## Lipnowski-Tsimerman: How large is A_g(F_p)?

Mike Lipnowski and Jacob Tsimerman have an awesome new preprint up, which dares to ask:  how many principally polarized abelian varieties are there over a finite field?

Well, you say, those are just the rational points of A_g, which has dimension g choose 2, so there should be about p^{(1/2)g^2} points, right?  But if you think a bit more about why you think that, you realize you’re implicitly imagining the cohomology groups in the middle making a negligible contribution to the Grothendieck-Lefchetz trace formula.  But why do you imagine that?  Those Betti numbers in the middle are huge, or at least have a right to be. (The Euler characteristic of A_g is known, and grows superexponentially in dim A_g, so you know at least one Betti number is big, at any rate.)

Well, so I always thought the size of A_g(F_q) really would be around p^{(1/2) g^2}, but that maybe humans couldn’t prove this yet.  But no!  Lipnowski and Tsimerman show there are massively many principally polarized abelian varieties; at least exp(g^2 log g).  This suggests (but doesn’t prove) that there is not a ton of cancellation in the Frobenius eigenvalues.  Which puts a little pressure, I think, on the heuristics about M_g in Achter-Erman-Kedlaya-Wood-Zureick-Brown.

What’s even more interesting is why there are so many principally polarized abelian varieties.  It’s because there are so many principal polarizations!  The number of isomorphism classes of abelian varieties, full stop, they show, is on order exp(g^2).  It’s only once you take the polarizations into account that you get the faster-than-expected-by-me growth.

What’s more, some abelian varieties have more principal polarizations than others.  The reducible ones have a lot.  And that means they dominate the count, especially the ones with a lot of multiplicity in the isogeny factors.  Now get this:  for 99% of all primes, it is the case that, for sufficiently large g:  99% of all points on A_g(F_p) correspond to abelian varieties which are 99% made up of copies of a single elliptic curve!

That is messed up.

## Shende and Tsimerman on equidistribution in Bun_2(P^1)

Very nice paper just posted by Vivek Shende and Jacob Tsimerman.  Take a sequence {C_i} of hyperelliptic curves of larger and larger genus.  Then for each i, you can look at the pushforward of a random line bundle drawn uniformly from Pic(C) / [pullbacks from P^1] to P^1, which is a rank-2 vector bundle.  This gives you a measure $\mu_i$ on Bun_2(P^1), the space of rank-2 vector bundles, and Shende and Tsimerman prove, just as you might hope, that this sequence of measures converges to the natural measure.

I think (but I didn’t think this through carefully) that this corresponds to saying that if you look at a sequence of quadratic imaginary fields with increasing discriminant, and for each field you write down all the ideal classes, thought of as unimodular lattices in R^2 up to homothety, then the corresponding sequence of (finitely supported) measures on the space of lattices converges to the natural one.

Equidistribution comes down to counting, and the method here is to express the relevant counting problem as a problem of counting points on a variety (in this case a Brill-Noether locus inside Pic(C_i)), which by Grothendieck-Lefschetz you can do if you can control the cohomology (with its Frobenius action.)  The high-degree part of the cohomology they can describe explicitly, and fortunately they are able to exert enough control over the low-degree Betti numbers to show that the contribution of this stuff is negligible.

In my experience, it’s often the case that showing that the contribution of the low-degree stuff, which “should be small” but which you don’t actually have a handle on, is often the bottleneck!  And indeed, for the second problem they discuss (where you have a sequence of hyperelliptic curves and a single line bundle on each one) it is exactly this point that stops them, for the moment, from having the theorem they want.

Error terms are annoying.  (At least when you can’t prove they’re smaller than the main term.)