NTS Fall 2012/Abstracts

September 13

Nigel Boston (UW–Madison)

Title: Non-abelian Cohen–Lenstra heuristics

Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian p-group A (p odd) arises as the p-class group of an imaginary quadratic field K is apparently proportional to 1/|Aut(A)|. The group A is isomorphic to the Galois group of the maximal unramified abelian p-extension of K. In work with Michael Bush and Farshid Hajir, I generalized this to non-abelian unramified p-extensions of imaginary quadratic fields. I shall recall all the above and describe a further generalization to non-abelian unramified p-extensions of H-extensions of Q, for any p, H, where p does not divide the order of H.

September 20

Simon Marshall (Northwestern)

Title: Multiplicities of automorphic forms on GL₂

Abstract: I will discuss some ideas related to the theory of p-adically completed cohomology developed by Frank Calegari and Matthew Emerton. If F is a number field which is not totally real, I will use these ideas to prove a strong upper bound for the dimension of the space of cohomological automorphic forms on GL₂ over F which have fixed level and growing weight.

September 27

Jordan Ellenberg (UW–Madison)

Title: Topology of Hurwitz spaces and Cohen-Lenstra conjectures over function fields

Abstract: We will discuss recent progress, joint with Akshay Venkatesh and Craig Westerland, towards the Cohen–Lenstra conjecture over the function field F_q(t). There are two key novelties, one topological and one arithmetic. The first is a homotopy-theoretic description of the "moduli space of G-covers with infinitely many branch points." The second is a description of the stable components of Hurwitz space over F_q, as a module for Gal(F_q/F_q). At least half the talk will be devoted to explaining why these objects are relevant to a very down-to-earth question like Cohen–Lenstra. If time permits, I'll explain what this has to do with the conjectures Nigel spoke about two weeks ago, and a bit about what Daniel is up to.

February 23

Sean Rostami (Madison)

Title: Centers of Hecke algebras

Abstract: The classification and construction of smooth representations of algebraic groups (over non-archimedean local fields) depends heavily on certain function algebras called Hecke algebras. The centers of such algebras are particularly important for classification theorems, and also turn out to be the home of some trace functions that appear in the Hasse–Weil zeta function of a Shimura variety. The Bernstein isomorphism is an explicit identification of the center of an Iwahori–Hecke algebra. I talk about all these things, and outline a satisfying direct proof of the Bernstein isomorphism (the theorem is old, the proof is new).