Algebra and Number Theory Seminar Fall 2013

October 2, 2013

Introduction to Stark's Conjectures

Alex Beloi, University of California, Santa Cruz

We'll use some classic analytic results on special values of Hurtwitz zeta functions to illustrate how Stark's Conjecture works in some simple cases. We then give a precise statement of Stark's conjecture in the rank-1 abelian case, as well as explain what those terms mean, and conclude by stating some of the known results in specific cases.
__________________________________________________________________________________________

October 9, 2013

On the Principal Minors of Gross's Regulator Matrix

Samit Dasgupta, University of California, Santa Cruz

In 1981, Gross stated a conjecture relating the leading term of the p-adic L-function of a character of a totally real field to a p-adic regulator of p-units in an abelian extension of the field. We will describe a refinement of the conjecture that gives a formula for the principal minors of Gross's regulator matrix. In the specific case of the diagonal matrix, we recover a conjectural formula for these units that I proposed earlier. This is joint work with Michael Spiess.
____________________________________________________________________________________________

October 16, 2013

There and Back Again: Serre's Modularity Conjecture

Mitchell Owen, University of California, Santa Cruz

In this talk I will briefly discuss how one associates mod l Galois representations to (elliptic) modular forms, and why one might hope to reverse the process. I will state a precise version of this reversal known as Serre's conjecture, which was proven by Khare and Witenberger in 2008.

_____________________________________________________________________________________________

October 23, 2013

Eisenstein ideals at non-prime levels

Ken Ribet, University of California, Berkeley

Barry Mazur's 1977 landmark "Eisenstein ideal" article provided a definitive discussion of Eisenstein-like behavior for the space of weight-two cusp forms on Gamma_0(N) where N is a prime number. What if the weight is no longer 2? What if the level is no longer prime? Faltings--Jordan taught us that transition to weights greater than 2 is relatively smooth: one needs to think in terms of crystalline representations rather than representations associated with finite flat group schemes. For non-prime level, the situation is currently far from clear. I will focus on the case where the level is a product of two distinct primes and explain what seems to have been established to date. In part, I will rely on the 2013 PhD thesis of H. Yoo, who focused especially on ``multiplicity'' (i.e., dimension) questions.

October 30, 2013

Grothendieck topologies and cohomology theories

Gabriel Martins, University of California, Santa Cruz

In this talk I'll say some words about how the approach of sheaf cohomology through the use of derived functors let's us define new cohomology theories by altering the open covers of a topological space X.
This leads us to the definition of a Grothendieck topology on a category, a far reaching generalization of the idea of a topology on a set.
Finally, I'll talk a little about one particular Grothendieck topology for schemes that gives us the infinitesimal site and state a result due to Grothendieck relating it's cohomology to the DeRham cohomology of the scheme.

November 6, 2013

Crystalline Cohomology of Superspace

Martin Luu, Stanford University

In physical theories involving superspaces it is often useful to have a comparison of the cohomology of superspace with the cohomology of the underlying commutative - i.e. "bosonic" - subspace. It has been shown in recent years that quantum field theoretic considerations over rings other than the complex numbers can be very useful. This motivates us to construct an analogue of crystalline cohomology in the setting of an algebro-geometric version of supergeometry. We then prove a comparison result between this super-crystalline cohomology of a superscheme with the usual cryalline cohomology of the underlying "bosonic subscheme".

November 13, 2013

Vector-valued modular forms and why you might be interested in them.

Geoff Mason, University of California, Santa Cruz

A quick introduction to vector-valued modular forms. I'll explain what I think is the main (open) arithmetic problem about these gadgets and what is known about the solution. (Some of this is joint work with Cameron Franc.)

Thursday, November 21, 2013

The view from a few steps behind Mason

Terry Gannon, University of Alberta

I'll give two snap shots of my approach to vector-valued modular forms: existence (in the spirit of Riemann-Hilbert) and freeness (by studying elliptic fixed points). Mason and collaborators had both years ago, using very different methods. My justification for reproving these are that my methods perhaps are simpler, and appear to give additional stuff for free. Hopefully I will be able to convey the beauty of the developing theory of vector-valued modular forms.

*PLEASE NOTE THERE HAS BEEN A DATE CHANGE*

December 4, 2013

Cameron Franc, University of California, Santa Cruz

Saturday, December 7, 2013

BAANTAG
Ashay Burungale, UCLA

Hilaf Hasson, Stanford

Mark Kisin, Harvard

Claus Sorensen, University of California, San Diego

John Voight, Dartmouth