Arithmetic Geometry intercity seminar Fall 2005


This seminar will have two sub-programs.
  1. Lectures by Johan de Jong (Kloosterman professor in Leiden in October and November) on the subject ``Brauer groups, Tsen's theorem, Graber-Harris-Starr and a generalisation'' (work in progress, joint with Jason Starr). He will speak 1 hour on this subject on the 4 dates (see below) in October and November. The lecture on October 7 will be the so-called Kloosterman lecture, accessible to a wider audience.

  2. Lectures to understand the recent work by Khare, Wintenberger and by Dieulefait on Serre's conjecture on modular Galois representations for small levels. This includes the results of Richard Taylor on potential modularity for 2-dimensional Galois representations, and many other aspects of the GL(2) case of Langlands's program. A good introduction to Serre's conjecture is given in Chapter 1 of the book by Ribet and Stein in the references below.

    We will try to have notes for all lectures in this program available on this webpage as soon as possible after the lecture has been given.

    The lectures on September 23 and October 7 are in common with the Intercity Number Theory Seminar, after that the two seminars go on seperately.

    The lectures will be accessible to algebraic geometers, (algebraic) number theorists and Lie group theorists who know about p-adic numbers.

    Comments and suggestions on this sub-program are welcome, suggestions for good references especially. Those who are interested to give a lecture should contact Bas Edixhoven (


September 23, Utrecht

This day is joint with the Intercity Number Theory Seminar. Room K11 of the Mathematical Institute.
Bas Edixhoven: Introduction to Serre's conjecture (in pdf).
The conjecture will be stated, and put in its historical context and in the wider context of the Langlands program. Serre's level and weight of a 2-dimensional mod p Galois representation will be defined. Khare's result will be stated. An overview will be given of what will be treated in the seminar.

Johan Bosman: Galois representations associated to modular forms.
Modular forms, Hecke operators and eigenforms will be defined. The existence of Galois representations associated to eigenforms will be stated. The construction of these representations will be postponed until later, in the more general case of Hilbert modular forms. Something about weight 2 being easier and about weight 1 being special can be said.

Gunther Cornelissen: Theorem G of Taylors article ``Remarks on a conjecture of Fontaine and Mazur''.
The following statement will be explained and proved.
Let K be a number field, and S a finite set of places of K. Let KS be a maximal algebraic extension of K in which all places in S are completely split. Let X be a smooth irreducible quasi-projective scheme over K, such that for every v in S the set X(Kv) is non-empty. Then X(KS) is Zariski dense in X.

October 7, Leiden

This day is joint with the Intercity Number Theory Seminar. Room 312 of the Mathematical Institute.
Sander Dahmen: Lower bounds for discriminants.
Lower bounds for discriminants of number fields will be proved, in terms of their degree only. See Odlyzko in the references below.

Frits Beukers: Upper bounds for discriminants.
Let p be prime. Suppose that r is an irreducible representation of the absolute Galois group of Q on a 2-dimensional vector space over an algebraic closure of Fp, with p in {2,3}. The image of such a representation is finite, hence determines a number field K that is Galois over Q. An upper bound for the discriminant of K will be proved that contradicts the lower bound of the previous lecture in the case that p is in {2,3} and r is unramified outside p. The audience will draw the logical conclusion. For p=2 this result is due to Tate, and for p=3 it is due to Serre.

Reference: see Moon and Taguchi in the references below. Or the very recent article by Moon and Ono.

Bas Edixhoven: Overview of Khare's proof.
An overview will be given of Khare's proof of Serre's conjecture in level one. Those who will not attend the rest of the seminar will have an idea of Khare's proof, and it is hoped that those who will attend the rest of the seminar will now be sufficiently motivated to digest the more technical parts that are to come.

Reference: Khare 2 in the references below.

Johan de Jong: Kloosterman lecture, Rational points and rational connectivity.
Recently, Graber, Harris and Starr proved that any family of rationally connected projective varieties over a smooth curve has a section. A complex projective variety (or manifold) M is rationally connected when every two points in M lie on a rational curve in M.

In this lecture we will explain a generalization of this result, joint with Jason Starr, to the case when the base of the family is a surface.

