Math 581C/582C
Introduction to L-functions

Ralph Greenberg

Autumn 2005/Winter 2006, Monday/Wednesday/Friday, 11:30
This course will be a survey of conjectures and theorems about various "L-functions" that occur in number theory. This topic is an important aspect of modern arithmetic algebraic geometry. The most familiar example of an L-function is the Riemann zeta function defined by ζ(s) = Σn=1∞ n-s (for Re(s) > 1). In the mid-19th century, Dirichlet introduced a certain generalization of ζ(s) (which he called L-functions) for the purpose of proving his famous theorem concerning the existence of infinitely many primes in arithmetic progressions. The course will start with this topic.

Another generalization to be discussed in the course is the natural analogue of ζ(s) which is associated to a curve over a finite field, and even more generally, algebraic varieties defined over a finite field. In the case of nonsingular algebraic varieties, one of the main theorems asserts that this zeta function is essentially a rational function and that one can determine the absolute values of zeros and poles (the analogue of the Riemann hypothesis). We will discuss this topic in some detail for curves and for other special types of algebraic varieties of higher dimension.

In a different direction, Artin found an important generalization of Dirichlet's L-functions in the 1920s. These are referred to as ``Artin L-functions. These functions can also be defined by convergent infinite series for Re(s) > 1. In attempting to prove the analytic continuation of these functions to the entire complex plane, Artin was led to his "general reciprocity law" - a far-reaching generalization of the law of quadratic reciprocity proved by Gauss. The analytic continuation of Artin L-functions is still an important open question. Considerable progress has been made. We will discuss all the necessary background to understand these functions.

The L-functions defined by Dirichlet and Artin can be expressed as "Euler products." For example, the Riemann zeta function can be defined by the infinite product ζ(s) = Π (1 - p-s ) for Re(s) > 1, where p varies over all the primes. An important generalization is associated to certain kinds of representations of the Galois group Gal(Q/Q), where Q. denotes the field of algebraic numbers - so called l-adic representations (where l is a prime number). The simplest example (after Artin L-functions which are one special case) occurs when one considers an elliptic curve $E$ defined over Q. In this case, the l-adic representation can be defined in terms of the points of order ln (for n >0) on the elliptic curve. This is usually called the "Hasse-Weil L-function" for E. Its properties have been the subject of study for a long time. One of the notable conjectures is the Birch and Swinnerton-Dyer conjecture. We will discuss this conjecture and the more general Bloch-Kato conjecture. These concern the values of the L-functions when s is an integer. A significant part of the course will concern the values of L-functions for integral values of s. This is an interesting topic even in the special case of the Riemann zeta function.

Prerequisites: A knowledge of some algebraic number theory and algebraic geometry would be helpful, although not essential. We will summarize the background that we need to discuss the various topics.