*Time: *Friday, May 18, 2012,
14h30-15h30

*Location: *salle W

## The arithmetic of hyperelliptic curves

**Benedict Gross**
(Harvard - Personal website)

Manjul Bhargava has recently made significant progress on the arithmetic of elliptic curves over Q. Together with his student Arul Shankar, he has calculated the average order of the n-Selmer group, for n = 2,3,4,5, and has obtained an upper bound on the average rank (which is less than one). To do this, they identify elements of the Selmer group with certain orbits in a representation of a semi-simple group over Q, and estimate the number of orbits of bounded height using the geometry of numbers. In this talk, which is a report on joint work with Bhargava, I will explain how these techniques can be extended to study the arithmetic of hyperelliptic curves of a fixed genus over Q, with a marked rational Weierstrass point.