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Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-theory

Monday, December 5, 2016 to Friday, December 9, 2016

CIRM, Luminy, France

Organizers: Laurent Clozel (Paris Sud), Francesco Calegari (Chicago)

CMI Enhancement and Partnership Program

Infomation

Overview

There has been exciting progress recently on two (three ?) connected topics: the extension of the Taylor-Wiles method to situations where the standard (im)parity condition fails, and the relation of deformation theory (starting with Hida’s idea) with the Leopoldt conjecture. The salient points are the paper(s) of Calegari and Geraghty, Calegari’s paper on stable (completed) homology for congruence groups, and the Khare-Thorne paper on Leopoldt; the construction of new Galois representations by Harris-Lan-Thorne-Taylor and Scholze, in particular for torsion classes, is also a crucial advance. Finally, Calegari’s paper reveals new relations with problems in K-theory on which our kowledge does not seem to have improved much since the Batelle meetings.

This workshop will be limited to the researchers involved in these developments. The aim is to obtain an up-to date integrated picture of these developments as well as their interrelations.  Connected theories such as completed cohomology should be reviewed, as well as some complements in K-theory. 

Related Links: 
Workshop webpage
CIRM

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Home  » Automorphic forms, Shimura varieties, Galois representations and L-functions

Workshop

Automorphic forms, Shimura varieties, Galois representations and L-functions

December 01, 2014 – December 05, 2014


Registration Deadline: December 05, 2014 over 3 years ago
To apply for Funding you must register by: September 01, 2014 almost 4 years ago
Parent Program:
  • New Geometric Methods in Number Theory and Automorphic Forms
Location: MSRI: Simons Auditorium, Atrium
Organizers
LEAD&nbsp Pierre Colmez (Institut de Mathématiques de Jussieu), Stephen Kudla (University of Toronto), Elena Mantovan (California Institute of Technology), Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure), Richard Taylor (Institute for Advanced Study)
Speaker(s)

Show List of Speakers

  • Frank Calegari (Northwestern University)
  • Ana Caraiani (Imperial College, London)
  • Gaetan Chenevier (Centre National de la Recherche Scientifique (CNRS))
  • Laurent Clozel (Université de Paris XI)
  • Pierre Colmez (Institut de Mathématiques de Jussieu)
  • Henri Darmon (McGill University)
  • Matthew Emerton (University of Chicago)
  • Laurent Fargues (Institut de Mathématiques de Jussieu)
  • Benjamin Howard (Boston College)
  • Kai-Wen Lan (University of Minnesota, Twin Cities)
  • Dipendra Prasad (Tata Institute of Fundamental Research)
  • A. Raghuram (Indian Institute of Science Education and Research)
  • Peter Scholze (Universität Bonn)
  • Christopher Skinner (Princeton University)
  • Akshay Venkatesh (Stanford University)
  • Marie-France Vigneras (Université de Paris VII (Denis Diderot))
  • Sarah Zerbes (University College)
  • Wei Zhang (Columbia University)
Description

L-functions attached to Galois representations coming from algebraic   geometry contain subtle arithmetic information (conjectures of Birch and Swinnerton-Dyer, Deligne, Beilinson, Bloch and Kato, Fontaine and Perrin-Riou). Langlands has predicted the existence of a correspondence relating these L-functions to L-functions of automorphic forms which are much better understood. The workshop will focus on recent developments related to Langlands correspondence (construction of Galois representations attached to automorphic forms via the cohomology of Shimura varieties, modularity of Galois representations…) and arithmetic of special values of L-functions.

It will be dedicated to Michael Harris as a tribute to his enormous influence on the themes of the workshop.

Bibliography (PDF)

 

Organized in partnership with:

Additional funding provided by:

ArShiFo

This workshop has received funding from the European Research Council under the European Community’s Seventh
Framework Programme (FP7/2007-2013) / ERC Grant agreement n¬° 290766 (AAMOT)

L-functions attached to Galois representations coming from algebraic   geometry contain subtle arithmetic information (conjectures of Birch and Swinnerton-Dyer, Deligne, Beilinson, Bloch and Kato, Fontaine and Perrin-Riou). Langlands has predicted the existence of a correspondence relating these L-functions to L-functions of automorphic forms which are much better understood. The workshop will focus on recent developments related to Langlands correspondence (construction of Galois representations attached to automorphic forms via the cohomology of Shimura varieties, modularity of Galois representations…) and arithmetic of special values of L-functions.

