One feature of the Poonen-Rains heuristics that might seem strange at first is that the dimension of the Selmer group isn’t 0 almost all the time. This is by contrast with the Cohen-Lenstra heuristic, where the p-torsion in the class group is indeed trivial about 1-1/p of the time. Instead, the Poonen-Rains heuristics predict that the p-Selmer rank is 0 about half the time and 1 about half the time, with about 1/p’s worth of measure devoted to ranks 2 or higher. Of course, given that we expect a random elliptic curve to have Mordell-Weil rank 1 half the time, it would be bad news for their heuristic if it predicted a lower frequency of positive Selmer rank!

But why is the intersection of two maximal isotropic subspaces 1 half the time and 0 half the time? You can get a nice picture of what’s going on by thinking about the case of a quadratic form Q in 4 variables. The vanishing of the quadratic form cuts out a quadric surface in P^3. A maximal isotropic subspace is a 2-dimensional space on which Q vanishes — in other words, a line on the quadric. The intersection of two maximal isotropics is o-dimensional if the corresponding lines are disjoint, 1-dimensional if the lines intersect at a point, and 2-dimensional when the lines coincide. So what’s the probability that two random lines on the quadric intersect? The key point is that there are *two families* of lines. If L1 and L2 come from different families, they intersect; if they come from the same family, they’re disjoint (except in the unlikely event they coincide.) So there you go — the intersection of the maximal isotropics is split 50-50 between 0-dimensional and 1-dimensional. More generally, the variety of maximal isotropic subspaces in an even-dimensional orthogonal space has two components, and this explains the leading term of Poonen-Rains.

It would be interesting to understand how to describe the “two types of maximal isotropics” in the infinite-dimensional F_p-vector space considered by Poonen-Rains, and to understand why the two maximal isotropics supplied by a given elliptic curve lie in the same family if and only if the L-function of E has even functional equation, which should lead one to expect that Sel_p(E) has even rank (or even, thanks to recent progress on the parity conjecture by Nekovar, Kim, los Dokchitsers, etc., *implies* that Sel_p(E) has even rank, subject to finiteness of Sha.)

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