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.

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What a nice paper!