It will be dedicated to Michael Harris as a tribute to his enormous influence on the themes of the workshop.

Bibliography (PDF)

 

Organized in partnership with:

Additional funding provided by:

ArShiFo

This workshop has received funding from the European Research Council under the European Community’s Seventh
Framework Programme (FP7/2007-2013) / ERC Grant 
agreement n¬° 290766 (AAMOT)

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Keywords and Mathematics Subject Classification (MSC)

Primary Mathematics Subject Classification

  • 11-xx – Number theory

Secondary Mathematics Subject Classification
No Secondary AMS MSC

Funding & Logistics
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Funding

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.’s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

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Lodging

MSRI has preferred rates at the Hotel Durant . Reservations may be made by calling 1-800-238-7268. When making reservations,
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enter the promo/corporate code 123MSRI. Our preferred rate is $139 per night for a Deluxe Queen/King, based
on availability.

MSRI has preferred rates of $149 – $189 plus tax at the
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availability. Guests can either call the hotel’s main line at 510-845-7300 and ask for the
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MSRI has preferred rates of $110 – $140 at the Berkeley Lab Guest House , depending on room availability. Reservations may be made by calling
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Select “I am an individual traveler affiliated with MSRI”.

Additional lodging options may be found on our
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Schedule, Notes/Handouts & Videos
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  • Agenda View
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Dec 01, 2014
Monday
09:15 AM – 09:30 AM

 

Welcome

Location
MSRI: Simons Auditorium
Video
Abstract
Supplements

Show Detail

09:30 AM – 10:30 AM

 

Parabolically induced mod p representations of reductive p-adic groups

Marie-France Vigneras (Université de Paris VII (Denis Diderot))
Location
MSRI: Simons Auditorium
Video
Abstract
Supplements
Title & Abstract
37.8 KB application/pdf
Vigneras Notes
156 KB application/pdf
Show Detail

10:30 AM – 11:00 AM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

11:00 AM – 12:00 PM

 

Moduli stacks of local Galois representations

Matthew Emerton (University of Chicago)
Location
MSRI: Simons Auditorium
Video
Abstract
Supplements
Title & Abstract
83.1 KB application/pdf
Notes Emerton
350 KB application/pdf
Show Detail

12:00 PM – 02:00 PM

 

Lunch

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

02:00 PM – 03:00 PM

 

Integral p-adic Hodge theory

Peter Scholze (Universität Bonn)
Location
MSRI: Simons Auditorium
Video
Abstract

We will discuss joint work with Bhargav Bhatt and Matthew Morrow which proves a comparison between integral p-adic etale cohomology and integral crystalline cohomology, using a geometric construction of Breuil-Kisin modules.

Supplements
Scholze Notes
289 KB application/pdf
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03:00 PM – 03:30 PM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

04:10 PM – 05:00 PM

 

MSRI/Evans Lecture

Pierre Colmez (Institut de Mathématiques de Jussieu)
Location
740 Evans Hall, UC Berkeley
Video
Abstract
Supplements
Notes Colmez
948 KB application/pdf
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Dec 02, 2014
Tuesday
09:30 AM – 10:30 AM

 

Special parahorics and exotic good reduction

Wei Zhang (Columbia University)
Location
MSRI: Simons Auditorium
Video
Abstract

I will report on joint work with M. Rapoport and B. Smithling, on an "arithmetic transfer conjecture" (ATC). Similar to the arithmetic fundamental lemma conjecture (for unramified unitary groups over a p-adic field), the transfer conjecture connects the derivative of a relative orbital integral to an intersection number on a formal moduli space of $p$-divisible groups associated to a special parahoric of a ramified unitary group. Miraculously, this formal moduli space has good reduction.

Supplements
Notes Zhang
242 KB application/pdf
Show Detail

10:30 AM – 11:00 AM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

11:00 AM – 12:00 PM

 

Cycles on Shimura varieties and applications to Faltings heights

Benjamin Howard (Boston College)
Location
MSRI: Simons Auditorium
Video
Abstract

I’ll speak about ongoing joint work with Bruinier, Kudla, Rapoport, and Yang on the modularity of certain generating series of cycles on unitary Shimura varieties, and applications to new cases of Colmez’s conjecture on the Faltings heights of CM abelian varieties.

