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.