 𝔽_{p}. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable.

We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over 𝔽_{p}. By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay’s heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

1. Introduction

1.1. Selmer groups. D. R. Heath-Brown [HB93, HB94], P. Swinnerton-Dyer [SD08], and D. Kane [Kan11] obtained the distribution for the nonnegative integer s(E) defined as the 𝔽_{2}-dimension of the 2-Selmer group Sel_{2} E minus the dimension of the rational 2-torsion group E(ℚ)[2], as E varies over quadratic twists of certain elliptic curves over ℚ. The distribution was the one for which

Prob (s(E) = d) = (∏_{j≥0}(1 + 2^{−j})^{−1}) (∏_{j=1}^{d} \frac{2}{2^{j} − 1}) .

In [HB94], it was reconstructed from the moments of 2^{s(E)}; in [SD08] and [Kan11], it arose as the stationary distribution for a Markov process.

Our work begins with the observation that this distribution coincides with a distribution arising naturally in combinatorics, namely, the limit as n → ∞ of the distribution of dim(Z ∩ W) where Z and W are random maximal isotropic subspaces inside a hyperbolic quadratic space of dimension 2n over 𝔽_{2}. Here it is essential that the maximal isotropic subspaces be isotropic not only for the associated symmetric bilinear pairing, but also for the

Date: May 17, 2011; minor correction August 6, 2012. 2010 Mathematics Subject Classification. Primary 11G10; Secondary 11G05, 11G30, 14G25, 14K15. Key words and phrases. Selmer group, Shafarevich-Tate group, maximal isotropic, quadratic space, Weil

pairing, theta characteristic.

B. P. was partially supported by NSF grant DMS-0841321. This article is to appear in J. Amer. Math. Soc.

quadratic form; otherwise, one would obtain the wrong distribution. That such a quadratic space might be relevant is suggested already by the combinatorial calculations in [HB94].

Is it just a coincidence, or is there some direct relation between Selmer groups and intersections of maximal isotropic subgroups? Our answer is that Sel_{2} E is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional quadratic space V over 𝔽_{2}. The fact that it could be obtained as an intersection of two subspaces that were maximal isotropic for a pairing induced by a Weil pairing is implicit in standard arithmetic duality theorems.

To make sense of our answer, we use the theory of quadratic forms on locally compact abelian groups as introduced by A. Weil in [Wei64]. The locally compact abelian group V in the application is the restricted direct product of the groups H^{1}(ℚ_{p}, E[2]) for p ≤ ∞ with respect to the subgroups of unramified classes. The quadratic form Q is built using D. Mumford’s Heisenberg group, using ideas of Yu. Zarhin [Zar74, §2]. Then arithmetic duality theorems are applied to show that the images of the compact group ∏_{p≤∞} E(ℚ_{p})/2E(ℚ_{p}) and the discrete group H^{1}(ℚ, E[2]) are maximal isotropic in (V, Q). Their intersection is Sel_{2} E.

1.2. Conjectures for elliptic curves. This understanding of the structure of Sel_{2} E suggests the following, in which we replace 2 by p and generalize to global fields:

Conjecture 1.1. Fix a global field k and a prime p.

(a) As E varies over all elliptic curves over k,

Prob (dim_{𝔽_{p}} Sel_{p} E = d) = (∏_{j≥0}(1 + p^{−j})^{−1}) (∏_{j=1}^{d} \frac{p}{p^{j} − 1}) .

(For the sake of definiteness, we define the probability by considering the finitely many elliptic curves y^{2} = x^{3} + ax + b where a, b ∈ k have height ≤ B, and looking at the limit of the probability as B → ∞; use a long Weierstrass equation if char k is 2 or 3.)

(b) The average of # Sel_{p} E over all E/k is p + 1.

(c) For m ∈ ℤ_{≥0}, the average of (# Sel_{p} E)^{m} over all E/k is ∏_{i=1}^{m}(p^{i} + 1).

Several results in the direction of Conjecture 1.1 are known:

• When p = 2, Heath-Brown [HB94, Theorem 2] proved the analogue of Conjecture 1.1 for the family of quadratic twists of y^{2} = x^{3} − x over ℚ, with the caveat that the distribution of dim Sel_{2} E is shifted by +2 to account for the contribution of E[2] (cf. Remark 4.17). Swinnerton-Dyer [SD08] and Kane [Kan11] generalized this result to the family of quadratic twists of any fixed elliptic curve E/ℚ with E[2] ⊆ E(ℚ) and no rational cyclic subgroup of order 4.

• For the family of all elliptic curves E/ℚ with E[2] ⊂ E(ℚ), G. Yu [Yu06, Theorem 1] built upon Heath-Brown’s approach to prove that the average size of Sel_{2} E is finite. (Strictly speaking, if the limit defining the average does not exist, the result holds with a lim sup.) See also [Yu05] for results for other families of elliptic curves over ℚ.

• For the family of all elliptic curves over 𝔽_{q}(t) with 3 ∤ q, A. J. de Jong [dJ02] proved that the average size of Sel_{3} E (in the lim sup sense) is at most 4 + O(1/q), where the O(1/q) is an explicit rational function of q. In fact, de Jong speculated that the truth was 4, and that the same might hold for number fields.

• For the family of all elliptic curves over ℚ, M. Bhargava and A. Shankar proved that the average size of Sel_{2} is 3 [BS10a] and the average size of Sel_{3} is 4 [BS10b].

• For E over a number field k with a real embedding and with E[2](k) = 0, B. Mazur and K. Rubin [MR10] and Z. Klagsbrun showed how to twist judiciously to obtain lower bounds on the number of quadratic twists of E (up to a bound) with prescribed dim Sel_{2}.

As has been observed by Bhargava, since rk E(k) ≤ dim_{𝔽_{p}} Sel_{p} E, Conjecture 1.1(b) would imply that Prob(rk E(k) ≥ 2) is at most (p+1)/p^{2}. If we assume this for an infinite sequence of primes p, we conclude that asymptotically 100% of elliptic curves over k have rank 0 or 1.

A priori, the average rank could still be greater than 1 if there were rare curves of very high rank, but Conjecture 1.1(b) for an infinite sequence of primes p implies also that the elliptic curves of rank ≥ 2 contribute nothing to the average value of rk E(k), and in fact nothing to the average value of p^{rk E(k)}. (Proof: Define e_{p} as the lim sup as B → ∞ of the sum of p^{rk E(k)} over curves of rank ≥ 2 of height bounded by B divided by the total number of curves of height bounded by B. If q is a larger prime, then e_{p} ≤ (p^{2}/q^{2})e_{q} ≤ (p^{2}/q^{2})(q+1), which tends to 0 as q → ∞, so e_{p} = 0.)

If in addition we assume that the parity of rk E(k) is equidistributed, we obtain the following well-known conjecture:

Conjecture 1.2. Fix a global field k. Asymptotically 50% of elliptic curves over k have rank 0, and 50% have rank 1. Moreover, the average rank is 1/2.

Remark 1.3. D. Goldfeld conjectured that the average rank in a family of quadratic twists of a fixed elliptic curve over ℚ was 1/2 [Gol79, Conjecture B]. Other evidence for Conjecture 1.2 was provided by the extensive study by N. Katz and P. Sarnak of a function field analogue [KS99a,KS99b].

Also, as has been observed by Rubin, the distribution in Conjecture 1.1(a) tends, as p → ∞, to the distribution assigning probability 50% to each of 0 and 1. Thus, even without assuming equidistribution of parity, Conjecture 1.1 for any infinite set of primes p would imply not only that 100% of elliptic curves have rank 0 or 1, but also that at least 50% have rank 0, and that the average rank is at most 1/2.

Conjecture 1.1(a) for a single p does not duplicate C. Delaunay’s prediction for dim_{𝔽_{p}} �(E)[p] [Del01,Del07]. Instead the predictions complement each other: we prove that the only distribution on rk E(ℚ) compatible with both predictions is the one in Conjecture 1.2, for which rk E(ℚ) is 0 or 1, with probability 1/2 each (see Theorem 5.2). A related result, that Conjectures 1.1(a) (for p = 2) and 1.2 together imply the �[2] predictions for rank 0 and 1, had been observed at the end of [Del07].

If we also use the heuristic that the dimensions of dim_{𝔽_{p}} Sel_{p} E for different p are independent except for the constraint that their parities are equal, we are led to the following generalization of Conjecture 1.1:

Conjecture 1.4. Fix a global field k and let n be a squarefree positive integer. Let ω(n) be the number of prime factors of n.

(a) Fix d_{p} ∈ ℤ_{≥0} for each prime p dividing n. As E varies over all elliptic curves over k,

Prob (Sel_{n} E ≃ ∏_{p|n}(ℤ/pℤ)^{d_{p}} ) = 2^{ω(n)−1} ∏_{p|n} ((∏_{j≥0}(1 + p^{−j})^{−1}) (∏_{j=1}^{d_{p}} \frac{p}{p^{j} − 1}))

if the d_{p} all have the same parity, and the probability is 0 otherwise.

(b) The average of # Sel_{n} E over all E/k is the sum of the divisors of n.

(c) For m ∈ ℤ_{≥0}, the average of (# Sel_{n} E)^{m} over all E/k is ∏_{p|n} ∏_{i=1}^{m}(p^{i} + 1).

