At the AIM workshop on Cohen-Lenstra heuristics last week I got to hear Bjorn Poonen give a terrific talk about his recent work with Eric Rains about the distribution of mod p Selmer groups in a quadratic twist family of elliptic curves.
Executive summary: if E is an elliptic curve, say in Weierstrass form y^2 = f(x), and d is a squarefree integer, then we can study the mod p Selmer group Sel_d(E) of the quadratic twist dy^2 = f(x), which sits inside the Galois cohomology H^1(G_Q, E_d[p]). This is a finite-dimensional vector space over F_p. And by analogy with the Cohen-Lenstra heuristics for class groups, we can ask whether these groups obey a probability distribution as d varies — that is, does
Pr(dim Sel_d(E) = r | d in [-B, … B])
approach a limit P_r as B goes to infinity, and if so, what is it?
The Poonen-Rains heuristic is based on the following charming observation. The product of the local cohomology groups H_1(G_v, E[p]) is an infinite-dimensional F_p-vector space endowed with a bilinear form coming from cup product. In here you have two subspaces: the image of global cohomology, and the image of local Mordell-Weil. Each one of these, it turns out, is maximal isotropic — and their intersection is exactly the Selmer group. So the Selmer group can be seen as the intersection of two maximal isotropic subspaces in a very large quadratic space.
Heuristically, one might think of these two subspaces as being randomly selected among maximal isotropic subspaces. This suggests a question: if P_{r,N} is the probability that the intersection of two random maximal isotropics in F_p^{2N} has dimension r, does P_{r,N} approach a limit as N goes to infinity? It does — and the Poonen-Rains heuristic then asks that the probability that dim Sel_d(E) = r approaches the same limit. This conjecture agrees with theorems of Heath-Brown, Swinnerton-Dyer, and Kane in the case p=2, and with results of Bhargava and Shankar when p <= 5 (Bhargava and Shankar work with a family of elliptic curves of bounded height, not a quadratic twist family, but it is not crazy to expect the behavior of Selmer to be the same.) And in combination with Delaunay’s heuristics for variation of Sha, it recovers Goldfeld’s conjecture that elliptic curves are half rank 0 and half rank 1.
Johan de Jong wrote about a similar question, concentrating on the function field case, in his paper “Counting elliptic surfaces over finite fields.” (This is the first place I know where the conjecture “Sel_p should have size 1+p on average” is formulated.) He, too, models the Selmer group by a “random linear algebra” construction. Let g be a random orthogonal matrix over F_p; then de Jong’s model for the Selmer group is coker(g-1). This is a natural guess in the function field case: if E is an elliptic curve over a curve C / F_q, then the Selmer group of E is a subquotient of the etale H^2 of an elliptic surface S over F_q; thus it is closely related to the coinvariants of Frobenius acting on the H^2 of S/F_qbar. This H^2 carries a symmetric intersection pairing, so Frobenius (after scaling by q) is an orthogonal matrix, which we want to think of as “random.” (As first observed by Friedman and Washington, the Cohen-Lenstra heurstics can be obtained in similar fashion, but the relevant cohomology is H^1 of a curve instead of H^2 of a surface; so the relevant pairing is alternating and the relevant statistics are those of symplectic rather than orthogonal matrices.)
But this presents a question: why do these apparently different linear algebra constructions yield the same prediction for the distribution of Selmer ranks?
Here’s one answer, though I suspect there’s a slicker one.
A nice way to describe the distributions that arise in problems of Cohen-Lenstra type is by computing their moments. But the usual moments (e.g. “expected kth power of dimension of Selmer” or “kth power of order of Selmer” tend not to behave so well. Better is to compute “expected number of injections from F_p^k into Selmer,” which has a cleaner answer in every case I know. If the size of the Selmer group is X, this number is
(X-1)(X-p)….(X-p^{k-1}).
Evidently, if you know these “moments” for all k, you can compute the usual moments E(X^k) (which are indeed computed explicitly in Poonen-Rains) and vice versa.
Now: let A be the random variable (valued in abelian groups!) “intersection of two random maximal isotropics in a 2N-dimensional quadratic space V” and B be “coker(g-1) where g is a random orthogonal N x N matrix.”
The expected number of injections from F_p^k to B is just the number of injections from F_p^k to F_p^N which are fixed by g. By Burnside’s lemma, this is the number of orbits of the orthogonal group on Inj(F_p^k, F_p^N). But by Witt’s Theorem, the orbit of an injection f: F_p^k -> F_p^N is precisely determined by the restriction of the orthogonal form to F_p^k; the number of symmetric bilinear forms on F_p^k is p^((1/2)k(k+1)) and so this is the expected value to be computed.
What about the expected number of injections from F_p^k to A? We can compute this as follows. There are about p^{Nk} injections from F_p^k to V. Of these, about p^{2Nk – (1/2)k(k+1)} have isotropic image. Call the image W; we need to know how often W lies in the intersection of the two maximal isotropics V_1 and V_2. The probability that W lies in V_1 is easily seen to be about p^{-Nk + (1/2)k(k+1)}, and the probability that W lies in V_2 is the same; these two events are independent, so the probability that W lies in A is about p^{-2NK + (1/2)k(k+1)}. Summing over all isotropic injections gives an expected number of p^{(1/2)k(k+1)} injections from F_p^k to A. Same answer!
(Note: in the above paragraph, “about” always means “this is the limit as N gets large with k fixed.”)
What’s the advantage of having two different “random matrix” formulations of the heuristic? The value of the “maximal isotropic intersection” model is clear — as Poonen and Rains show, the Selmer really is an intersection of maximal isotropic subspaces in a quadratic space. One value of the “orthogonal cokernel” model is that it’s clear what it says about the Selmer group mod p^k.
Question: What does the orthogonal cokernel model predict about the mod-4 Selmer group of a random elliptic curve? Does this agree with the theorem of Bhargava and Shankar, which gives the first moment of Sel_4 in a family of elliptic curves ordered by height?
Quomodocumque 