London Mathematical Society - EPSRC Durham Symposium
Automorphic forms and Galois representations
Monday 18th July - Thursday 28th July 2011

List of abstracts

Victor Abrashkin (Durham University) Tuesday 26th July 09:15
A semi-stable case of the Shafarevich Conjecture
Suppose F is the quotient field of the ring of Witt vectors with coeffcients in an algebraically closed field k of odd characteristic p. We construct an integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0;p). This modification of Breuil's theory results in the following application in the spirit of the Shafarevich Conjecture. If Y is a projective algebraic variety over rational numbers with good reduction away from 3 and semi-stable reduction modulo 3, then for the Hodge numbers of the complexification YC of Y it holds h2(YC) = h{1,1}(YC).

Laurent Berger (ENS de Lyon) Tuesday 19th July 09:15
In this talk, I will review the definition and basic properties of (phi,Gamma)-modules. The contents include : rings of power series, localization maps, (phi,Gamma)-modules, p-adic representations, p-adic Hodge theory, trianguline representations, representations of B2(Qp).

Christophe Breuil (Orsay University & CNRS) Wednesday 20th July 11:45
Extensions between Galois characters and mod. p local-global compatibility for GL2
Let F be a finite unramified extension of Qp. We give a way to recover (non-split) extensions between two characters of Gal(Q̅ p/F) over F̅ p in terms of the GL2(F)-action inside the cohomology mod. p of Shimura curves. This is joint work with F. Diamond.

Francis Brown (Jussieu) Tuesday 19th July 17:15
Mixed Tate motives over Z and fundamental group of P1 minus 3 points
In this talk, I will outline a proof of the following theorem: the category of mixed Tate motives over Z is generated by the fundamental group of the projective line minus 3 points. This implies a conjecture due to Deligne and Ihara on the action of the absolute Galois group on the pro-l fundamental group. The proof also implies a conjecture due to M. Hoffman, which states that every multiple zeta value ζ(n1,...,nr) is a Q-linear combination of ζ(a1,..,as) where ai=2 or 3.

David Burns (KCL) Tuesday 26th July 11:45
On main conjectures in geometric Iwasawa theory and related conjectures
We discuss a proof of the main conjecture of non-commutative Iwasawa theory for flat, smooth sheaves on schemes that are separated and of finite type over a finite field. We also describe various consequences of this result including an equivariant refinement of a leading term formula of Lichtenbaum, the proof of conjectures of Gross, of Tate, of Rubin and of Chinburg and a non-abelian generalisation of Deligne's proof of the Brumer-Stark Conjecture.

Colin Bushnell (KCL) Monday 25th July 17:15
To an effective local Langlands correspondence
This talk focuses on the local Langlands correspondence for complex representations. If F is a non-Archimedean local field, the correspondence takes irreducible n-dimensional representations of the Weil group of F to irreducible cuspidal representations of GL(n,F). The latter are known in great detail, from the classification of CB and Phil Kutzko. The talk addresses the way in which this detail is reflected in the structure of representations of the Weil group. The results are complete and explicit outside of one particular area.

Kevin Buzzard (Imperial) Monday 25th July 09:15
Reductions of 2-dimensional crystalline representations.
If p is a prime, k≥2 is an integer and a is an element of Q̅ p with norm less than 1, then (assuming a2 is not 4pk-1) there is a unique 2-dimensional crystalline representation of the absolute Galois group of Qp with Hodge-Tate weights 0 and k-1, determinant cyclok-1 and trace of Frobenius equal to a. However the natural proof of this statement unsurprisingly goes via big rings like Bcris and in practice we know less about this representation than we would like to. For example, what is the semisimplification of the mod p reduction of this representation? There are several approaches to answering such questions. I will give a survey of some approaches, both local and global, including some recent new results, and explain what is known and what is conjectured about the problem, plus why we are interested in it (i.e. applications).

Pierre-Henri Chaudouard (Paris-Sud) Thursday 21st July 11:45
Geometry of the fundamental lemma
I will present the main geometric objects that appear in the proof of the fundamental lemma by Ngô.

Gaetan Chenevier (Ecole Polytechnique) Friday 22nd July 11:45
Kneser neighbours and orthogonal Galois representations in dimensions 16 and 24 (joint with Jean Lannes)

Pierre Colmez (CNRS, Universite Paris 6) Wednesday 20th July 9:15, Thursday 21st July 9:15
(phi,Gamma)-modules and representations of GL(2,Qp), I, II
In these two talks, we will give an overview of the p-adic local Langlands correspondence for GL(2,Qp).