The factor of 2^{ω(n)−1} arises in (a), because only 2 of the 2^{ω(n)} choices of parities for p | n are constant sequences.

Remark 1.5. Based on investigations for n ≤ 5, Bhargava and Shankar have proposed Conjecture 1.4(b) for all positive integers n, at least for k = ℚ.

Remark 1.6. As was noticed during a discussion with Bhargava and Kane, if we combine Delaunay’s heuristics for �[n] with Conjecture 1.2 for varying E/ℚ, we can predict the distribution for the abelian group Sel_{n} E for any fixed positive integer n. Namely, E(ℚ)_{tors} = 0 with probability 1, and in that case the term on the left in

0 → \frac{E(ℚ)}{nE(ℚ)} → Sel_{n} E → �(E)[n] → 0

is free, so the sequence of ℤ/nℤ-modules splits; thus, given Conjecture 1.2, the distribution of Sel_{n} E can be deduced from knowing the distribution of �(E)[n] for rank 0 curves and for rank 1 curves.

1.3. Conjectures for abelian varieties. In fact, our theorems are proved in a more general context, with [2]: E → E replaced by any self-dual isogeny λ: A → Â that is of odd degree or that comes from a symmetric line sheaf on an abelian variety A over a global field k. (See Theorem 4.14.) In this setting, we have a surprise: it is not Sel_{λ} A itself that is the intersection of maximal isotropic subgroups, but its quotient by �^{1}(k, A[λ]), and the latter group is sometimes nonzero, as we explain in Section 3. Moreover, for certain families, such as the family of all genus 2 curves, there may be “causal” subgroups of Sel_{λ} A, which increase its expected size by a constant factor. Taking these into account suggests the following:

Conjecture 1.7. Fix a global field k of characteristic not 2, and fix a positive integer g. Let f ∈ k[x] range over separable polynomials of degree 2g + 1 (with coefficients of height bounded by B, with B → ∞). Let C be the smooth projective model of y^{2} = f(x). Construct the Jacobian A := Jac C. Then the analogues of Conjectures 1.1, 1.2, and 1.4 hold for Sel_{p} A and Sel_{n} A, with the same distributions. They hold also with 2g + 1 replaced by 2g + 2 if n is odd.

Conjecture 1.8. Fix a global field k of characteristic not 2, and fix an even positive integer g. Let f ∈ k[x] range over polynomials of degree 2g + 2 (with coefficients of height bounded by B, with B → ∞). Let C be the smooth projective model of y^{2} = f(x), and let A = Jac C.

(a) If X_{Sel_{2}} is the ℤ_{≥0}-valued random variable predicted by Conjecture 1.1(a) to model dim_{𝔽_{2}} Sel_{2} E, then the analogous random variable for Sel_{2} A is X_{Sel_{2}} + 1.

(b) The average of # Sel_{2} A is 6 (instead of 3).

(c) For m ∈ ℤ_{≥0}, the average of (# Sel_{2} A)^{m} is 2^{m} ∏_{i=1}^{m}(2^{i} + 1).

Example 4.20 will explain the rationale for the +1 in Conjecture 1.8(a), and will explain why we do not venture to make an analogous conjecture for odd g.

Remark 1.9. Although we have formulated conjectures for the family of all curves of a specified type, our model makes sense also for more limited families (e.g., a family of quadratic twists). One should take into account systematic contributions to the Selmer group, however, as was necessary in Conjecture 1.8.

2. Random maximal isotropic subspaces

2.1. Quadratic modules. See [Sch85, 1.§6 and 5.§1] for the definitions of this section. Let V and T be abelian groups. Call a function Q: V → T a (T -valued) quadratic form if Q is a quadratic map (i.e., the symmetric pairing ⟨ , ⟩: V × V → T sending (x, y) to Q(x + y) − Q(x) − Q(y) is bilinear) and Q(av) = a^{2}Q(v) for every a ∈ ℤ and v ∈ V. Then (V, Q) is called a quadratic module.

Remark 2.1. A quadratic map Q satisfying the identity Q(−v) = Q(v) is a quadratic form. (Taking x = y = 0 shows that Q(0) = 0, and then Q(av + (−v)) − Q(av) − Q(−v) = a(Q(0) − Q(v) − Q(−v)) computes Q(av) for other a ∈ ℤ by induction.)

Lemma 2.2. Let (V, Q) be a quadratic module. Suppose that v ∈ V and ℓ ∈ ℤ are such that ℓv = 0. If ℓ is odd, then ℓQ(v) = 0. If ℓ is even, then 2ℓQ(v) = 0.

Proof. We have ℓ^{2}Q(v) = Q(ℓv) = 0, and 2ℓQ(v) = ℓ⟨v, v⟩ = ⟨ℓv, v⟩ = 0. □

Given a subgroup W ⊆ V, let W^{⊥} := {v ∈ V : ⟨v, w⟩ = 0 for all w ∈ W }. Call W a maximal isotropic subgroup of (V, Q) if W^{⊥} = W and Q|_{W} = 0. Let I_{V} be the set of maximal isotropic subgroups of (V, Q).

Remark 2.3. Say that W is maximal isotropic for the pairing ⟨ , ⟩ if W^{⊥} = W. If W = 2W or T [2] = 0, then W^{⊥} = W implies Q|_{W} = 0, but in general Q|_{W} = 0 is a nonvacuous extra condition.

Call a quadratic module (V, Q) nondegenerate if Q is ℝ/ℤ-valued and V is finite (we will relax this condition in Section 2.4) and the homomorphism V → V^{∗} := Hom(V, ℝ/ℤ) defined by v ↦ (w ↦ ⟨v, w⟩) is an isomorphism. Call (V, Q) weakly metabolic if it is nondegenerate and contains a maximal isotropic subgroup. (Metabolic entails the additional condition that the subgroup be a direct summand.)

Remark 2.4. Suppose that (V, Q) is a nondegenerate quadratic module, and X is an isotropic subgroup of (V, Q). Then

(a) The quotient X^{⊥}/X is a nondegenerate quadratic module under the quadratic form Q_{X} induced by Q.

(b) If W ∈ I_{V}, then (W ∩X^{⊥})+X ∈ I_{V}, and ((W ∩X^{⊥})+X)/X ∈ I_{X^{⊥}/X}. Let π^{V,X^{⊥}/X}(W) denote this last subgroup, which is the image of W ∩ X^{⊥} in X^{⊥}/X.

Remark 2.5. If (V, Q) is a nondegenerate quadratic module with #V < ∞, the obstruction to V being weakly metabolic is measured by an abelian group WQ ≃ ⨁_{p} WQ(p) called the Witt group of nondegenerate quadratic forms on finite abelian groups [Sch85, 5.§1]. The obstruction for (V, Q) equals the obstruction for X^{⊥}/X for any isotropic subgroup X of (V, Q) (cf. [Sch85, Lemma 5.1.3]).

2.2. Counting subspaces.

Proposition 2.6. Let (V, Q) be a 2n-dimensional weakly metabolic quadratic space over F := 𝔽_{p}, with Q taking values in \frac{1}{p}ℤ/ℤ ≃ F.

(a) All fibers of π^{V,X^{⊥}/X}: I_{V} → I_{X^{⊥}/X} have size ∏_{i=1}^{dim X}(p^{n−i} + 1).

(b) We have #I_{V} = ∏_{j=0}^{n−1}(p^{j} + 1).

(c) Let W be a fixed maximal isotropic subspace of V. Let X_{n} be the random variable dim(Z ∩ W), where Z is chosen uniformly at random from I_{V}. Then X_{n} is a sum of independent Bernoulli random variables B_{1}, …, B_{n} where B_{i} is 1 with probability 1/(p^{i−1} + 1) and 0 otherwise.

(d) For 0 ≤ d ≤ n, let a_{d,n} := Prob(X_{n} = d), and let a_{d} := lim_{n→∞} a_{d,n}. Then

∑_{d≥0} a_{d,n}z^{d} = ∏_{i=0}^{n−1} \frac{z + p^{i}}{1 + p^{i}} = ∏_{i=0}^{n−1} \frac{1 + p^{−i}z}{1 + p^{−i}} .

∑_{d≥0} a_{d}z^{d} = ∏_{i=0}^{∞} \frac{1 + p^{−i}z}{1 + p^{−i}} .

(e) For 0 ≤ d ≤ n, we have

a_{d,n} = ∏_{j=0}^{n−1}(1 + p^{−j})^{−1} ∏_{j=1}^{d} \frac{p}{p^{j} − 1} ∏_{j=0}^{d−1}(1 − p^{j−n}).

(f) For d ≥ 0, we have

a_{d} = c ∏_{j=1}^{d} \frac{p}{p^{j} − 1},

where

c := ∏_{j≥0}(1 + p^{−j})^{−1} = \frac{1}{2} ∏_{i≥0}(1 − p^{−(2i+1)}).

Proof.

(a) Choose a full flag in X; then π^{V,X^{⊥}/X} factors into dim X maps of the same type, so we reduce to the case dim X = 1. Write X = F v with v ∈ V. For Z ∈ I_{V}, let \overline{Z} be its image in I_{X^{⊥}/X}. There is a bijection \{Z ∈ I_{V} : v ∈ Z\} → I_{X^{⊥}/X} defined by Z ↦ \overline{Z} = Z/X.

