# 2009 Chern Lectures

*The 2009 Chern Lectures will be delivered by Richard Taylor on April 14, 15, 16 and 17, 2009.*

Department of Mathematics, University of California, Berkeley, presents

The 2009 Chern Lectures

Richard Taylor

Department of Mathematics

Harvard University

**Reciprocity Laws and Density Theorems**

**Talk 1: **

*April 14, 10 Evans Hall, 4:00-5:00pm*

(To download PDF files of the slides for this talk, click the corresponding link at the bottom of this page.)

The first lecture will present a leisurely historical introduction to
reciprocity laws and density theorems starting with Gauss' law of
quadratic reciprocity and finishing with the Sato-Tate conjecture. This
lecture will sketch Serre's approach to deducing density theorems from
reciprocity laws.

**Talk 2: **

*April 15, 10 Evans Hall, 4:00-5:00pm*

(To download PDF files of the slides for this talk, click the corresponding link at the bottom of this page.)

The second lecture will introduce a more general language for discussing
reciprocity laws (Galois representations and automorphic forms) and
will formulate somewhat more precisely the principal conjectures.

**Talk 3: **

*April 16, 60 Evans Hall, 4:00-5:00pm*

The third lecture will focus on the main techniques we have for proving
reciprocity laws: the construction of Galois representations for
automorphic forms, the theory of base change and modularity lifting
theorems. I will discuss the current state of our knowledge and the
principal barriers to further progress.

**Talk 4: **

*April 17, 740 Evans Hall, 4:00-5:00pm*

The final lecture will discuss potential reciprocity laws. In
particular, the lecture will explain why these laws suffice for
applications to density theorems and will sketch the use of the Dwork
family to prove a series of potential reciprocity laws. Finally, the
lecture will discuss applications and again indicate the main obstacles
to further progress.

** Abstract:** Much attention has been paid to the counting of the
number of solutions to polynomial congruences, i.e. the number of
solutions to the polynomial equations in the integers modulo m for some
m. Some of the hardest questions involve fixing the equation(s) and
letting the modulus m vary over prime numbers. In this case one can ask
for an algorithm which given the modulus

*m*predicts the number of solutions (reciprocity laws) or about the distribution of the number of solutions as m varies (density theorems). The two sorts of questions are not surprisingly related. Reciprocity laws go back to Gauss' celebrated law of quadratic reciprocity and density theorems to Dirichlet's theorem on primes in an arithmetic progression. Until recently almost all the results known concerned a single polynomial in one variable. Following Wiles' breakthrough in the early 90's, much progress has been made on more complicated reciprocity laws like the Shimura-Taniyama conjecture on elliptic curves (a single cubic equation in two variables). More recently still, the corresponding density theorem (the Sato-Tate conjecture) has been proved for elliptic curves.

In my first lecture I will present a leisurely historical introduction to reciprocity laws and density theorems starting with Gauss' law of quadratic reciprocity and finishing with the Sato-Tate conjecture. I will sketch Serre's approach to deducing density theorems from reciprocity laws. In my second I will introduce a more general language for discussing these questions (Galois representations and automorphic forms) and formulate somewhat more precisely the principal conjectures. In my third lecture I will discuss the main techniques we have for proving reciprocity laws: the construction of Galois representations for automorphic forms, the theory of base change and modularity lifting theorems. I will try to indicate the current state of our knowledge and what I see as the principal barriers to further progress. In my final lecture I will discuss potential reciprocity laws. I will explain why they suffice for applications to density theorems and I will sketch the use of the Dwork family to prove a series of potential reciprocity laws. I will discuss applications and again indicate what I see as the main obstacles to further progress.