2012-2013 Bowen Lectures

The 2012-13 Bowen Lectures will be delivered by Benedict Gross (Harvard University) on October 9th, 10th, and 11th, 2012. Each lecture begins at 4:10pm and ends at 5:00pm.

Series Title: On the arithmetic of hyperelliptic curves

Tuesday October 9th
Lecture 1: The rank of elliptic curves
50 Birge Hall 

I will review why elliptic curves are somewhat special, define the rank of a curve over the rational numbers, and motivate the conjecture of Birch and Swinnerton-Dyer on the L-function. After stating what is known, I will briefly introduce a new method, due to Manjul Bhargava, for studying the average rank.

Wednesday October 10th
Lecture 2: Hyperelliptic curves with a rational Weierstrass point
105 North Gate Hall

I will review the equations that define these curves, and describe the subgroup of 2-torsion in their Jacobians. After a review of the 2-descent, I will state the results which Bhargava and I have obtained on the average value of the order of the 2-Selmer group, and give some corollaries on the rational points of the curve and its Jacobian.

Thursday October 11th
Lecture 3: Arithmetic invariant theory
60 Evans Hall

I will describe a representation of the orthogonal group whose stable orbits are related to the 2-descent on hyperelliptic curves of a fixed genus with a rational Weierstrass point, and construct the principal homogeneous spaces for the Jacobian using pencils of quadrics.