In the other three lectures (October 14, November 11 and November 25) we will explain the idea behind the proof of the theorem, the analogy with Tsen's theorem, the connection with weak approximation and the connection with the period-index problem for Brauer groups.

Reception following the Kloosterman lecture.

October 14, Amsterdam (UvA)

Building B (Roetersstraat), room 318. Attention: this is not the mathematics building P, also known as Euclides building.
11:00-12:00 and 13:00-14:00
Gerard van der Geer: Hilbert modular forms and Hilbert modular varieties, I and II.
Let F be a totally real number field. Hilbert modular forms on Hilbert modular varieties and Hecke operators, all associated to F, will be described, adelically. A precise statement on existence of Galois representations will be given, without proof.

References: Taylor 4 and T. Saito in the references below.

Bart de Smit: General theory of deformations of Galois representations.
Reference: Lenstra-de Smit in the references below.

Johan de Jong: Rational points and rational connectivity, II.

November 11, Leiden

Room 312 of the Mathematical Institute.

Gebhard Boeckle: Deformation rings of Galois representations in nice cases
Abstract for the lecture and the text of a lecture in Strasbourg.
Reference: Boeckle in the references below.

Bas Edixhoven: Hilbert modular forms and local Langlands.
The automorphic representation associated to an eigenform will be described, as well as its local factors at the places of F. The statement that the local factor of the automorphic representation determines the local Galois representation (after F-semisimplification) will be explained, and an explicit description will be given in at least the case of a principal series local representation.

References: Taylor 4, Carayol, Bump and T. Saito in the references below.

Fabio Mainardi: Automorphic theory for GL2.
Jacquet-Langlands's switch to quaternion algebras and solvable base change will be explained, just the results, no proofs.

Reference: Langlands in the references below.

Johan de Jong: Rational points and rational connectivity, III.

November 25, Nijmegen

Room: collegezaal N7 (N1004) (first lecture) and Colloquiumkamer (HG01.028) (remaining lectures).
Theo van den Bogaart: Construction of Galois representations in cohomology of Shimura curves.
Reference: Carayol and T. Saito in the references below.

Johan de Jong: Strictly compatible families of Galois representations.
Reference: Taylor 3 and T. Saito in the references below.

Robin de Jong: Overview of modularity lifting.
Reference: section 6 of Khare 2 in the references below.

Johan de Jong: Rational points and rational connectivity, IV.

December 9, Utrecht

Room 160 in the Buys Ballot lab.
Gerard van der Geer: Taylor's potential modularity
Reference: Taylor 1, 2 and 3 in the references below.

13:00-14:00, 14:15-15:15 and 15:30-16:30
Jean-Pierre Wintenberger: Existence of minimal lifts and compatible families, the proof in the weight 2 case, Khare's proof in the general case.
Here is the text of his Bourbaki lecture of November 2005.
Reference: Dieulefait 1, Khare 1, Khare 2 and Khare-Wintenberger in the references below.


On the isomorphism....

Automorphic forms and representations.

Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert.

Dieulefait 1
The level 1 weight 2 case of Serre's conjecture

Khare 1
On Serre's modularity conjecture for 2-dimensional mod p representations of G_Q unramified outside p

Khare 2
Serre's modularity conjecture (survey)

Khare, Wintenberger
On Serre's reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q

Base change for GL2

Lenstra-de Smit
Explicit construction of universal deformation rings.

Moon, Ono
2-adic properties of certain modular forms and their applications to arithmetic functions.

Moon, Taguchi
Refinement of Tate's discriminant bound and non-existence theorems for mod p Galois representations

Discriminant bounds

Ribet, Stein
Lectures on Serre's conjectures

T. Saito
Hilbert modular forms and p-adic Hodge theory (preliminary version)

Taylor 1
On the meromorphic continuation of degree two L-functions.

Taylor 2
Remarks on a conjecture of Fontaine and Mazur.

Taylor 3
Galois representations (extended version of his ICM notes).

Taylor 4
On Galois representations associated to Hilbert modular forms.

La conjecture de modularité de Serre: le cas de conducteur 1 (d'après C. Khare).

Bas Edixhoven <>
Last modified: Thu Jan 5 16:36:31 CET 2006