Supplements
Notes Howard
309 KB application/pdf
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12:00 PM – 02:00 PM

 

Lunch

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

02:00 PM – 03:00 PM

 

On the Hodge-Tate period map for Shimura varieties of Hodge type

Ana Caraiani (Imperial College, London)
Location
MSRI: Simons Auditorium
Video
Abstract

The Hodge-Tate period map is an important, new tool for studying the geometry of Shimura varieties, p-adic automorphic forms and torsion classes in the cohomology of Shimura varieties. It is a G(A_f)-equivariant map from a perfectoid Shimura variety into a flag variety with only an action of G(Q_p) and can be thought of as a p-adic analogue of the Borel embedding. In this talk, I will describe a canonical construction of the Hodge-Tate period map and of automorphic vector bundles for Shimura varieties of Hodge type. This is part of ongoing joint work with Peter Scholze.

Supplements
Notes Caraiani
305 KB application/pdf
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03:00 PM – 03:30 PM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

03:30 PM – 04:30 PM

 

Higher Koecher’s principle

Kai-Wen Lan (University of Minnesota, Twin Cities)
Location
MSRI: Simons Auditorium
Video
Abstract

I will report on a generalization of the classical Koecher’s principle for the coherent cohomology of Shimura varieties.

Supplements

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04:30 PM – 05:00 PM

 

Poster Session

Mladen Dimitrov (Université de Lille I (Sciences et Techniques de Lille Flandres Artois)), Volker Heiermann (Université d’Aix-Marseille (AMU)), Valentin Hernandez (Polytechnical University of Cataluña (Barcelona)), Jean-Pierre Labesse (Centre International de Recontres Mathématiques (CIRM), Luminy), Arthur-César Le Bras (Rheinische Friedrich-Wilhelms-Universität Bonn), Jie LIN (Université de Paris VII (Denis Diderot)), Marc-Hubert Nicole (Université d’Aix-Marseille (AMU))
Location
MSRI: Atrium
Video
Abstract
Supplements

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05:00 PM – 09:00 PM

 

Banquet at Hong Kong East Ocean Seafood Restaurant

Location
Video
Abstract
Supplements

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Dec 03, 2014
Wednesday
09:00 AM – 10:00 AM

 

Special Values of automorphic L-functions and congruences

A. Raghuram (Indian Institute of Science Education and Research)
Location
MSRI: Simons Auditorium
Video
Abstract

Hida proved in 1981 that if a prime p divides the algebraic part of the value at s = 1 of the adjoint L-function of a holomorphic cusp form f, then there is another cusp form g such that f is congruent to g modulo the prime p. This result was generalized to Hilbert modular forms by Eknath Ghate and Mladen Dimitrov, and to certain cusp forms on GL(2) over an imaginary quadratic field by Eric Urban, and recently to cusp forms on GL(2) over any number field by Namikawa. In this talk, I will discuss further generalizations of this phenomenon to the context adjoint L-values for cohomological cuspidal automorphic representations of GL(n) over any number field. This is joint work with Baskar Balasubramanyam

Supplements
Notes Raghuram
268 KB application/pdf
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10:00 AM – 10:30 AM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

10:30 AM – 11:30 AM

 

The  exterior algebra in the cohomology of an arithmetic group

Akshay Venkatesh (Stanford University)
Location
MSRI: Simons Auditorium
Video
Abstract
Supplements
Title & Abstract
92.5 KB application/pdf
Notes Venkatesh
260 KB application/pdf
Show Detail

11:30 AM – 12:00 PM

 

Break

Location
MSRI: Atrium
Video
Abstract
Supplements

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12:00 PM – 01:00 PM

 

Branching laws and period integrals for non-tempered representations.

Dipendra Prasad (Tata Institute of Fundamental Research)
Location
MSRI: Simons Auditorium
Video
Abstract

Questions about decomposition of representations of groups over local fields to subgroups, such as from SO(n) to SO(n-1), has been of  considerable interest in the recent past with impressive works of Waldspurger, Moeglin-Waldspurger, and Raphael Beuzart-Plessis. Corresponding global question for unitary groups has been partly settled by Wei Zhang. 

These questions have so far been considered only for tempered representations (of the group and the subgroup). In this lecture, I will discuss the situation with non-tempered representations.  This is a report on work done with B. Gross and Wee Teck Gan.
 

