Summer Graduate School

New Geometric Techniques in Number Theory

July 01, 2013 - July 12, 2013

Parent Program:	New Geometric Methods in Number Theory and Automorphic Forms
Location:	SLMath: Eisenbud Auditorium

Organizers

Toby Gee (Imperial College, London), LEAD Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure), David Nadler (University of California, Berkeley), Peter Scholze (Universität Bonn)

Teaching Assistants(s)

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George Boxer (Harvard University)
Gabriel Dospinescu (École Normale Supérieure de Lyon)
Penghui Li (University of California, Berkeley)
Timo Richarz (Max-Planck-Institut für Mathematik)

Speaker(s)

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Arunabha Biswas (Texas Tech University)
George Boxer (Harvard University)
Kestutis Cesnavicius (Centre National de la Recherche Scientifique (CNRS))
Gabriel Dospinescu (École Normale Supérieure de Lyon)
Toby Gee (Imperial College, London)
Karol Koziol (University of Alberta)
Jaclyn Lang (Université de Paris XIII (Paris-Nord))
Penghui Li (University of California, Berkeley)
Judith Ludwig (Max-Planck-Institut für Mathematik)
Gilbert Moss (University of Texas, Austin)
Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure)
David Nadler (University of California, Berkeley)
Timo Richarz (Max-Planck-Institut für Mathematik)
Peter Scholze (Universität Bonn)
Shrenik Shah (Institute for Defense Analyses (CCR-LJ))
Carl Wang Erickson (University of Pittsburgh)

Description

The branches of number theory most directly related to automorphic forms have seen enormous progress over the past five years. Techniques introduced since 2008 have made it possible to prove many new arithmetic applications. The purpose of the current workshop is to drow the attention of young students or researchers to new questions that have arisen in the course of bringing several chapters in the Langlands program and related algebraic number theory to a close. We will focus especially on some precise questions of a geometric nature, or whose solutions seem to require new geometric insights. A graduate level in Number Theory is expected.

This two-week workshop will be devoted to the following subjects: Automorphy lifting theorems, p-adic local Langlands program, Characters of categorical representations and Hasse-Weil zeta function. During the first week, the lecturers present an open question and related mathematical objects. The first exercice sessions serve to direct the participants to an appropriate subject depending on their level. During the second week, the lecturers give some more advanced lectures on the field.

Bibliography

Keywords and Mathematics Subject Classification (MSC)

Primary Mathematics Subject Classification No Primary AMS MSC

Secondary Mathematics Subject Classification No Secondary AMS MSC

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Schedule, Notes/Handouts & Videos

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Jul 01, 2013
Monday

09:00 AM - 09:15 AM		Welcome
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
09:15 AM - 10:45 AM		Lecture David Nadler (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Lecture Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture Peter Scholze (Universität Bonn)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Asymptotic behaviors of higher Mahler Measure Arunabha Biswas (Texas Tech University)
Location SLMath: Eisenbud Auditorium Video -- Abstract We study behaviors of higher Mahler measure which is a generalization of classical Mahler measure. Apart from giving a broad overview of classical Mahler measure and higher Mahler measure, in this talk we shall show several new asymptotic properties of zeta Mahler measure (of a specific class of x-a) involving partitions of integers, the Rieman zeta function, the Dirichlet η-function and π. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session Gabriel Dospinescu (École Normale Supérieure de Lyon)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 02, 2013
Tuesday

09:15 AM - 10:45 AM		Lecture Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Penghui Li (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture David Nadler (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Representations and Pseudorepresentations Carl Wang Erickson (University of Pittsburgh)
Location SLMath: Eisenbud Auditorium Video -- Abstract We will explain what a pseudorepresentation is, and describe a few examples of the appearance of this concept in number theory. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session Timo Richarz (Max-Planck-Institut für Mathematik)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 03, 2013
Wednesday

09:15 AM - 10:45 AM		Lecture Peter Scholze (Universität Bonn)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 01:45 PM		BBQ
Location Tilden Park Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Problem Session Gabriel Dospinescu (École Normale Supérieure de Lyon)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
02:45 PM - 03:45 PM		Problem Session Timo Richarz (Max-Planck-Institut für Mathematik)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session Penghui Li (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 05, 2013
Friday

09:15 AM - 10:45 AM		Lecture David Nadler (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Lecture Peter Scholze (Universität Bonn)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
12:45 PM - 01:45 PM		Lunch
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Selmer groups and class groups Kestutis Cesnavicius (Centre National de la Recherche Scientifique (CNRS))
Location SLMath: Eisenbud Auditorium Video -- Abstract Let A be an abelian variety over a number field K. If A has nontrivial (resp. full) K-rational p-torsion for a prime p, exploiting the fppf cohomological approach to Selmer groups, we obtain inequalities bounding the size of the p-Selmer group of A from below (resp. above) in terms of the size of the p-torsion subgroup of the ideal class group of K. When K varies in a family of field extensions, these inequalities relate the growth of Selmer groups to that of class groups; I will discuss such relations in several different settings. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session Penghui Li (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 08, 2013
Monday

