Southern California Number Theory Day at UCI, 10/27/2007

Abstracts

Elena Mantovan

Integral models for toroidal compactifications of Shimura varieties

In the case of good reduction, smooth integral models for Shimura varieties of PEL type have been constructed by Faltings and Chai. In my talk I'll describe how their construction can be extended to the cases of bad reduction at unramified primes, and discuss the geometry of the resulting spaces. A useful tool in this context is provided by the language of 1-motives. This is joint work with Ben Moonen.

Chandrashekhar Khare

Serre's conjecture

I will spend most of the time formulating Serre's conjecture and explaining some of its applications: for instance, it implies Artin's conjecture for 2-dimensional odd complex representations of the absolute Galois group of Q. I will sketch some of the main ideas in the recent proof of the conjecture in joint work with Wintenberger, as completed by Kisin. I will also explain, if time permits, how our work offers a fresh perspective on Wiles' proof of Fermat's Last Theorem (FLT). For instance it gives a diiferent, in a sense more elementary, aproach to Ribet's level lowering results, a key ingredient in the proof of FLT.

Audrey Terras

Fun with zeta and L-functions of graphs

I will present an introduction to zeta and L-functions of graphs by comparison with the zeta and L-functions of number theory. Basic properties will be discussed, including: the Ihara formula saying that the zeta function is the reciprocal of a polynomial. I will then explore graph analogs of the Riemann hypothesis, the prime number theorem, Chebotarev's density theorem, zero (pole) spacings, and connections with expander graphs and quantum chaos. References include my joint papers with Harold Stark in Advances in Mathematics. There is also a book I am writing on my website.

Vinayak Vatsal

Special values of L functions modulo p

It has been known since Euler that the values of the Riemann zeta function at negative integers are certain rational numbers, namely the Bernoulli numbers B_k. Similarly, the values of Dirichlet L-functions at s=0 are related to class numbers of certain number fields. These are simple instances of a common phenomenon, namely that the values of L-functions at critical points are algebraic, up to a simple factor, and that these algebraic numbers are related to algebraic quantities such as class numbers and Selmer groups. The present talk will be a survey talk on the algebraicity of special values of L-functions and their divisibility properties modulo primes.