Harald Helfgott

Abstract:   Siegel's theorem on integer points on curves was originally non-explicit; Faltings' theorem still is. For many applications, what is needed is quantitative versions of Siegel's and Faltings' theorems; certain simplifying assumptions may be justified, but the quantitative results must be as strong as possible. We will see how to bound the number of integer points on elliptic curves and the number of rational points of bounded height on curves of higher genus by means of sphere packings. The said bounds allow us to strengthen the existing results on the square-free sieve. Joint work by the speaker and A. Venkatesh has brought about two further applications: the number of elliptic curves of conductor N is shown to be at most O(N^0.2239), whereas the number of 3-torsion elements in the class group of Q(\/-N) is at most O(N^0.442). (In both cases, the best previous bounds were of the order of O(N^1/2+epsilon).) We obtain these bounds by a refinement of the simplest approach based on sphere packings; this refinement allows us to go beyond the Bombieri-Pila bound for certain kinds of equations.