While I was traveling this past week, there was a conference held here entitled L-functions and Automorphic Forms, which was a celebration of the 60th birthday of my math department colleague Dorian Goldfeld. From all I’ve heard the conference was a great success, well attended, with lots of interesting talks. But by far the biggest excitement was due to one talk in particular, that of Lucien Szpiro on “Finiteness Theorems for Dynamical Systems”. Szpiro, a French mathematician who often used to be a visitor at Columbia, but is now permanently at the CUNY Graduate Center, claimed in his talk to have a proof of the abc conjecture (although I gather that, due to Szpiro’s low-key presentation, not everyone in the audience realized this…).

The abc conjecture is one of the most famous open problems in number theory. There are various slightly different versions, here’s one:

For each $\epsilon >0$ there exists a constant $C_\epsilon$ such that, given any three positive co-prime integers a,b,c satisfying a+b=c, one has

$$ c < C_ \epsilon R(abc)^{1+\epsilon}$$ where $R(abc)$ is the product of all the primes that occur in a,b,c, each counted only once.

The abc conjecture has a huge number of implications, including Fermat’s Last Theorem, as well as many important open questions in number theory. Before the proof by Wiles, probably quite a few people thought that when and if Fermat was proved it would be proved by first proving abc. For a very detailed web-site with information about the conjecture (which leads off with a quotation from Dorian “The abc conjecture is the most important unsolved problem in diophantine analysis”), see here. There are lots of expository articles about the subject at various levels, for two by Dorian, see here (elementary) and here (advanced).

As far as I know, Szpiro does not yet have a manuscript with the details of the proof yet ready for distribution. Since I wasn’t at the talk I can only relay some fragmentary reports from people who were there. Szpiro has been teaching a course last semester which dealt a bit with the techniques he has been working with, here’s the syllabus which includes:

We will then introduced the canonical height associated to a dynamical system on the Riemann Sphere. We will study such dynamical systems from an algebraic point of view. In particular we will look at the dynamics associated to the multiplication by 2 in an elliptic curve . We will relate these notions and the questions they raised to the abc conjecture and the Lehmer conjecture.

For more about these techniques, one could consult some of Szpiro’s recent papers, available on his web-site.

The idea of his proof seems to be to use a and b to construct an elliptic curve E, then show that if abc is wrong you get an E with too many torsion points over quadratic extensions of the rational numbers. The way he gets a bound on the torsion is by studying the “algebraic dynamics” given by the iterated map on the sphere coming from multiplication by 2 on the elliptic curve. I’m not clear about this, but it also seems that what Szpiro was proving was not quite the same thing as abc (his exponent was larger than 1+ε, something which doesn’t change many of the important implications).

Maybe someone else who was there can explain the details of the proof. I suspect that quite a few experts are now looking carefully at Szpiro’s arguments, and whether or not he actually has a convincing proof will become clear soon.

Update: I’m hearing from some fairly authoritative sources that there appears to be a problem with Szpiro’s proof.