Southern Califorina Number Theory Day 2012

The compatibility between the local and global Langlands correspondences on a certain reductive group can sometimes be reduced to the corresponding compatibility over some GL(n), with the aid of converse theorems. Such converse theorems are typically effective only for (globally) generic representations. In the case of holomorphic Siegel modular forms we discuss the interplay between a less restrictive converse theorem (obtained explicitly) and p-adic families to compensate for the fact that such forms are not generic.

Pablo Peleaz (Universtiy of Paris XIII) The motivic spectral sequence via birational invariants

The motivic spectral sequence which relates algebraic K-theory to motivic cohomology is the analogue in algebraic geometry of the Atiyah-Hirzebruch spectral sequence which relates topological K-theory to singular cohomology. In this talk, we will present an approach to the motivic spectral sequence via birational invariants, and show that it is equivalent to Voevodsky's approach via the slice filtration.

Kannan Soundararajan (Stanford University) Moments and the distribution of values of L-functions

I will discuss work on the value distribution and moments of families of L-functions. We will start with values to the right of the critical line, where the problem can be well modeled by random Euler products. This fails on the critical line, and the L-values here have a different flavor with Selberg's theorem on log-normality being a representative result.

Spencer Bloch (University of Chicago), "p-adic Hodge Conjecture"

The p-adic Hodge conjecture says that for a smooth variety over the p-adic numbers, an algebraic cycle on the closed fibre (i.e. mod p) lifts to an algebraic cycle if and only if its crystalline cohomology class is in the appropriate Hodge filtration level. Recent progress on this conjecture (joint with M. Kerz and H. Esnault) involving continuous K_0 will be explained.