Fix \overline{W} ∈ I_{X^{⊥}/X}, and let W ∈ I_{V} be such that \overline{W} = W/X. We want to show that #\{Z ∈ I_{V} : \overline{Z} = \overline{W} \} = p^{n−1} + 1. This follows once we show that the map

\{Z ∈ I_{V} : \overline{Z} = \overline{W} \} → \{codimension 1 subspaces of W not containing v\} ∪ \{W \}
Z ↦ Z ∩ W

is a bijection. If v ∈ Z, then \overline{Z} = \overline{W} implies that Z = W. If v ∉ Z, then \overline{Z} = \overline{W} implies that Z ∩ W has codimension 1 in W (and does not contain v). Conversely, for a given W_{1} of codimension 1 in W not containing v, the Z ∈ I_{V} containing W_{1} are in bijection with the maximal isotropic subspaces of the weakly metabolic 2-dimensional space W_{1}^{⊥}/W_{1}, which is isomorphic to (F^{2}, xy), so there are two such Z: one of them is W, and the other satisfies Z ∩ W = W_{1} and \overline{Z} = \overline{W}. Thus we have the bijection.

(b) Apply (a) to a maximal isotropic X.

(c) If n > 0, fix a nonzero v in W, and define \overline{Z} as in the proof of (a). Then

dim Z ∩ W = dim \overline{Z} ∩ \overline{W} + δ_{v∈Z},

where δ_{v∈W} is 1 if v ∈ Z and 0 otherwise. The term dim \overline{Z} ∩ \overline{W} has the distribution X_{n−1}. Conditioned on the value of \overline{Z}, the term δ_{v∈Z} is 1 with probability 1/(p^{n−1} + 1) and 0 otherwise, since there are p^{n−1} + 1 subspaces Z ∈ I_{V} with the given \overline{Z}, and only one of them (namely, the preimage of \overline{Z} under V → V/F v) contains v. Thus X_{n} is the sum of X_{n−1} and the independent Bernoulli random variable B_{n}, so we are done by induction on n.

(d) The generating function for X_{n} is the product of the generating functions for B_{1}, …, B_{n}; this gives the first identity. The second follows from the first.

(e) This follows from (d) and Cauchy’s q-binomial theorem (which actually goes back to [Rot11] and is related to earlier formulas of Euler). Namely, set t = 1/p in formula (18) of [Cau43], and divide by ∏_{j=0}^{n−1}(1 + p^{−j}).

(f) Take the limit of (e) as n → ∞. The alternative formula for c follows from substituting

1 + p^{−j} = \frac{1 − p^{−2j}}{1 − p^{−j}}

for j ≠ 0 and cancelling common factors. □

Remark 2.7. There is a variant for finite-dimensional vector spaces V over a finite field F of non-prime order. One can define the notion of weakly metabolic quadratic form Q: V → F, and then prove Proposition 2.6 with q in place of p.

If we consider only even-dimensional nondegenerate quadratic spaces over F, then the obstruction analogous to that in Remark 2.5 takes values in a group of order 2. The obstruction is the discriminant in F^{×}/F^{×2} if char F ≠ 2, and the Arf invariant (see [Sch85, 9.§4]) if char F = 2.

Remark 2.8. By Lemma 2.2, a quadratic form on a 2-torsion module will in general take values in the 4-torsion of the image group. Thus we need an analogue of Proposition 2.6(c) for a \frac{1}{4}ℤ/ℤ-valued quadratic form Q on a 2n-dimensional 𝔽_{2}-vector space V such that Q(V) ⊄ \frac{1}{2}ℤ/ℤ, or equivalently such that ⟨x, x⟩ = 2Q(x) is not identically 0.

The map x ↦ ⟨x, x⟩ is a linear functional V → \frac{1}{2}ℤ/ℤ ≃ 𝔽_{2} since

⟨x + y, x + y⟩ = ⟨x, x⟩ + 2⟨x, y⟩ + ⟨y, y⟩ = ⟨x, x⟩ + ⟨y, y⟩.

Hence there exists a nonzero c ∈ V such that ⟨x, x⟩ = ⟨x, c⟩ for all x ∈ V. This equation shows that for any maximal isotropic subspace W of V, we have c ∈ W^{⊥} = W. The map W ↦ W/𝔽_{2}c defines a bijection between the set of maximal isotropic subspaces of V and the set of maximal isotropic subspaces of (𝔽_{2}c)^{⊥}/𝔽_{2}c, which is a \frac{1}{2}ℤ/ℤ-valued quadratic space. So the random variable dim(Z ∩ W) for V is 1 plus the corresponding random variable for the \frac{1}{2}ℤ/ℤ-valued quadratic space of dimension dim V − 2.

Definition 2.9. Given a prime p, let X_{Sel_{p}} be a ℤ_{≥0}-valued random variable such that for any d ∈ ℤ_{≥0}, the probability Prob(X_{Sel_{p}} = d) equals the a_{d} in Proposition 2.6(d).

In the notation of Proposition 2.6(c), we can also write

X_{Sel_{p}} = lim_{n→∞} X_{n} = ∑_{n=1}^{∞} B_{n}.

Remark 2.10. The distribution of X_{Sel_{2}} agrees with the distribution of s(E) mentioned at the beginning of Section 1.1.

2.3. Some topology. To interpret a_{r} as a probability and not only a limit of probabilities, we are led to consider infinite-dimensional quadratic spaces. The naïve dual of such a space V is too large to be isomorphic to V, so we consider spaces with a locally compact topology and use the Pontryagin dual. In order to define a probability measure on the set of maximal isotropic subspaces, we need additional countability constraints. This section proves the equivalence of several such countability constraints.

For a locally compact abelian group G, define the Pontryagin dual G^{∗} := Hom_{conts}(G, ℝ/ℤ). Recall that a topological space is σ-compact if it is expressible as a union of countably many compact subspaces, first-countable if each point has a countable basis of neighborhoods, second-countable if the topology admits a countable basis, and separable if it has a countable dense subset.

Proposition 2.11. Let G be a locally compact abelian group. The following are equivalent:

(a) G^{∗} is σ-compact.

(b) G is first-countable.

(c) G is metrizable.

Moreover, G is second-countable if and only if G and G^{∗} are both σ-compact.

Proof. After peeling off a direct factor ℝ^{n} from G, we may assume that G contains a compact open subgroup K, by the Pontryagin–van Kampen structure theorem [vK35, Theorem 2]. Each of (a), (b), (c) holds for G if and only if it holds for K, and for K the three conditions are equivalent to second-countability by [Kak43, Theorem 2 and the bottom of page 366]. To prove the final statement, observe that G is second-countable if and only if K is secondcountable and G/K is countable. By the above, K is second-countable if and only G^{∗} is σ-compact; on the other hand, G/K is countable if and only if G is σ-compact. □

Corollary 2.12. Let G be a locally compact abelian group such that G ≃ G^{∗}. Then the following are equivalent:

(a) G is σ-compact.

(b) G is first-countable.

(c) G is metrizable.

(d) G is second-countable.

(e) G is separable.

Proof. Proposition 2.11 formally implies the equivalence of (a), (b), (c), (d). To obtain (d) ⟹ (e), choose one point from each nonempty set in a countable basis. To prove (e) ⟹ (a), reduce to the case that G contains a compact open subgroup K; then separability implies that G/K is finite, so G is σ-compact. □

2.4. Quadratic forms on locally compact abelian groups.

Definition 2.13. A locally compact quadratic module (V, Q) is a locally compact abelian group V equipped with a continuous quadratic form Q: V → ℝ/ℤ. (This notion was introduced in [Wei64, p. 145], where Q was called a “caractère du second degré”; for him, the codomain of Q was the group of complex numbers of absolute value 1, because he was interested in the Fourier transforms of such Q.)

The definitions of maximal isotropic and nondegenerate extend to this setting.

Definition 2.14. Call a nondegenerate locally compact quadratic module (V, Q) weakly metabolic if it contains a compact open maximal isotropic subgroup W; we then say also that (V, Q, W) is weakly metabolic.

Remark 2.15. In Definition 2.14, it would perhaps be more natural to require the subgroup W to be only closed, not necessarily compact and open. Here we explain that the two definitions are equivalent when V contains a compact open subgroup, which is not a strong hypothesis, since by the Pontryagin–van Kampen structure theorem, if V is a locally compact abelian group, then V ≃ ℝ^{n} ⊕ V′ as topological groups, where V′ contains a compact open subgroup.

If (V, Q) is a nondegenerate locally compact quadratic module containing a compact open subgroup K, then X := K ∩ K^{⊥} is a compact open subgroup that is isotropic for the pairing. Then Q restricts to a continuous linear map X → \frac{1}{2}ℤ/ℤ, and its kernel Y is a compact open subgroup that is isotropic for Q. Next, if W is any closed maximal isotropic subgroup of (V, Q), then (W ∩ Y^{⊥}) + Y is a compact open maximal isotropic subgroup of (V, Q) (cf. Remark 2.4(b)).

Remark 2.16. If (V, Q) is a nondegenerate locally compact quadratic module containing a compact open isotropic subgroup X, then the obstruction to (V, Q) containing a maximal isotropic closed subgroup is the same as that for X^{⊥}/X, so the obstruction is measured by an element of WQ that is independent of X (cf. Remark 2.5).

