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Graduate Summer School

Arithmetic of L-functions    — Application DEADLINE January 28, 2009

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 background material on group representations and L-functions I will explain Stark's original conjectures, some analogues, and the enormous amount of evidence for them, theoretical and computational. Most of what I say is in my book "Les conjectures de Stark sur les fonctions L d'Artin en $s=0$".

David Burns, King's College London   and   Guido Kings, UniversitäRegensburg

The Equivariant Tamagawa Number Conjecture

This is a course on the Equivariant Tamagawa Number Conjecture (ETNC) for Dirichlet L-functions and L-functions associated to elliptic curves.The course will focus on:

1. stating the conjecture;
2. presenting evidence in support of the conjecture;
3. proving that the integral and p-adic refinements of Stark's Conjecture (a la Rubin and Gross) are consequences of the ETNC for Dirichlet L-functions and that the Birch and Swinnerton-Dyer  Conjecture is a consequence of the ETNC for L-functions associated to elliptic curves.

Manfred Kolster, McMaster University   and   Cristian Popescu, University of California-San Diego

Integral Abelian Stark-type Conjectures

Topics:

1. Integral refinements of Stark's Conjecture for abelian L-functions of arbitrary order of vanishing at s=0 and consequences.
2. p-adic refinements of Stark's Conjecture for abelian L-functions and consequences.
3. The conjectures of Lichtenbaum and Coates-Sinnott on special values of abelian L-functions at negative integers.
4. An equivariant main conjecture in Iwasawa theory and consequences.

David Rohrlich, Boston University

Root Numbers

After a general survey of L-functions, functional equations, and epsilon factors, we shall focus on the connections between root numbers and the arithmetic of elliptic curves. Some attention will be paid throughout to the relevant issues in the representation theory of finite groups.

Karl Rubin, University of California-Irvine

Euler Systems

Euler systems were introduced by Kolyvagin as a new tool for bounding the size of ideal class groups and Selmer groups, and for relating the sizes of those groups to special values of L-functions.  In this course we will describe the basic Euler system machinery, and apply it in the fundamental cases of cyclotomic fields (class number formulas) and elliptic curves (the Birch and Swinnerton-Dyer conjecture).

Douglas Ulmer, University of Arizona

The Birch and Swinnerton-Dyer Conjecture over Function Fields

We know a lot more about ranks of elliptic curves and the conjecture of Birch and Swinnerton-Dyer over function fields than we do over number fields.  I plan to discuss how one can prove special cases of the the BSD conjecture over function fields as well as how one can construct elliptic curves with large Mordell-Weil groups.  Much of this also applies to higher dimensional Jacobians.  The lectures should be accessible to anyone with a first course in algebraic geometry and some acquaintance with elliptic curves.

Vinayak Vatsal, University of British Columbia

Complex Multiplication and Heegner Points

We will start by discussing the Kronecker-Weber theorem, which gives a description of the abelian extensions of the rational
numbers in terms of points of finite order on the circle group. We then move to the theory of complex multiplication, which
gives an analogous description of the abelian extensions of imaginary quadratic fields in terms of point of finite order on certain special elliptic curves, the so called CM elliptic curves. From this we move on to Heegner points, namely, the points on modular curves associated to these CM elliptic curves.  We will discuss their basic properties, and some of their surprising applications to number fields and the arithmetic of all elliptic curves. If time permits, we will discuss some of the many generalizations of CM points to higher dimensions, other number fields, and p-adic settings.

 

Participants in the Graduate Summer School also may wish to become involved in the Undergraduate Summer School, attend parts of the Research Program, or participate in the programs of the Education component. Graduate students are expected to participate in Institute-wide activities such as the "Cross Program Activities" and may be asked to contribute some time to volunteer projects related to running the Summer Session.

A limited number of graduate students who have not completed the basic courses may attend. These students will attend some graduate level courses and may be involved as teaching assistants in other programs or work as audio-visual assistants.

The Graduate Summer School is supported by National Science Foundation grant no. 0437137.