## 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.

## Expander graphs, gonality, and variation of Galois representations

Suppose you have a 1-dimensional family of polarized abelian varieties — or, just to make things concrete, an abelian variety A over Q(t) with no isotrivial factor.

You might have some intuition that abelian varieties over Q don’t usually have rational p-torsion points — to make this precise you might ask that A_t[p](Q) be empty for “most” t.

In fact, we prove (among other results of a similar flavor) the following strong version of this statement.  Let d be an integer, K a number field, and A/K(t) an abelian variety.  Then there is a constant p(A,d) such that, for each prime p > p(A,d), there are only finitely many t such that A_t[p] has a point over a degree-d extension of K.

The idea is to study the geometry of the curve U_p parametrizing pairs (t,S) where S is a p-torsion point of A_t.  This curve is a finite cover of the projective line; if you can show it has genus bigger than 1, then you know U_p has only finitely many K-rational points, by Faltings’ theorem.

But we want more — we want to know that U_p has only finitely many points over degree-d extensions of K.  This can fail even for high-genus curves:  for instance, the curve

C:   y^2 = x^100000 + x + 1

has really massive genus, but choosing any rational value of x yields a point on C defined over a quadratic extension of Q.  The problem is that C is hyperelliptic — it has a degree-2 map to the projective line.  More generally, if U_p has a degree-d map to P^1,  then U_p has lots of points over degree-d extensions of K.  In fact, Faltings’ theorem can be leveraged to show that a kind of converse is true.

So the relevant task is to show that U_p admits no map to P^1 of degree less than d; in other words, its gonality is at least d.

Now how do you show a curve has large gonality?  Unlike genus, gonality isn’t a topological invariant; somehow you really have to use the geometry of the curve.  The technique that works here is one we learned from an paper of Abramovich; via a theorem of Li and Yau, you can show that the gonality of U_p is big if you can show that the Laplacian operator on the Riemann surface U_p(C) has a spectral gap.  (Abramovich uses this technique to prove the g=1 version of our theorem:  the gonality of classical modular curves increases with the level.)

We get a grip on this Laplacian by approximating it with something discrete.  Namely:  if U is the open subvariety of P^1 over which A has good reduction, then U_p(C) is an unramified cover of U(C), and can be identified with a finite-index subgroup H_p of the fundamental group G = pi_1(U(C)), which is just a free group on finitely many generators g_1, … g_n.  From this data you can cook up a Cayley-Schreier graph, whose vertices are cosets of H_p in G, and whose edges connect g H with g_i g H.  Thanks to work of Burger, we know that this graph is a good “combinatorial model” of U_p(C); in particular, the Laplacian of U_p(C) has a spectral gap if and only if the adjacency matrix of this Cayley-Schreier graph does.

At this point, we have reduced to a spectral problem having to do with special subgroups of free groups.  And if it were 2009, we would be completely stuck.  But it’s 2010!  And we have at hand a whole spray of brand-new results thanks to Helfgott, Gill, Pyber, Szabo, Breuillard, Green, Tao, and others, which guarantee precisely that Cayley-Schreier graphs of this kind, (corresponding to finite covers of U(C) whose Galois closure has Galois group a perfect linear group over a finite field) have spectral gap; that is, they are expander graphs. (Actually, a slightly weaker condition than spectral gap, which we call esperantism, is all we need.)

Sometimes you think about a problem at just the right time.  We would never have guessed that the burst of progress in sum-product estimates in linear groups would make this the right time to think about Galois representations in 1-dimensional families of abelian varieties, but so it turned out to be.  Our good luck.