Example 2.17. If W is a locally compact abelian group, then V := W ×W^{∗} may be equipped with the quadratic form Q((w, f)) := f(w). If W contains a compact open subgroup Y, then its annihilator in W^{∗} is a compact open subgroup Y′ of W^{∗}, and X := Y × Y′ is a compact open subgroup of V, so Remark 2.15 shows that (V, Q) is weakly metabolic.

Example 2.18 (cf. [Bra48, Théorème 1]). Suppose that (V_{i}, Q_{i}, W_{i}) for i ∈ I are weakly metabolic. Define the restricted direct product

V := \begin{matrix}∏ & ′ \\ i∈I \end{matrix}(V_{i}, W_{i}) := \{(v_{i})_{i∈I} ∈ ∏_{i∈I} V_{i} : v_{i} ∈ W_{i} for all but finitely many i\} .

Let W := ∏_{i∈I} W_{i}. As usual, equip V with the topology for which W is open and has the product topology. For v := (v_{i}) ∈ V, define Q(v) = ∑_{i∈I} Q_{i}(v_{i}), which makes sense since Q_{i}(v_{i}) = 0 for all but finitely many i. Then (V, Q, W) is another weakly metabolic locally compact quadratic module.

Moreover, if I is countable and each V_{i} is second-countable, then V is second-countable too. (Proof: Use Corollary 2.12 to replace second-countable by σ-compact. If each V_{i} is σ-compact, then each V_{i}/W_{i} is countable, so V/W ≃ ⨁_{i∈I} V_{i}/W_{i} is countable, so V is σ-compact.)

Let (V, Q) be a locally compact quadratic module. Let I_{V} be the set of maximal isotropic closed subgroups of (V, Q). Let X_{V} be the poset of compact open isotropic subgroups of (V, Q), ordered by (reverse) inclusion.

Theorem 2.19. Let (V, Q, W) be a second-countable weakly metabolic locally compact quadratic module.

(a) The set X_{V} is a countable directed poset.

(b) The finite sets I_{X^{⊥}/X} for X ∈ X_{V} with the maps π^{X_{1}^{⊥}/X_{1},X_{2}^{⊥}/X_{2}} for X_{1} ⊆ X_{2} (cf. Remark 2.4(b)) form an inverse system.

(c) If ⋂_{X∈X_{V}} X = 0, then the collection of maps π^{V,X^{⊥}/X} induces a bijection

I_{V} → \underleftarrow{lim}_{X∈X_{V}} I_{X^{⊥}/X}.

Equip I_{V} with the inverse limit topology.

(d) In the remaining parts of this theorem, assume that p is a prime such that pV = 0. Then there exists a unique probability measure μ on the Borel σ-algebra of I_{V} such that for every compact open isotropic subgroup X of (V, Q), the push-forward π_{∗}^{V,X^{⊥}/X}μ is the uniform probability measure on the finite set I_{X^{⊥}/X}.

(e) The measure μ is invariant under the orthogonal group Aut(V, Q).

(f) If Z is distributed according to μ, then Prob(Z is discrete) = 1 and Prob(Z∩W is finite) = 1. If moreover dim_{𝔽_{p}} V is infinite, then the distribution of dim(Z ∩ W) is given by X_{Sel_{p}} (see Definition 2.9).

Proof.

(a) The intersection of two compact open isotropic subgroups of V is another one, so X_{V} is a directed poset. To prove that X_{V} is countable, first consider the bijection

{compact open subgroups of W } → {finite subgroups of V/W }  (1)
X ↦ X^{⊥}/W.

Since V/W is a countable discrete group, both sets above are countable. The map

X_{V} → {compact open subgroups of W }
X ↦ X ∩ W

has finite fibers, since the X ∈ X_{V} containing a given compact open subgroup Y of W are in bijection with the isotropic subgroups of the finite group Y^{⊥}/Y. Thus X_{V} is countable.

(b) Given X_{1} ⊆ X_{2} ⊆ X_{3} the maps π^{X_{i}^{⊥}/X_{i},X_{j}^{⊥}/X_{j}} for i < j behave as expected under composition.

(c) The same computation proving (b) shows that the map is well-defined. If X ∈ X_{V}, then X^{⊥} is another compact open subgroup of V since it contains X as a finite-index open subgroup. The group (X^{⊥})^{∗} ≃ V/X is a discrete 𝔽_{p}-vector space, so it equals the direct limit of its finite-dimensional subspaces. Taking duals shows that X^{⊥} is the inverse limit of its finite quotients, i.e., of the groups X^{⊥}/Y where Y ranges over open subgroups of X^{⊥}. Moreover, every open subgroup of X^{⊥} contains an open subgroup of X (just intersect with X), so it suffices to take the latter.

Now the inverse map (Z_{X}) ↦ Z is constructed as follows: given (Z_{X}) ∈ \underleftarrow{lim}_{X∈X_{V}} I_{X^{⊥}/X}, let

Z̃_{X} := \underleftarrow{lim}_{\begin{matrix}Y ∈X_{V} \\ Y ⊆X \end{matrix}} (Z_{Y} ∩ \frac{X^{⊥}}{Y} ) ⊆ \underleftarrow{lim}_{\begin{matrix}Y ∈X_{V} \\ Y ⊆X \end{matrix}} \frac{X^{⊥}}{Y} = X^{⊥}

Z := ⋃_{X∈X_{V}} Z̃_{X}.

The maps in the inverse system defining Z̃ are surjections, so the image of Z̃_{X} in X^{⊥}/X equals Z_{X}. If X, X′ ∈ X_{V} and X′ ⊆ X, then Z̃_{X} = Z̃_{X′} ∩ X^{⊥}, so Z ∩ X^{⊥} = Z̃_{X}. Since each Z_{Y} is isotropic in Y^{⊥}/Y, the group Z̃_{X} is isotropic, so Z is isotropic. If z ∈ Z^{⊥}, then we have z ∈ X^{⊥} for some X, and then for any Y ⊆ X, the element z mod Y ∈ Y^{⊥}/Y is perpendicular to π^{V,Y^{⊥}/Y} (Z) = Z_{Y}, but Z_{Y}^{⊥} = Z_{Y}, so z mod Y ∈ Z_{Y}, and also z mod Y ∈ X^{⊥}/Y; this holds for all Y ⊆ X, so z ∈ Z. Thus Z^{⊥} = Z; i.e., Z ∈ I_{V}.

Now we show that the two constructions are inverse to each other. If we start with (Z_{X}), then the Z produced by the inverse map satisfies π^{V,X^{⊥}/X}(Z) = Z_{X}. Conversely, if we start with Z, and define Z_{X} := π^{V,X^{⊥}/X}(Z), then the inverse map applied to (Z_{X}) produces Z′ such that Z ∩X^{⊥} ⊆ Z′ for all X, so Z ⊆ Z′, but Z and Z′ are both maximal isotropic, so Z = Z′.

(d) Since V/W is a discrete 𝔽_{p}-vector space of dimension at most ℵ_{0}, we may choose a cofinal increasing sequence of finite-dimensional subspaces of V/W, and this corresponds under (1) to a cofinal decreasing sequence Y_{1}, Y_{2}, … of compact open subgroups of W whose intersection is 0. Thus (c) applies. Each map in the inverse system has fibers of constant size, by Proposition 2.6(a), so the uniform measures on these finite sets are compatible. By [Bou04, III.§4.5, Proposition 8(iv)], the inverse limit measure exists.

(e) The construction is functorial with respect to isomorphisms (V, Q) → (V′, Q′).

(f) Since ∑_{r=0}^{∞} a_{r} = 1, it suffices to prove the last statement, that Prob(dim(Z ∩ W) = r) = a_{r}. Let Y_{i} be as in the proof of (d). Then dim(Z ∩ W) is the limit of the increasing sequence of nonnegative integers dim(π^{V,Y_{i}^{⊥}/Y_{i}}(Z) ∩ π^{V,Y_{i}^{⊥}/Y_{i}}(W)). By Proposition2.6(c) and its proof, the difference of consecutive integers in this sequence is a sum of independent Bernoulli random variables. Since ∑_{j≥1} Prob(B_{j} = 1) converges, the Borel-Cantelli lemma implies that

Prob (dim(Z ∩ W) ≠ dim(π^{V,Y_{i}^{⊥}/Y_{i}}(Z) ∩ π^{V,Y_{i}^{⊥}/Y_{i}}(W))) → 0

as i → ∞. In particular, Prob (dim(Z ∩ W) = ∞) is 0. On the other hand, dim Y_{i}^{⊥}/Y_{i} → ∞ as i → ∞, so

Prob (dim(Z ∩ W) = d) = lim_{n→∞} a_{d,n} = a_{d} = Prob (X_{Sel_{p}} = d) .  □

Remark 2.20. There is only one infinite-dimensional second-countable weakly metabolic locally compact quadratic 𝔽_{p}-vector space (V, Q), up to isomorphism. Inside V we are given a compact open maximal isotropic subspace W, and Theorem 2.19(f) implies the existence of a discrete maximal isotropic closed subspace Z with Z ∩ W = 0. Since V is infinite and second-countable, dim_{𝔽_{p}} Z = ℵ_{0}, so the isomorphism type of Z as locally compact abelian group is determined. The pairing Z × W → ℝ/ℤ defined by (z, w) ↦ Q(z + w) puts Z and W in Pontryagin duality. Now the summing map Z × W → V is an isomorphism under which Q corresponds to the standard quadratic form on Z × Z^{∗} defined in Example 2.17.

