3. Postdoctoral Program - MSRI

IAS/PCMI Summer Workshop: The Arithmetic of L-Functions
June 28, 2009 to July 18, 2009
Organized by: Cristian Popescu (University of California, San Diego), Karl Rubin* (University
of California, Irvine), Alice Silverberg (University of California, Irvine)
The Graduate Summer School bridges the gap between a general graduate education in
mathematics and the specific preparation necessary to do research on problems of current
interest. In general, these students will have completed their first year, and in some cases, may
already be working on a thesis. While a majority of the participants will be graduate students,
some postdoctoral scholars and researchers may also be interested in attending.
Prerequisite is a course in algebraic number theory, or equivalent. Familiarity with the language
and methods of algebraic geometry would also be helpful for some of the courses.
The main activity of the Graduate Summer School will be a set of intensive short lectures offered
by leaders in the field, designed to introduce students to exciting, current research in
mathematics. These lectures will not duplicate standard courses available elsewhere. Each course
will consist of lectures with problem sessions. Course assistants will be available for each lecture
series. The participants of the Graduate Summer School meet three times each day for lectures,
with one or two problem sessions scheduled each day as well.
Course Titles and Descriptions:
The 2009 Summer Session in Arithmetic of L-functions will consist of eight graduate level
lecture series. On any day during the summer session, three lectures will be offered. Graduate
students are asked to attend the lectures as well as two daily problem sessions associated with the
lecture and led by a graduate TA.
Benedict Gross, Harvard University
Introduction to the Birch and Swinnerton-Dyer Conjecture
These lectures will give an overview of the Birch and Swinnerton-Dyer conjecture, which relates
the L-function of an elliptic curve at s=1 to arithmetic information about the curve. We will
formulate the conjecture, discuss the current state of progress on it, and describe some methods
for attacking it.
John Tate, University of Texas-Austin
Introduction to Stark's Conjectures
The conjectures concern the leading term $c(\chi)s^{r(\chi)}$ of the Taylor expansion at $s=0$
of the Artin L-function $L(s,\chi,K/k)$ attached to a character $\chi$ of the Galois group $G$ of
a finite Galois extension $K/k$ of number fields. In the case of the zeta function $(K=k,
\chi=1)$, the coefficient $c(\chi)$ is given by the so-called class number formula. Stark's great
achievement in the 1970's was to give an analogue for an arbitrary L-function. After some