Supplements
Notes Prasad
1.49 MB application/pdf
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Dec 04, 2014
Thursday
09:30 AM – 10:30 AM

 

Modularity of non-semisimple Galois representations

Henri Darmon (McGill University)
Location
MSRI: Simons Auditorium
Video
Abstract
Supplements
Title & Abstract
89.3 KB application/pdf
Notes Darmon
1.09 MB application/pdf
Show Detail

10:30 AM – 11:00 AM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

11:00 AM – 12:00 PM

 

The Bloch Kato Conjecture for some Adjoint Selmer Groups.

Frank Calegari (Northwestern University)
Location
MSRI: Simons Auditorium
Video
Abstract

We prove some new vanishing results for the Bloch-Kato Selmer groups of Adjoint motives. This is joint work with David Geraghty.

Supplements
Notes Calegari
273 KB application/pdf
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12:00 PM – 02:00 PM

 

Lunch

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

02:00 PM – 03:00 PM

 

Euler System

Sarah Zerbes (University College)
Location
MSRI: Simons Auditorium
Video
Abstract

I show how Beilinson’s Eisenstein symbol give rise to motivic cohomology classes attached to pairs of modular forms of weight >= 2. These motivic cohomology classes can be used to construct an Euler system — a compatible family of global cohomology classes — attached to pairs of modular forms, related to the critical values of the corresponding Rankin-Selberg L-function. This is joint work with Kings and Loeffler, extending my previous work with Lei and Loeffler for weight 2 forms. This Euler system has several arithmetic applications, including one divisibility in the Iwasawa main conjecture for modular forms over imaginary quadratic fields, and cases of the finiteness of Tate–Shafarevich groups for elliptic curves twisted by dihedral Artin representations. 

Supplements
Notes Zerbes
311 KB application/pdf
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03:00 PM – 03:30 PM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

03:30 PM – 04:30 PM

 

p-adic L-functions, Iwasawa theory, and elliptic curves

Christopher Skinner (Princeton University)
Location
MSRI: Simons Auditorium
Video
Abstract

This talk will describe some recent applications of p-adic L-functions and Iwasawa theory to the arithmetic of elliptic curves, especially to establishing the p-parts of the the Birch-Swinnerton-Dyer formula in cases of rank at most one.

Supplements
Notes Skinner
195 KB application/pdf
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Dec 05, 2014
Friday
09:00 AM – 10:00 AM

 

G-bundles and the local Langlands correspondence

Laurent Fargues (Institut de Mathématiques de Jussieu)
Location
MSRI: Simons Auditorium
Video
Abstract

Given a reductive group G over the p-adic numbers I will explain a classification result for G-bundles over the curve defined in my joint work with Fontaine. I will then state and explain a conjecture linked to this classification

Supplements
Notes Fargues
880 KB application/pdf
Show Detail

10:00 AM – 10:30 AM

 

Tea

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

10:30 AM – 11:30 AM

 

On conductor 1 algebraic automorphic representations of GL(n) over Q, and applications

Gaetan Chenevier (Centre National de la Recherche Scientifique (CNRS))
Location
MSRI: Simons Auditorium
Video
Abstract

In this talk, I will present some results concerning the classification of the cuspidal automorphic representations of GL(n) over Q whose archimedean component is “ algebraic ‘’ and whose non archimedean components are all unramified.  As an application, I will sketch a proof of the following result : there are exactly 23 Siegel modular cusp forms of weight 12, for the full Siegel modular group, in genus at most 12.

Supplements
Notes Chenevier
1.09 MB application/pdf
Show Detail

11:30 AM – 12:00 PM

 

Break

Location
MSRI: Atrium
Video
Abstract
Supplements

Show Detail

12:00 PM – 01:00 PM

 

P-radicial base change,mod p

Laurent Clozel (Université de Paris XI)
Location
MSRI: Simons Auditorium
Video
Abstract

A new method introduced by Treumann, Venkatesh and the speaker makes it possible to prove cases of Langlands functoriality for cohomology classes mod p. I will show how this implies, in this context, automorphic induction for radicial extensions E=F(x^(1/p)) of number fields. I will also discuss the application to endoscopy for unitary groups.

Supplements
Notes Clozel
258 KB application/pdf
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