UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421


Fall Quarter 2014

For previous quarters' schedule, click here.


September 11

Peter Schneider (University of Munster and MSRI)
Rigid character groups, Lubin-Tate theory, and (phi,Gamma)-modules

The talk will describe joint work with L. Berger and B. Xie in which we build, for a finite extension L of Q_p, a new theory of (phi,Gamma)-modules whose coefficient ring is the ring of holomorphic functions on the rigid character variety of the additive group o_L, resp. a "Robba" version of it.

October 2

NO SEMINAR

October 9

NO SEMINAR

October 16

NO SEMINAR

October 23

Michelle Manes (University of Hawai'i)
Galois theory of quadratic rational functions

Given a global field K and a rational function f(x) defined over K, one may take pre-images of 0 under successive iterates of f, and thus obtain an infinite tree by assigning edges according to the action of f. The absolute Galois group of K acts on the tree, giving a subgroup of the group of all tree automorphisms.
Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. The analogy here is to Serre's finite index results for Galois representations arising from elliptic curves.
I will discuss the contributions of several researchers, including Boston and Jones, along with my own work (joint with Jones) on these questions.

October 30

Valentijn Karemaker (University of Utrecht)
Hecke algebra isomorphisms and adelic points on algebraic groups

Let G denote an algebraic group over Q and K and L two number fields. Assume that there is a group isomorphism of points on G over the adeles of K and L, respectively. We establish conditions on the group G, related to the structure and the splitting field of its Borel groups, under which K and L have isomorphic adele rings. Under these conditions, if K or L is a Galois extension of Q and G(A_K) and G(A_L) are isomorphic, then K and L are isomorphic as fields. As a corollary, we show that an isomorphism of Hecke algebras for GL(n) (for fixed n > 1), which is an isometry in the L^1 norm over two number fields K and L that are Galois over Q, implies that the fields K and L are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over Q.

November 6

Dorian Goldfeld (Columbia University)
TBA

TBA

November 13

Kim Laine (Berkeley)
Security in genus 3

TBA

November 20

TBA
TBA

TBA

November 27

THANKSGIVING BREAK

December 4

TBA
TBA

TBA

December 11

TBA
TBA

TBA