Remark 2.21. Suppose that V is as in Theorem 2.19(f), and that Z ∈ I_{V} is chosen at random. The probability that Z contains a given nonzero vector v of V then equals 0, because for any compact open isotropic subgroup X ≤ V small enough that v ∈ X^{⊥} −X, if dim X^{⊥}/X = 2n, then the probability that π^{V,X^{⊥}/X}(Z) contains the nonzero element π^{V,X^{⊥}/X}(v) is 1/(p^{n} + 1), which tends to 0 as X shrinks. Now, if we fix a discrete Z_{0} ∈ I_{V} and choose Z ∈ I_{V} at random, then dim(Z ∩ Z_{0}) = 0 with probability 1 by the previous sentence applied to each nonzero vector of the countable set Z_{0}.

2.5. Moments. Given a random variable X, let 𝔼(X) be its expectation. So if m ∈ ℤ_{≥0}, then 𝔼(X^{m}) is its m^{th} moment.

Proposition 2.22. Fix a prime p and fix m ∈ ℤ_{≥0}. Let X_{n} be as in Proposition 2.6(c), and let X_{Sel_{p}} be as in Definition 2.9. Then

(a) We have

𝔼 ((p^{X_{n}})^{m}) = ∏_{i=1}^{m} \frac{p^{i} + 1}{1 + p^{−(n−i)}}

𝔼 ((p^{X_{Sel_{p}}})^{m}) = ∏_{i=1}^{m}(p^{i} + 1).

In particular, 𝔼(p^{X_{Sel_{p}}}) = p + 1.

(b) We have Prob(X_{n} is even) = 1/2 for each n > 0, and Prob(X_{Sel_{p}} is even) = 1/2.

(c) If we condition on the event that X_{Sel_{p}} has a prescribed parity, the moments in (a) remain the same. The same holds for the m^{th} moment of p^{X_{n}} if m < n.

Proof.

(a) Substitute z = p^{m} in Proposition 2.6(d). The products telescope.

(b) Substitute z = −1 in Proposition 2.6(d).

(c) Substitute z = −p^{m} in Proposition 2.6(d).

□

3. Shafarevich-Tate groups of finite group schemes

3.1. Definitions. For each field k, choose an algebraic closure \overline{k} and a separable closure k_{s} ⊆ \overline{k}, and let G_{k} := Gal(k_{s}/k). A local field is a nondiscrete locally compact topological field; each such field is a finite extension of one of ℝ, ℚ_{p}, or 𝔽_{p}((t)) for some prime p. A global field is a finite extension of ℚ or 𝔽_{p}(t) for some prime p.

In the rest of Section 3, k denotes a global field. Let Ω be the set of nontrivial places of k. For v ∈ Ω, let k_{v} be the completion of k at v, so k_{v} is a local field; if v is nonarchimedean, let O_{v} be the valuation ring in k_{v}.

For a sheaf of abelian groups M on the big fppf site of Spec k, define

�^{1}(k, M) := ker (H^{1}(k, M) → ∏_{v∈Ω} H^{1}(k_{v}, M)) .  (2)

Remark 3.1. If M is represented by a smooth finite-type group scheme over k, such as the kernel of an isogeny of degree not divisible by char k, then we may interpret the cohomology groups as Galois cohomology groups: H^{1}(k, M) = H^{1}(G_{k}, M(k_{s})) and so on.

Definition 3.2. Let C be a smooth projective curve of genus g over k. Let A = Jac C. The degree g − 1 component of the Picard scheme of C contains a closed subscheme T parametrizing line sheaves on C whose square is isomorphic to the canonical sheaf of C. This T is a torsor under A[2], called the theta characteristic torsor. Let c_{T} ∈ H^{1}(k, A[2]) be its class.

3.2. Vanishing criteria. The following criteria for vanishing of �^{1}(k, M) will be especially relevant for Theorem 4.14(b).

Proposition 3.3. Suppose that M is a finite étale group scheme over k, so we identify M with the finite G_{k}-module M(k_{s}).

(a) If M = ℤ/nℤ, then �^{1}(k, M) = 0.

(b) If M is a direct summand of a direct sum of permutation ℤ/nℤ-modules arising from finite separable extensions of k, then �^{1}(k, M) = 0.

(c) Let G be the image of G_{k} in Aut M(k_{s}). Identify H^{1}(G, M) with its image under the injection H^{1}(G, M) ↪ H^{1}(k, M). Then

�^{1}(k, M) ⊆ ⋂_{cyclic H ≤ G} ker (H^{1}(G, M) → H^{1}(H, M)) .

(d) If p is a prime such that pM = 0 and the Sylow p-subgroups of Aut M(k_{s}) are cyclic, then �^{1}(k, M) = 0.

(e) If E is an elliptic curve, and p ≠ char k, then �^{1}(k, E[p]) = 0.

(f) If char k ≠ 2, and A is the Jacobian of the smooth projective model of y^{2} = f(x), where f ∈ k[x] is separable of odd degree, then �^{1}(k, A[2]) = 0.

Proof.

(a) See [Mil06, Example I.4.11(i)].

(b) Combine (a) with Shapiro’s lemma [AW67, §4, Proposition 2] to obtain the result for a finite permutation module (ℤ/nℤ)[G_{k}/G_{L}] for a finite separable extension L of k. The result for direct summands of direct sums of these follows.

(c) This is a consequence of the Chebotarev density theorem: see [BPS10].

(d) Let G be the image of G_{k} → Aut M(k_{s}). Then any Sylow p-subgroup P of G is cyclic. But the restriction H^{1}(G, M) → H^{1}(P, M) is injective [AW67, §6, Corollary 3], so (c) shows that �^{1}(k, M) = 0.

(e) Any Sylow-p-subgroup of GL_{2}(𝔽_{p}) is conjugate to the group of upper triangular unipotent matrices, which is cyclic. Apply (d).

(f) The group A[2] is a direct summand of the permutation ℤ/2ℤ-module on the set of zeros of f. Apply (b). □

3.3. Jacobians of hyperelliptic curves. See [PR11, Example 3.12(b)] for a 2-dimensional Jacobian A with �^{1}(ℚ, A[2]) ≠ 0. Such examples are rare: a special case of Proposition 3.4 below shows that asymptotically 100% of 2-dimensional Jacobians A over ℚ have �^{1}(ℚ, A[2]) = 0.

Proposition 3.4. Fix g ≥ 1 and a prime p. For random f(x) ∈ ℤ[x] of degree 2g + 1 with coefficients in [−B, B], if A is the Jacobian of the smooth projective model C of y^{2} = f(x), the probability that �^{1}(ℚ, A[p]) = 0 tends to 1 as B → ∞. The same holds if 2g + 1 is replaced by 2g + 2 (the general case for a genus g hyperelliptic curve).

Proof. By Proposition 3.3(e) we may assume g ≥ 2. We may assume that f is separable.

First consider the case p ≠ 2. Generically, the image of G_{ℚ} → Aut A[p] is as large as possible given the existence of the Weil pairing e_{p}, i.e., isomorphic to GSp_{2g}(𝔽_{p}). By the Hilbert irreducibility theorem, the same holds for asymptotically 100% of the polynomials f. By Proposition 3.3(c), �^{1}(ℚ, A[p]) is contained in the subgroup H^{1}(GSp_{2g}(𝔽_{p}), A[p]) of H^{1}(ℚ, A[p]), and that subgroup is 0 because the central element −I ∈ GSp_{2g}(𝔽_{p}) has no fixed vector (cf. [Fou64, Lemma 14.4] and [Pol71, Theorem 2.3]).

Now suppose that p = 2. In the degree 2g + 1 case, we are done by Proposition 3.3(f). So assume that deg f = 2g + 2. Let Δ be the set of zeros of f in \overline{ℚ}, so #Δ = 2g + 2. For m ∈ ℤ/2ℤ, let W_{m} be the quotient of the sum-m part of the permutation module 𝔽_{2}^{Δ} ≃ 𝔽_{2}^{2g+2} by the diagonal addition action of 𝔽_{2}. Then the G_{ℚ}-module A[2] may be identified with W_{0}, and W_{m} is a torsor under W_{0}.

Again by the Hilbert irreducibility theorem, we may assume that the group Gal(ℚ(A[2])/ℚ) ≃ Gal(f) is as large as possible, i.e., equal to S_{2g+2}. Then �^{1}(ℚ, A[2]) ⊆ H^{1}(S_{2g+2}, W_{0}) ⊂ H^{1}(ℚ, A[2]). The group H^{1}(S_{2g+2}, W_{0}) is of order 2, generated by the class c_{W_{1}} of W_{1} [Pol71, Theorem 5.2]. Computations as in [PS99, §9.2] show that for each prime ℓ, the probability that f(x) factors over ℤ_{ℓ} into irreducible polynomials of degree 2g and 2 defining unramified and ramified extensions of ℚ_{ℓ}, respectively, is of order 1/ℓ (not smaller) as ℓ → ∞, and in this case no point in W_{1} is G_{ℚ_{ℓ}}-invariant, so c_{W_{1}} has nonzero image in H^{1}(ℚ_{ℓ}, A[2]). Since the conditions at finitely many ℓ are asymptotically independent as B → ∞, and since ∑ 1/ℓ diverges, there will exist such a prime ℓ for almost all f, and in this case c_{W_{1}} ∉ �^{1}(ℚ, A[2]), so �^{1}(ℚ, A[2]) = 0. □

Remark 3.5. Proposition 3.4 can easily be extended to an arbitrary global field of characteristic not equal to 2 or p.

