Algebraic Geometry Seminar

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Special Divisors on Hyperelliptic Curves

Speaker: Jacob Tsimerman, Harvard University

Location: Warren Weaver Hall 1314

Date: Tuesday, March 11, 2014, 3:30 p.m.

Synopsis:

Let X(1) be the moduli space of elliptic curves over C. Points on X(1) corresponding to elliptic curves with complex multiplication are known as Heegner points. The Andre-Oort conjecture - which was proven in this case by Pila in 2009 - describes how products of Heegner points are distributed in X(1)^n for the Zariski topology. We will first describe a strengthening of this conjecture by Zhang which describes how Galois orbits of Heegner points are distributed in X(1)^n for the Euclidean topology. We will then explain a natural function field analogue to Zhang's conjecture. In the simplest case, this analogue has an interpretation in terms of counting certain line bundles on hyper-elliptic curves over finite fields, and establishing it amounts to estimating the number of points on intersections of the theta divisor and its translates. Using intersection cohomology methods we are able to reduce the conjecture to an assertion that the total cohomology of such an intersection is exponentially bounded by the genus, and time permitting we explain how to prove this conjecture over the complex numbers. This is joint work with Vivek Shende.