Henri Darmon (McGill) Tuesday 19th July 15:45
A p-adic Gross-Zagier formula for Garrett triple product L-functions

Luis Dieulefait (Barcelona) Tuesday 26th July 17:15
Non-solvable base change for GL(2)
We will explain the proof of base change for classical cusp forms of odd level to any Galois totally real number field assuming that a few small primes are split in the field. This is achieved through a method of propagation of modularity based on Modularity Lifting Theorems and the theory of congruences between modular forms (including cases of level raising and change of weight).

Mladen Dimitrov (University Paris Diderot) Saturday 23rd July 15:45
On the Eigencurve at classical weight one points
In this talk I will present a joint work with Joel Bellaiche on the geometry of the Eigencurve at classical points of weight one. A well-known result of Hida asserts that the Coleman-Mazur Eigencurve is etale over the weight space at all classical ordinary points of weight two or more, therefore is smooth at those points. Under a regularity assumption, we prove that smoothness equally holds at classical points of weight one and give a full description of the cases where etaleness holds too.

Matthew Emerton (Northwestern) Friday 22nd July 15:45
Moduli of potentially semi-stable Galois representations (joint with Toby Gee)
If K is a finite extension of Qp and n is a positive integer then we construct a stack X0 over Fp whose F-points (for any finite extension F of Fp) correspond to continuous representations GK → GLn(F). If tau is an n-dimensional type and k is a regular n-tuple of Hodge-Tate weights, then we construct a p-adic formal stack X(tau,k) over Zp whose O-valued points (for the ring of integers O in any finite extension of Qp) correspond to lattices in potentially semi-stable representations of GK of dimension n, type tau, and Hodge-Tate weights k. The underlying reduced Fp-stack of X(tau,k) is a union of irreducible components of X0, and the resulting specialization map X(tau,k) → X0 corresponds precisely to passage to the residual representation. We will explain how this framework provides a natural interpretation and generalization of the Breuil--Mezard conjecture. In the case n = 2 and K = Qp, we will describe X0 quite explicitly.

Laurent Fargues (Orsay) Tuesday 19th July 10:45, Wednesday 20th July 10:45
Curves and vector bundles in p-adic Hodge theory, I, II

Jean-Marc Fontaine (Paris 11) Friday 22nd July 10:45, Saturday 23rd July 10:45
Curves and vector bundles in p-adic Hodge theory, III, IV

Hidekazu Furusho (Nagoya University) Wednesday 20th July 17:15
Around Associators
I will review several topics around associators and explain recent developments related to them.

Toby Gee (Northwestern) Monday 25th July 15:45, Tuesday 26th July 15:45
Potential automorphy: the theorems
These are the second and third in a series of four talks on joint work of Tom Barnet-Lamb, Toby Gee, David Geraghty and Richard Taylor which shows that various Galois representations of arbitrary dimension become automorphic after a finite base change.

In Talk 2 (First Theorems) we will discuss the deformation theory of Galois representations and introduce `connectivity' and `potential diagonalizability'. We will then state the basic automorphy lifting theorems and explain briefly how they can be applied to obtain the basic potential automorphy theorem.

In Talk 3 (Main Theorems) we will describe automorphy lifting theorems and potential automorphy theorems in the potentially diagonalizable case as well as giving some idea of the proof (Harris' tensor product trick). We will also discuss applications of the same method to the weight part of Serre's conjecture and its generalizations.

Thomas Haines (University of Maryland) Tuesday 19th July 11:45
Introduction to endoscopic transfer
I will review, to the extent possible without detailed discussion of transfer factors, basic notions and definitions pertaining to endoscopic groups, the fundamental lemma, endoscopic transfer, and endoscopic character identities. If time permits I will discuss the (conjectural) endoscopic transfer of the stable Bernstein center.