Remark 3.6. Let C be the smooth projective model of y^{2} = f(x), where f(x) ∈ k[x] is separable of degree 2g + 2. As torsors under A[2] ≃ W_{0}, we have T ≃ W_{g−1} (cf. [Mum71, p. 191]).

3.4. Jacobians with generic 2-torsion. Suppose that C is a curve of genus g ≥ 2 over a global field k of characteristic not 2 such that the image G of G_{k} → Aut A[2] is as large as possible, i.e., Sp_{2g}(𝔽_{2}). (This forces C to be non-hyperelliptic if g ≥ 3.) By [Pol71, Theorems 4.1 and 4.8], the group H^{1}(G, A[2]) ⊆ H^{1}(G_{k}, A[2]) is of order 2, generated by c_{T}. So Proposition 3.3(c) shows that �^{1}(k, A[2]) is of order 2 or 1, according to whether the nonzero class c_{T} lies in �^{1}(k, A[2]) or not.

4. Selmer groups as intersections of two maximal isotropic subgroups

4.1. Quadratic form arising from the Heisenberg group. Let A be an abelian variety over a field k. Let Â be its dual abelian variety. Let λ: A → Â be an isogeny equal to its dual. The exact sequence

0 → A[λ] → A →^{λ} Â → 0

gives rise to the “descent sequence”

0 → \frac{Â(k)}{λA(k)} →^{δ} H^{1}(A[λ]) → H^{1}(A)[λ] → 0,  (3)

where H^{1}(A)[λ] is the kernel of the homomorphism H^{1}(λ): H^{1}(A) → H^{1}(Â).

Since λ is self-dual, we obtain an (alternating) Weil pairing

e_{λ}: A[λ] × A[λ] → 𝔾_{m}

identifying A[λ] with its own Cartier dual (cf. [Mum70, p. 143, Theorem 1]). Composing the cup product with H^{1}(e_{λ}) gives a symmetric pairing

∪_{e_{λ}}: H^{1}(A[λ]) × H^{1}(A[λ]) → H^{2}(𝔾_{m})

and its values are killed by deg λ. It is well-known (especially when λ is separable) that the image of the natural map Â(k)/λA(k) → H^{1}(A[λ]) is isotropic with respect to ∪_{e_{λ}}.

In the rest of Section 4, we will assume that we are in one of the following cases:

I. The self-dual isogeny λ has odd degree. II. The self-dual isogeny λ is of the form ϕ_{ℒ} for some symmetric line sheaf ℒ on A.

(Symmetric means [−1]^{∗}ℒ ≃ ℒ. For the definition of ϕ_{ℒ} , see [Mum70, p. 60 and Corollary 5 on p. 131].)

In each of these cases, we will construct a natural quadratic form q whose associated bilinear pairing is ∪_{e_{λ}}. Moreover, in each case, we will show that the image of Â(k)/λA(k) → H^{1}(A[λ])

is isotropic with respect to q.

Remark 4.1. We need Case II in addition to the easier Case I since, for example, we want to study Sel_{p} E even when p = 2.

Remark 4.2. For any symmetric line sheaf ℒ, the homomorphism ϕ_{ℒ} is self-dual [Pol03, p. 116]. If moreover ℒ is ample, then ϕ_{ℒ} is an isogeny (see [Mum70, p. 124 and Corollary 5 on p. 131]).

Remark 4.3. If A is an elliptic curve and λ is multiplication-by-n for some positive integer n, then Case II applies. Namely, let P be the origin of A, and let ℒ be the symmetric line sheaf 𝒪(nP); then ϕ_{ℒ} = λ.

Remark 4.4. The obstruction to expressing a self-dual isogeny λ as ϕ_{ℒ} is given by an element c_{λ} ∈ H^{1}(Â[2]). For example, if λ is an odd multiple of the principal polarization on a Jacobian of a curve with no rational theta characteristic, then c_{λ} ≠ 0. See [PR11, §3] for these facts, and for many criteria for the vanishing of c_{λ}.

Remark 4.5. If λ = 2μ for some self-dual isogeny μ: A → Â, then λ is of the form ϕ_{ℒ} , since c_{λ} = 2c_{μ} = 0. Explicitly, take ℒ := (1, μ)^{∗}𝒫, where 𝒫 is the Poincaré line sheaf on A × Â (see [Mum70, §20, proof of Theorem 2]).

We now return to the construction of the quadratic form.

Case I: deg λ is odd.

Then q(x) := −\frac{1}{2} (x ∪_{e_{λ}} x) is a quadratic form whose associated bilinear pairing is −∪_{e_{λ}}.

(The sign here is chosen to make the conclusion of Corollary 4.7 hold for q.) Since the image of Â(k)/λA(k) → H^{1}(A[λ]) is isotropic with respect to ∪_{e_{λ}}, it is isotropic with respect to q too.

Case II: there is a symmetric line sheaf ℒ on A such that λ = ϕ_{ℒ} . (This hypothesis remains in force until the end of Section 4.1.)

When λ is separable, Zarhin [Zar74, §2] constructed a quadratic form q: H^{1}(A[λ]) → H^{2}(𝔾_{m}) whose associated bilinear pairing was ∪_{e_{λ}}; for elliptic curves, C. O’Neil showed that

the image of Â(k)/λA(k) → H^{1}(A[λ]) is isotropic for q (this is implicit in [O’N02, Proposition 2.3]). Because we wish to include the inseparable case, and because we wish to prove isotropy of the quadratic form for abelian varieties of arbitrary dimension, we will give a detailed construction and proof in the general case.

The pairs (x, ϕ) where x ∈ A(k) and ϕ is an isomorphism from ℒ to τ_{x}^{∗}ℒ form a (usually nonabelian) group under the operation

(x, ϕ)(x′, ϕ′) = (x + x′, (τ_{x′}ϕ)ϕ′).

The same can be done after base extension, so we get a group functor. Automorphisms of ℒ induce the identity on this group functor, so it depends only on the class of ℒ in Pic A.

Proposition 4.6 (Mumford).

(a) This functor is representable by a finite-type group scheme H(ℒ), called the Heisenberg group (or theta group or Mumford group).

(b) It fits in an exact sequence

1 → 𝔾_{m} → H(ℒ) → A[λ] → 1,  (4)

where the two maps in the middle are given by t ↦ (0, multiplication by t) and (x, ϕ) to x. This exhibits H(ℒ) as a central extension of finite-type group schemes.

(c) The induced commutator pairing

A[λ] × A[λ] → 𝔾_{m}

is the Weil pairing e_{λ}.

Proof. See [Mum91, pp. 44–46]. □

Corollary 4.7. The connecting homomorphism q: H^{1}(A[λ]) → H^{2}(𝔾_{m}) induced by (4) is a quadratic form whose associated bilinear pairing H^{1}(A[λ])×H^{1}(A[λ]) → H^{2}(𝔾_{m}) sends (x, y)

to −x ∪_{e_{λ}} y.

Proof. Applying [PR11, Proposition 2.9] to (4) shows that q is a quadratic map giving rise to the bilinear pairing claimed. By Remark 2.1, it remains to prove the identity q(−v) = q(v). Functoriality of (4) with respect to the automorphism [−1] of A gives a commutative diagram

[figure: 1; 𝔾_{m}; H(ℒ); A[λ]; −1; 1; 1; 𝔾_{m}; H([−1]^{∗}ℒ); A[λ]; 1.]

But [−1]^{∗}ℒ ≃ ℒ, so both rows give rise to q. Functoriality of the connecting homomorphism gives q(−v) = q(v) for any v ∈ H^{1}(A[λ]). □

Remark 4.8. The proof of the next proposition involves a sheaf of sets on the big fppf site of Spec k, but in the case char k ∤ deg λ, it is sufficient to think of it as a set with a continuous G_{k}-action.

Proposition 4.9. Identify Â(k)/λA(k) with its image W under δ in (3). Then q|_{W} = 0.

Proof. Let 𝒫 be the Poincaré line sheaf on A × Â. For y ∈ Â(k), let 𝒫_{y} be the line sheaf on A obtained by restricting 𝒫 to A × {y}. For any y_{1}, y_{2} ∈ Â(k), there is a canonical isomorphism ι_{y_{1},y_{2}} : 𝒫_{y_{1}} ⊗ 𝒫_{y_{2}} → 𝒫_{y_{1}+y_{2}}, satisfying a cocycle condition [Pol03, §10.3].

The group H(ℒ)(k) acts on the left on the set of triples (x, y, ϕ) where x ∈ A(k), y ∈ Â(k), and ϕ: ℒ ⊗ 𝒫_{y} → (τ_{x}^{∗}ℒ) as follows:

(x, ϕ)(x′, y′, ϕ′) = (x + x′, y′, (τ_{x′}ϕ)ϕ′).