09:15 AM - 10:45 AM		Lecture Toby Gee (Imperial College, London)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Gabriel Dospinescu (École Normale Supérieure de Lyon)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture David Nadler (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Towards a Langlands correspondence for Hecke modules of SL_n in characteristic p Karol Koziol (University of Alberta)
Location SLMath: Eisenbud Auditorium Video -- Abstract We show how to realize the pro-p-Iwahori-Hecke algebra of SL_n as a subalgebra of the pro-p-Iwahori-Hecke algebra of GL_n. Using the interplay between these two algebras, we deduce a numerical Langlands correspondence between "packets" of Hecke modules and mod-p projective Galois representations. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session George Boxer (Harvard University)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 09, 2013
Tuesday

09:15 AM - 10:45 AM		Lecture Ariane Mézard (Institut de Mathématiques de Jussieu; École Normale Supérieure)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Timo Richarz (Max-Planck-Institut für Mathematik)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture Toby Gee (Imperial College, London)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		p-adic functoriality Judith Ludwig (Max-Planck-Institut für Mathematik)
Location SLMath: Eisenbud Auditorium Video -- Abstract I will explain a notion of p-adic functoriality. Roughly speaking the goal is to extend classical Langlands functoriality to a transfer of p-adic families of automorphic representations. As these p-adic families can be organized into mysterious geometric objects, so called eigenvarieties, one aims at giving this p-adic functoriality a geometric structure as well. I will explain what this means and if time permits discuss some results in the context of unitary groups. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session George Boxer (Harvard University)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 10, 2013
Wednesday

09:15 AM - 10:45 AM		Lecture Peter Scholze (Universität Bonn)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Gabriel Dospinescu (École Normale Supérieure de Lyon)
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Lecture Toby Gee (Imperial College, London)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		On images of Galois representations associated to non-CM Hida families of modular forms Jaclyn Lang (Université de Paris XIII (Paris-Nord))
Location SLMath: Eisenbud Auditorium Video -- Abstract In the 1980s Ribet and Momose determined the image of Galois representations associated to non-CM classical modular forms in terms of conjugate self-twists of those modular forms. Recent work of Hida shows that Galois representations associated to non-CM Hida families have large image. We define conjugate self-twists in the context of Hida families and combine the above mentioned results to obtain a description of the images of Galois representations associated to Hida families. We will attempt to motivate and introduce the local Langlands conjecture for GL(2) in families proposed by Emerton and Helm, and discuss how the classical theory of zeta integrals and local constants works in this setting. Supplements -- Show Detail
03:15 PM - 03:45 PM		Local constants for families of representations of GL(2) of a local field Gilbert Moss (University of Texas, Austin)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session George Boxer (Harvard University)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 11, 2013
Thursday

09:15 AM - 10:45 AM		Lecture Toby Gee (Imperial College, London)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Timo Richarz (Max-Planck-Institut für Mathematik)
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Problem Session Gabriel Dospinescu (École Normale Supérieure de Lyon)
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Problem Session Penghui Li (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Interpolating Periods Shrenik Shah (Institute for Defense Analyses (CCR-LJ))
Location SLMath: Eisenbud Auditorium Video -- Abstract We study the interpolation of Hodge-Tate and de Rham periods in families of Galois representations. Given a Galois representation on a coherent locally free sheaf over a reduced rigid space and a bounded range of weights, we obtain a stratification of this space by locally closed subvarieties where the Hodge-Tate and bounded de Rham periods (within this range) form locally free sheaves. At every thickened geometric point within one of the strata, we obtain a corresponding number of unbounded de Rham periods. If the number of interpolated de Rham periods is the number of fixed Hodge-Tate-Sen weights, we prove similar statements for non-reduced affinoid spaces. We also prove strong vanishing results for higher cohomology. These results encapsulate a robust theory of interpolation for Hodge-Tate and de Rham periods that simultaneously generalizes results of Berger-Colmez and Kisin. Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Problem Session George Boxer (Harvard University)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail

Jul 12, 2013
Friday

09:15 AM - 10:45 AM		Problem Session Peter Scholze (Universität Bonn)
Location SLMath: Eisenbud Auditorium Video Abstract -- Supplements -- Show Detail
09:15 AM - 10:45 AM		Problem Session Timo Richarz (Max-Planck-Institut für Mathematik)
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
10:45 AM - 11:15 AM		Coffee Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:15 AM - 12:15 PM		Problem Session Penghui Li (University of California, Berkeley)
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
12:15 PM - 01:45 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
01:45 PM - 02:45 PM		Group Presentation
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
02:45 PM - 03:15 PM		Group Presentation
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail
03:15 PM - 03:45 PM		Tea Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
03:45 PM - 04:45 PM		Group Presentation
Location SLMath: Eisenbud Auditorium Video -- Abstract -- Supplements -- Show Detail