Expander graphs, gonality, and variation of Galois representations

New paper on the arXiv with Chris Hall and Emmanuel Kowalski.

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.

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5 thoughts on “Expander graphs, gonality, and variation of Galois representations

  1. My version of the story is there

  2. Harald says:

    Bonege! Gratulojn!

    Chu iu ajn el viaj legantoj scias chu la implikacio (polilogaritma diametro -> esperantismo) validas ne nur por la grafoj de Cayley, sed ankau por la grafoj de Cayley-Schreier?

    (La implikacio alidirekta funkcias por chiu grafo de grado barita.)

  3. Harald says:

    PS. I think it might be good for certain abilities of yours in possible disuse if you took upon yourself to translate the above for the nationally limited.

  4. “…you might ask that A_t[p](Q) be empty for ‘most’ t.” Empty?? I’m pretty sure that’s too much to ask.

    Nitpicking aside, this is a really cool paper — I wish I had been involved with it somehow. (Not that I almost was or that there was any reasonable way I would have been — hence “I wish.”) I plan on inviting Chris Hall to UGA to talk about it in the near future. Or perhaps I just did?

  5. I have just posted some remarks by P. Sarnak on this paper here (with his permission).

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