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!