Tag Archives: expander graphs

JMM, Golsefidy, Silverman, Scanlon

Like Emmanuel, I spent part of last week at the Joint Meetings in New Orleans, thanks to a generous invitation from Alireza Salefi Golsefidy and Alex Lubotzky to speak in their special session on expander graphs.  I was happy that Alireza was willing to violate a slight taboo and speak in his own session, since I got to hear about his work with Varju, which caps off a year of spectacular progress on expansion in quotients of Zariski-dense subgroups of arithmetic groups.  Emmanuel’s Bourbaki talk is your go-to expose.

I think I’m unlike most mathematicians in that I really like these twenty-minute talks.  They’re like little bonbons — you enjoy one and then before you’ve even finished chewing you have the next in hand!  One nice bonbon was provided by Joe Silverman, who talked about his recent work on Lehmer’s conjecture for polynomials satisfying special congruences.  For instance, he shows that a polynomial which is congruent mod m to a multiple of a large cyclotomic polynomial can’t have a root of small height, unless that root is itself a root of unity.  He has a similar result where the implicit G_m is replaced by an elliptic curve, and one gets a lower bound for algebraic points on E which are congruent mod m to a lot of torsion points.  This result, to my eye, has the flavor of the work of Bombieri, Pila, and Heath-Brown on rational points.  Namely, it obeys the slogan:  Low-height rational points repel each other. More precisely — the global condition (low height) is in tension with a bunch of local conditions (p-adic closeness.)  This is the engine that drives the upper bounds in Bombieri-Pila and Heath-Brown:  if you have too many low-height points, there’s just not enough room for them to repel each other modulo every prime!

Anyway, in Silverman’s situation, the points are forced to nestle very close to torsion points — the lowest-height points of all!  So it seems quite natural that their own heights should be bounded away from 0 to some extent.  I wonder whether one can combine Silverman’s argument with an argument of the Bombieri-Pila-Heath-Brown type to get good bounds on the number of counterexamples to Lehmer’s conjecture….?

One piece of candy I didn’t get to try was Tom Scanlon’s Current Events Bulletin talk about the work of Pila and Willkie on problems of Manin-Mumford type.  Happily, he’s made the notes available and I read it on the plane home.  Tom gives a beautifully clear exposition of ideas that are rather alien to most number theorists, but which speak to issues of fundamental importance to us.  In particular, I now understand at last what “o-minimality” is, and how Pila’s work in this area grows naturally out of the Bombieri-Pila method mentioned above.  Highly recommended!

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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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