Generalized ideal class groups can be described adelically in terms of a coset space for the group GL1, and this in turn leads to a notion of "class number" (as the size of a certain set, if finite) for an arbitrary affine algebraic group over a global field. Related to this is the notion of the "Tate-Shafarevich set" of an algebraic group, which is tied up with questions relating global and local information. Finiteness of class numbers and Tate-Shafarevich sets for affine algebraic groups was proved by Borel and his coworkers over number fields, andif one grants the finiteness of Tate-Shafarevich groups for abelian varieties then Mazur showed how to get such finiteness for all algebraic group varieties over number fields (which has applications to the local-to-global principle for projective varieties over number fields).

The above methods do not apply over global function fields. After reviewing some history, I will explain the content of a recent classification theorem of "pseudo-reductive groups" proved jointly with Gabber and G. Prasad that makes it possible to prove the analogous finiteness theorems in the function field case away from characteristic 2. If time permits I will say something about how this classification theorem is used to get such results.

Mathematical Systems Biology - Spatial Dynamics and Growth and Signaling

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.

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 Bk. 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.

XVth Southern California Geometric Analysis Seminar