A Preview of Barbados/Bellairs

This post is probably not so interesting unless you plan to travel to the Caribbean in a few weeks. The website for the conference is offline, so I thought I might update attendees on what might be happening, at least those who read my blog.

There are two hours of talks in the morning by me and two hours of talks in the evening. Warning: the paragraphs below are not necessarily in one-to-one correspondence with talks.

Part I: I will give an overview of the Taylor-Wiles method in something approaching its original formulation (so without Kisin’s modifications). I may give the circular proof of modularity for \(\mathrm{GL}(1)\) as an example. I will then start talking about modular forms of weight one. I will give the details of local-global compatibility as proved in my paper with David, first in the irreducible case, and then via a modification of this method in the general case (using results which will be in Joel Specter’s thesis).

Background I: Jared and Peter will give a background talk on the geometry of Shimura varieties, with an emphasis on the case of modular curves, and (possibly) also that of Siegel 3-folds (threefolds?).

Part II: I will introduce the general strategy developed by myself and David to prove modularity lifting in the \(\ell_0 = 1\) and \(\ell_0 = \ell_0\) situations, in particular, the details of our patching lemma. I will outline how the method naturally breaks up into several different constituent problems (constructing Galois representations, proving local-global compatibility, proving vanishing of cohomology outside certain ranges, representation theoretic problems arising from Taylor-Wiles primes). I will then apply these strategies to prove minimal modularity lifting theorems for weight one modular forms in the residually irreducible setting.

Background II: David(?) will talk about Kisin’s modification of the Taylor-Wiles method. Toby Gee will talk about how to prove local-global compatibility and what that means in a (somewhat) general setting.

Part III: I will discuss the geometry of local deformation rings for \(\mathrm{GL}(2)\). Topics to be covered here include classical questions of multiplicity one and two, as well as non-minimal modularity lifting theorems in weight one.

Background III: Sug Woo will discuss the relation between cohomology and automorphic forms and how the Eichler-Shimura isomorphism generalizes to higher dimensions. Jack Thorne will discuss Taylor-Wiles primes for \(\mathrm{GL}(n)\).

Part IV: I will talk about my work with David concerning minimal modularity lifting theorems for low weight Siegel modular forms. This will consist of generalizing some of the ingredients from \(\mathrm{GL}(2)\), such as local-global compatibility results, and vanishing results of Lan-Suh. I also discuss an approach to Taylor-Wiles primes in the torsion setting for \(\mathrm{GL}(n)\).

Part V: I will talk about completed cohomology in low degree. I shall explain my results with Matt on the stability of completed cohomology, and the computation of these groups using \(K\)-theory.

Related Research: I have asked a number of people to talk about their recent work on topics related to this conference. This includes George Boxer who has agreed to talk about coherent cohomology and generalized Hasse invariants, and Ila Varma who will talk about local-global compatibility for non-self-dual representations at \(\ell \ne p\).