The same holds after base extension, and we get an fppf-sheaf of sets G(ℒ) on which H(ℒ) acts freely. There is a morphism G(ℒ) → Â sending (x, y, ϕ) to y, and this identifies Â with the quotient sheaf H(ℒ)\G(ℒ). There is also a morphism G(ℒ) → A sending (x, y, ϕ) to x, and the action of H(ℒ) on G(ℒ) is compatible with the action of its quotient A[λ] on A. Thus we have the following compatible diagram:

\xymatrix{1 \ar[r] & H(ℒ) \ar@{.>}[r] \ar[d] & G(ℒ) \ar[r] \ar[d] & Â \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A[λ] \ar[r] & A \ar[r]^{λ} & Â \ar[r] & 0,}

where the first row indicates only that H(ℒ) acts freely on G(ℒ) with quotient being Â. This is enough to give a commutative square of pointed sets

[figure: H^{0}(Â); H^{1}(H(ℒ)); H^{0}(Â); H^{1}(A[λ]),]

so H^{0}(Â) → H^{1}(A[λ]) factors through H^{1}(H(ℒ)). But the sequence H^{1}(H(ℒ)) → H^{1}(A[λ]) → H^{2}(𝔾_{m}) from (4) is exact, so the composition H^{0}(Â) → H^{1}(A[λ]) → H^{2}(𝔾_{m}) is 0. □

Remark 4.10. Proposition 4.9 can be generalized to an abelian scheme over an arbitrary base scheme S. The proof is the same.

4.2. Local fields. Let k_{v} be a local field. The group H^{1}(k_{v}, A[λ]) has a topology making it locally compact, the group H^{2}(k_{v}, 𝔾_{m}) may be identified with a subgroup of ℚ/ℤ and given the discrete topology, and the quadratic form q: H^{1}(k_{v}, A[λ]) → H^{2}(k_{v}, 𝔾_{m}) is continuous (cf. [Mil06, III.6.5]; the same arguments work even though (4) has a nonabelian group in the middle). The composition

H^{1}(k_{v}, A[λ]) →^{q} H^{2}(k_{v}, 𝔾_{m}) ↪ ℚ/ℤ

is a quadratic form q_{v}. By local duality [Mil06, I.2.3, I.2.13(a), III.6.10], q_{v} is nondegenerate. Moreover, H^{1}(k_{v}, A[λ]) is finite if char k_{v} ∤ deg λ [Mil06, I.2.3, I.2.13(a)], and σ-compact in general [Mil06, III.6.5(a)], so it is second-countable by Corollary 2.12.

Proposition 4.11. Let k_{v} be a local field. In (3) for k_{v}, the group W ≃ Â(k_{v})/λA(k_{v}) is a compact open maximal isotropic subgroup of (H^{1}(k_{v}, A[λ]), q_{v}), which is therefore weakly metabolic.

Proof. By Proposition 4.9, q_{v} restricts to 0 on W, so it suffices to show that W^{⊥} = W. Let A(k_{v})_{•} be A(k_{v}) modulo its connected component (which is nonzero only if k_{v} is ℝ or ℂ). Then W is the image of Â(k_{v})_{•} → H^{1}(k_{v}, A[λ]), so W^{⊥} is the kernel of the dual map, which by Tate local duality [Mil06, I.3.4, I.3.7, III.7.8] is H^{1}(k_{v}, A[λ]) → H^{1}(k_{v}, A). This kernel is W, by exactness of (3). □

Suppose that k_{v} is nonarchimedean. Let O_{v} is its valuation ring and let 𝔽_{v} be its residue field. Suppose that A has good reduction, i.e., that it extends to an abelian scheme (again denoted A) over O_{v}. Then the fppf-cohomology group H^{1}(O_{v}, A[λ]) is an open subgroup of H^{1}(k_{v}, A[λ]).

Remark 4.12. If moreover char 𝔽_{v} ∤ deg λ, then we may understand H^{1}(O_{v}, A[λ]) in concrete terms as the kernel H^{1}(k_{v}, A[λ])_{unr} of the restriction map

H^{1}(k_{v}, A[λ]) → H^{1}(k_{v}^{unr}, A[λ])

of Galois cohomology groups.

Proposition 4.13. Suppose that k_{v} is nonarchimedean. Suppose that A extends to an abelian scheme over O_{v}. Then the subgroups W ≃ Â(k_{v})/λA(k_{v}) and H^{1}(O_{v}, A[λ]) in H^{1}(k_{v}, A[λ]) are equal. In particular, H^{1}(O_{v}, A[λ]) is a maximal isotropic subgroup.

Proof. By [Mil80, III.3.11(a)] and [Lan56], respectively, H^{1}(O_{v}, A) ≃ H^{1}(𝔽_{v}, A) = 0. The valuative criterion for properness [Har77, II.4.7] yields A(O_{v}) = A(k_{v}) and Â(O_{v}) = Â(k_{v}). So taking cohomology of (3) over O_{v} gives the result. □

4.3. Global fields. Let k be a global field. For any nonempty subset S of Ω containing the archimedean places, define the ring of S-integers O_{S} := {x ∈ k : v(x) ≥ 0 for all v ∉ S}.

Let A be an abelian variety over k, Let λ: A → Â be a self-dual isogeny as in Case I or II of Section 4.1. Choose a nonempty finite S containing all bad places, by which we mean that S contains all archimedean places and A extends to an abelian scheme A over O_{S}. In Example 2.18 take I = Ω, V_{i} = H^{1}(k_{v}, A[λ]), Q_{i} = q_{v}, and W_{i} = Â(k_{v})/λA(k_{v}), which is valid by Proposition 4.11. The resulting restricted direct product

V := \begin{matrix}∏ & ′ \\ v∈Ω \end{matrix} (H^{1}(k_{v}, A[λ]), \frac{Â(k_{v})}{λA(k_{v})})

equipped with the quadratic form

Q: \begin{matrix}∏ & ′ \\ v∈Ω \end{matrix} (H^{1}(k_{v}, A[λ]), \frac{Â(k_{v})}{λA(k_{v})}) → ℚ/ℤ
(ξ_{v})_{v∈Ω} ↦ ∑_{v} q_{v}(ξ_{v})

is a second-countable weakly metabolic locally compact quadratic module. Proposition 4.13, which applies for all but finitely many v, shows that

V = \begin{matrix}∏ & ′ \\ v∈Ω \end{matrix} (H^{1}(k_{v}, A[λ]), H^{1}(O_{v}, A[λ])) .

(The subgroup H^{1}(O_{v}, A[λ]) is defined and equal to Â(k_{v})/λA(k_{v}) only for v ∉ S, but that is enough.)

As usual, define the Selmer group

Sel_{λ} A := ker (H^{1}(k, A[λ]) → ∏_{v∈Ω} H^{1}(k_{v}, A)) .

Below will appear �^{1}(k, A[λ]), which is a subgroup of Sel_{λ} A, and is not to be confused with the Shafarevich-Tate group �(A) = �^{1}(k, A).

Theorem 4.14.

(a) The images of the homomorphisms

\xymatrix{ & H^{1}(k, A[λ]) \ar[d] \\ _{∏_{v∈Ω} \frac{Â(k_{v})}{λA(k_{v})}} \ar[r] & \begin{matrix}∏ & ′ \\ v∈Ω \end{matrix}(H^{1}(k_{v}, A[λ]), H^{1}(O_{v}, A[λ]))}

are maximal isotropic subgroups with respect to Q.

(b) The vertical map induces an isomorphism from Sel_{λ} A/�^{1}(k, A[λ]) to the intersection of these two images. (See Section 3 for information about �^{1}(k, A[λ]), which is often 0.)

Proof.

(a) The subgroup ∏_{v∈Ω} Â(k_{v})/λA(k_{v}) (or rather its image W under the horizontal injection)

is maximal isotropic by construction.

The vertical homomorphism H^{1}(k, A[λ]) → ^{∏′}_{v∈Ω}(H^{1}(k_{v}, A[λ]), H^{1}(O_{v}, A[λ])) is well-

defined since each element of H^{1}(k, A[λ]) belongs to the subgroup H^{1}(O_{T} , A[λ]) for some finite T ⊆ Ω containing S, and O_{T} ⊆ O_{v} for all v ∉ T . Let W be the image. Suppose that s ∈ H^{1}(k, A[λ]), and let w ∈ W be its image. The construction of the quadratic form of Corollary 4.7 is functorial with respect to base extension, so Q(w) can be computed by evaluating the global quadratic form

q: H^{1}(k, A[λ]) → H^{2}(k, 𝔾_{m})

on s, and afterwards summing the local invariants. Exactness of

0 → H^{2}(k, 𝔾_{m}) → ⨁_{v∈Ω} H^{2}(k_{v}, 𝔾_{m}) \underrightarrow{∑ inv} ℚ/ℤ → 0

in the middle (the reciprocity law for the Brauer group: see [GS06, Remark 6.5.6] for references) implies that the sum of the local invariants of our global class is 0. Thus Q|_{W} = 0.

It remains to show that W is its own annihilator. Since e_{λ} identifies A[λ] with its own Cartier dual, the middle three terms of the 9-term Poitou-Tate exact sequence ([Mil06, I.4.10(c)] and [GA09, 4.11]) give the self-dual exact sequence

H^{1}(k, A[λ]) ⟶^{β_{1}} \begin{matrix}∏ & ′ \\ v∈Ω \end{matrix}(H^{1}(k_{v}, A[λ]), H^{1}(O_{v}, A[λ])) ⟶^{γ_{1}} H^{1}(k, A[λ])^{∗},

where ∗ denotes Pontryagin dual. Since W = im(β_{1}) and the dual of β_{1} is γ_{1},

W^{⊥} = ker(γ_{1}) = im(β_{1}) = W.

