Benedict H. Gross (Harvard University)
The arithmetic of hyperelliptic curves
Hyperelliptic curves first arose in the work of Abel, who generalized some of Euler's results on elliptic integrals. They were studied over the real and complex numbers by Legendre, Jacobi, and Riemann. I will review some of this material, then turn to the study of rational points on hyperelliptic curves over Q. I will describe a beautiful recent result of B. Poonen and M. Stoll on curves with a rational Weierstrass point. I will also will show that a positive proportion of hyperelliptic curves of a fixed genus g >= 2 over Q have no points over any number field of odd degree, a result that is joint work with M. Bhargava and X. Wang.