# Galois Representations and Automorphic Forms

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The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology. We discuss a...

The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology. We discuss a...

(Introduction to the Lecture Series and and overview for those unable to attend the whole Lecture Series)

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine--Mazur conjecture, and to a conjecture of Kisin.

Let F be a locally compact non-Archimedean field, p its residue characteristic and G a connected reductive algebraic group over F . The classical Satake isomorphism describes the Hecke algebra (over the field of complex numbers) of double...

I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for...

The usual Katz-Mazur model for the modular curve $X(p^n)$ has horribly singular reduction. For large n there isn't any model of $X(p^n)$ which has good reduction, but after extending the base one can at least find a semistable model, which means...

Let p be an odd prime number and let F be a totally real field. Let F_cyc be the cyclotomic extension of F generated by the roots of unity of order a power of p . From the maximal abelian extension of F_cyc which is unramified (resp. unramified...

Let p and l be two distinct prime numbers, and fix a positive integer d . I will explain how the F_l-cohomology complex of the Lubin-Tate tower of height d of a p-adic field K realizes mod l versions of both the semi-simple Langlands correspondence...

Iwasawa developed his theory for class groups in towers of cyclotomic fields partly in analogy with Weil's theory of curves over finite fields. In this talk, we present another such conjectural analogy. It seems intertwined with Leopoldt's...

To a regular algebraic cuspidal representation of GL(2) over a quadratic imaginary field, whose central character is conjugation invariant, Taylor et al. associated a two dimensional Galois representation which is unramified at l different from p...

These two talks will be about automorphic cohomology in the non-classical

case.

https://www.ias.edu/math/files/seminars/GriffithsTwoTalks.pdf

https://www.ias.edu/math/files/seminars/GriffithsTwoTalks.pdf

Let G be a connected reductive group over Q such that G(R) has discrete series representations. I will report on some statistical results on the Satake parameters (w.r.t. Sato-Tate distributions) and low-lying zeros of L-functions for families of...

We attach Galois representations to automorphic representations on unitary groups whose weight (=component at infinity) is a holomorphic limit of discrete series. The main innovation is a new construction of congruences, using the Hasse Invariant...

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F_1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic...

GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR

Some automorphic forms, despite the fact they are algebraic, do not have any interpretation as cohomology classes on a Shimura variety: therefore nothing is known at present on their expected...

GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR

Note: (joint work with O. Brinon and A. Mokrane)

The Bernstein center plays a role in the representation theory of locally profinite groups analogous to that played by the center of the group ring in the representation theory of finite groups. When F is a finite extension of Q_p, we discuss...

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS
SEMINAR**

Suppose that G is a connected, quasisplit, orthogonal or symplectic group over a field F of characteristic 0. We shall describe a classification of the irreducible representations of G(F) if F is...

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS
SEMINAR**

We shall recall the spectral terms from the trace formula for G and its stabilaization, as well as corresponding terms from the twisted trace formula for GL(N). We shall then discuss aspects of...

Periods of automorphic forms over spherical subgroups tend to: (1) distinguish images of functorial lifts and (2) give information about L-functions.

This raises the following questions, given a spherical variety X=H\G: Locally, which irreducible...

A well known result of Coleman says that p-adic overconvergent (ellitpic) eigenforms of small slope are actually classical modular forms. Now consider an overconvergent p-adic Hilbert eigenform F for a totally real field L. When p is totally...