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.