Yuichiro Hoshi (RIMS, Kyoto) Friday 22nd July 17:15, Saturday 23rd July 17:15
Classical anabelian geometry/Grothendieck conjecture over local fields --- from "relative" to "absolute" ---
In the first talk, I will discuss works by Nakamura, Pop, Tamagawa, Mochizuki, Stix, ..... In the second talk, I will discuss the following topics:

-relative GC (Grothendieck Conjecture)
-absolute GC for canonical curves
-absolute GC for curves of Belyi type
-Section Conjecture-type result implies absolute GC.
Wieslawa Niziol (Utah) Wednesday 20th July 15:45
Comparison theorems: the open case
I will present a construction of the p-adic comparison morphism for semistable varieties (allowing horizontal divisors at infinity) that uses p-adic regulators into log-etale and log-syntomic cohomology.

Vytautas Paskunas (Universitaet Bielefeld) Friday 22nd July 09:15
On the Breuil-Mezard conjecture
The Breuil-Mezard Conjecture is a theorem of Kisin, which describes the Hilbert-Samuel multiplicities of potentially semi-stable deformation rings of 2-dimensional representations of the absolute Galois group of Qp. We will explain a purely local approach to it.

Florian Pop (UPenn) Thursday 21st July 10:45 and 17:15
General theory of outer representations of the Galois group/Grothendieck's Section Conjecture
In the first talk, I will discuss the following topics:

-The Grothendieck-Teichmuller group
-Faithfulness of such outer rep
-Ihara/Oda-Matsumoto (I/OM)
-Pro-l abelian-by-central I/OM
-Comparing profinite outer rep. with pro-l outer rep.

In the second talk, I will discuss Grothendieck's Section Conjecture.
Peter Schneider (Mathematisches Institut) Saturday 23rd July 09:15
From etale (phi,Gamma)-modules to equivariant sheaves

Sugwoo Shin (MIT) Saturday 23rd July 11:45, Monday 25th July 11:45
The trace formula and its applications
This is a 2-hour expository talk on the trace formula and its stabilization biased towards arithmetic. We will discuss applications to Langlands functoriality, cohomology of Shimura varieties and so on.

Benoit Stroh (Paris 13) Wednesday 27th July 10:45
Classicity and overconvergence (joint with Piloni)
Coleman proved that any overconvergent modular form with a sufficiently big weight on a modular curve is classical. We will explain and generalize this statement to many Shimura varieties. The Kottwitz-Rapoport stratification and the p-rank formula of Genestier-Ngô play a key role in our proof.

Richard Taylor (Harvard) Monday 25th July 10:45, Wednesday 27th July 15:45
Potential automorphy: Introduction and applications
These are the first and last in a series of talks on joint work of Tom Barnet-Lamb, Toby Gee, David Geraghty and Richard Taylor which shows that various Galois representations of arbitrary dimension become automorphic after a finite base change. This has many applications.

In the first talk I will recall the conjectural relationship between automorphic forms and Galois representations, using the 1 dimensional case as an example. I will introduce regularity and polarizability, assumptions without which we are able to make very little progress. Finally I will state the best theorem currently available attaching Galois representations to automorphic forms.

In the last talk I will discuss a general potential automorphy theorem, and explain various applications to the Sato-Tate conjecture, irreducibility of Galois representations associated to automorphic forms, and local-global compatibility.

Yichao Tian (Morningside Center of Math) Tuesday 26th July 10:45
Classicality of overconvergent Hilbert modular forms in the quadratic inert case
A famous theorem of Coleman says that an overconvergent p-adic elliptic modular form of small slope is classical. Now let F be a quadratic real field, and p be a rational prime that is inert in F. In this talk, I will explain that an overconvergent p-adic Hilbert modular form for F of integer weights (k1,k2) and slope < min{k1,k2}-2 is actually classical.

Jacques Tilouine (Paris 13) Wednesday 27th July 9:15
Overconvergent Igusa tower and overconvergent Siegel forms
In a joint work with O. Brinon and A. Mokrane, we construct, by using Brinon-Mokrane's overconvergent Igusa tower, modules of overconvergent Siegel modular forms over certain affinoids of the weight space. This is an alternative construction to the recent work by Andreatta-Iovita-Pilloni. This provides part of the eigenvariety for (eigen) holomorphic Siegel modular forms.

Jean-Pierre Wintenberger (Universite De Strasbourg) Wednesday 27th July 11:45
Ramification and Iwasawa modules
In a first part, we prove that Leopoldt and non vanishing of higher regulators conjectures are equivalent to the existence of Zp-extensions satisfying some properties of ramification. In a second part, we discuss where we are in an automorphic construction of these extensions.