Poonen-Rains and lines on a quadric surface

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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