(b) This follows from the exactness of (3) for S = Spec k_{v} for each v ∈ Ω. □

Remark 4.15. There is a variant of Theorem 4.14 in which the infinite restricted direct product is taken over only a subset S of Ω containing all bad places and all places of residue characteristic dividing deg λ. If S is finite, then the restricted direct product becomes a finite direct product. The same proof as before shows that the images of ∏_{v∈S} Â(k_{v})/λA(k_{v}) and H^{1}(O_{S}, A[λ]) are maximal isotropic. The intersection of the images equals the image of Sel_{λ} A.

Remark 4.16. Suppose that A = Jac C and λ is multiplication-by-2, with ℒ as in Remark 4.5. Let c_{T} be as in Definition 3.2, and let c_{T ,v} be its image in H^{1}(k_{v}, A[2]). It follows from [PS99, Corollary 2] that c_{T} ∈ Sel_{2} A. By [PR11, Theorem 3.9],

x ∪_{e_{2}} x = x ∪_{e_{2}} c_{T}  (5)

for all x ∈ H^{1}(k_{v}, A[2]). This, with Remark 2.8, implies that q_{v} takes values in \frac{1}{2}ℤ/ℤ (instead of just \frac{1}{4}ℤ/ℤ) if and only if c_{T ,v} = 0. Thus

Q takes values in \frac{1}{2}ℤ/ℤ ⟺ c_{T} ∈ �^{1}(k, A[2]).

For an example with c_{T} ∈ Sel_{2} A − �^{1}(k, A[2]), and another example with 0 ≠ c_{T} ∈ �^{1}(k, A[2]), see [PR11, Example 3.12].

Remark 4.17. Suppose that we are considering a family of abelian varieties with a systematic subgroup G of Sel_{λ} A coming from rational points (e.g., a family of elliptic curves with rational 2-torsion). Let X be the image of G in ^{∏′}_{v∈Ω}(H^{1}(k_{v}, A[λ]), H^{1}(O_{v}, A[λ])). Then our model for Sel_{λ} A should be that its image in X^{⊥}/X is an intersection of random maximal isotropic subgroups. In particular, the size of Sel_{λ} A/�^{1}(k, A[λ]) should be distributed as #X times the size of the random intersection.

Example 4.18. Suppose that char k ≠ p and A is an elliptic curve E: y^{2} = f(x). The theta divisor Θ on E is the identity point with multiplicity 1. Let ℒ = 𝒪(pΘ). Then λ is E →^{p} E, and Sel_{λ} A is the p-Selmer group Sel_{p} E. Moreover, �^{1}(k, E[p]) = 0 by Proposition 3.3(e). Thus Theorem 4.14 identifies Sel_{p} E as an intersection of two maximal isotropic subspaces in an 𝔽_{p}-vector space. Moreover, the values of the quadratic form on that space are killed by p, even when p = 2, since c_{T} = 0 in (5). In particular, dim Sel_{p} E should be expected to be distributed according to X_{Sel_{p}}, with the adjustment given by Remark 4.17 when necessary for the family at hand. This is evidence for Conjecture 1.1(a). The rest of Conjecture 1.1, concerning moments, is plausible given Proposition 2.22(a).

Example 4.19. The same reasoning applies to the p-Selmer group of the Jacobian of a hyperelliptic curve y^{2} = f(x) over a global field of characteristic not 2 in the following cases:

• f is separable of degree 2g + 1, and p is arbitrary;

• f is separable of degree 2g + 2, and p is odd.

(Use Propositions 3.3(f) and 3.4, and Remark 3.5.) This suggests Conjecture 1.7.

Example 4.20. Consider y^{2} = f(x) over a global field k of characteristic not 2 with deg f = 2g + 2 for even g ≥ 2. Proposition 3.4 and Remark 3.5 show that �^{1}(k, A[p]) is 0 with probability 1 for each p ≠ char k. But the Hilbert irreducibility theorem shows that c_{T} ≠ 0 with probability 1, so Remarks 4.16 and 2.8 suggest that dim Sel_{2} A now has the distribution X_{Sel_{2}} + 1. This suggests Conjecture 1.8.

In the analogous situation with g odd, it is less clear what to predict for Sel_{2} A: using techniques in [PS99, §9.2] one can show that the probability that c_{W_{1}} lies in Sel_{2} A is strictly between 0 and 1, and the existence of this element may invalidate the random model.

5. Relation to heuristics for � and rank

The Hilbert irreducibility theorem shows that asymptotically 100% of elliptic curves (ordered by naïve height) have E(ℚ)[p] = 0. (For much stronger results, see [Duk97] and [Jon10].) So for statistical purposes, when letting E run over all elliptic curves, we may ignore contributions of torsion to the p-Selmer group.

In analogy with the Cohen-Lenstra heuristics [CL84], Delaunay has formulated a conjecture describing the distribution of Shafarevich-Tate groups of random elliptic curves over ℚ. We now recall his conjectures for dim_{𝔽_{p}} �[p]. For each prime p and r ∈ ℤ_{≥0}, let X_{�[p],r} be a random variable taking values in 2ℤ_{≥0} such that

Prob (X_{�[p],r} = 2n) = p^{−n(2r+2n−1)} \frac{∏_{i=n+1}^{∞}(1 − p^{−(2r+2i−1)})}{∏_{i=1}^{n}(1 − p^{−2i})}.

The following conjecture is as in [Del01, Example F and Heuristic Assumption], with the correction that u/2 in the Heuristic Assumption is replaced by u (his u is our r). This correction was suggested explicitly in [Del07, §3.2] for rank 1, and it seems natural to make the correction for higher rank too.

Conjecture 5.1 (Delaunay). Let r, n ∈ ℤ_{≥0}. If E ranges over elliptic curves over ℚ of rank r, up to isomorphism, ordered by conductor, then the fraction with dim_{𝔽_{p}} �(E)[p] = 2n equals Prob (X_{�[p],r} = 2n).

If the “rank” r itself is a random variable R, viewed as a prior distribution, then the distribution of dim Sel_{p} E should be given by R + X_{�[p],R} . On the other hand, Theorem 4.14 suggests that dim Sel_{p} E should be distributed according to X_{Sel_{p}}. Let R_{conjectured} be the random variable taking values 0 and 1 with probability 1/2 each.

Theorem 5.2. For each prime p, the unique ℤ_{≥0}-valued random variable R such that X_{Sel_{p}} and R + X_{�[p],R} have the same distribution is R_{conjectured}.

Proof. First we show that R_{conjectured} has the claimed property. This follows from the following identities for n ∈ ℤ_{≥0}:

Prob (X_{Sel_{p}} = 2n) = c ∏_{j=1}^{2n} \frac{p}{p^{j} − 1}

= \frac{1}{2} ∏_{i≥0}(1 − p^{−(2i+1)}) · p^{−n(2n−1)} ∏_{j=1}^{2n}(1 − p^{−j})^{−1}

= \frac{1}{2}p^{−n(2n−1)} ∏_{i≥n+1}(1 − p^{−(2i−1)}) ∏_{i=1}^{n}(1 − p^{−2i})^{−1}

= \frac{1}{2} Prob (X_{�[p],0} = 2n)

Prob (X_{Sel_{p}} = 2n + 1) = c ∏_{j=1}^{2n+1} \frac{p}{p^{j} − 1}

= \frac{1}{2} ∏_{i≥0}(1 − p^{−(2i+1)}) · p^{−n(2n+1)} ∏_{j=1}^{2n+1}(1 − p^{−j})^{−1}

= \frac{1}{2}p^{−n(2n+1)} ∏_{i≥n+1}(1 − p^{−(2i+1)}) ∏_{i=1}^{n}(1 − p^{−2i})^{−1}

= \frac{1}{2} Prob (X_{�[p],1} = 2n) .

Next we show that any random variable R with the property has the same distribution as R_{conjectured}. For r ∈ ℤ_{≥0}, define a function f_{r}: ℤ_{≥0} → ℝ by f_{r}(s) := Prob(X_{�[p],r} = s − r). The assumption on R implies that

∑_{r=0}^{∞} Prob(R = r)f_{r}(s) = ∑_{r=0}^{∞} Prob (R_{conjectured} = r) f_{r}(s).

Thus to prove that R and R_{conjectured} have the same distribution, it will suffice to prove that the functions f_{r} are linearly independent in the sense that for any sequence of real numbers (α_{r})_{r≥0} with ∑_{r≥0} |α_{r}| < ∞, the equality ∑_{r=0}^{∞} α_{r}f_{r} = 0 implies that α_{r} = 0 for all r ∈ ℤ_{≥0}. In fact, α_{r} = 0 by induction on r, since f_{r}(s) = 0 for all s > r, and f(r, r) > 0. □

Acknowledgements

We thank Christophe Delaunay, Benedict Gross, Robert Guralnick, Karl Rubin, and the referee for comments.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

E-mail address: poonen@math.mit.edu
URL: http://math.mit.edu/~poonen/

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125

E-mail address: rains@caltech.edu