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.