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.