# 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

Abstract:

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

Abstract:

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

Abstract:

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.