The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

1000-1100 P. Kassaei: Analytic continuation of p-adic Hilbert modular forms and applications to the Artin conjecture-- The unramified case.

1130-1230 Y. Tian: Analytic continuation and modularity lifting in parallel weight one in the tamely ramified case.

1400-1500 C. Johansson: Proving control theorems for overconvergent forms using rigid cohomology.

1515-1615: V. Pilloni: Sur la conjecture de Fontaine-Mazur en poids 0.

Abstracts:

Kassaei:

We will present our proof of the Artin conjecture in the unramified case. In the course of doing so, we will also present an overview of the recent developments in analytic continuation of p-adic Hilbert modular forms, as well as generalizations of our work on the Artin conjecture by Pilloni, Stroh, Sasaki, and Tian.

Tian:

Let F be a totally real field, and p>2 be a prime unramified in F. Suppose given a 2-dimensional totally odd continuation p-adic Galois representation on F, which is residually modular and tamely ramified (up to twists by characters) at all places above p. We will show that, up to twist, such a Galois representation comes from a holomorphic Hilbert modular form in parallel weight one. A key ingredient in the proof is an analytic continuation result, which claims that any overconvergent Hilbert eigneform of finite slope can be extended to a very large area on the rigid analytic Hilbert modular variety of Iwahoric level. This is a joint work with Payman Kassaei and Shu Sasaki.

Johansson:

After the invention and use of the analytic continuation method by Buzzard and Taylor to prove icosahedral cases of the Artin conjecture, Kassaei realized that it might also be used (together with Kassaei's "gluing lemma") to reprove Coleman's theorem that an overconvergent U_p-eigenform of weight k>1 and slope <k-1 is classical. Coleman's original method used an analysis of the rigid cohomology of the ordinary locus in the modular curve. In this talk I will discuss recent work on extending this "cohomological" approach to higher dimensional Shimura varieties. Part of this is joint work with Vincent Pilloni.

Pilloni:

Nous démontrerons, sous les hypothéses techniques usuelles, que certaines représentations de Galois des corps totalement réels sont associées à des formes modulaires de Hilbert propre de poids 1. Le résultat clé est une caractérisation des espaces de formes modulaires classiques dans l'espaces des formes modulaires p-adique. Nous essayerons enfin de faire le lien avec le travail de Calegari-Geraghty sur ces questions. (Travail avec B. Stroh).

First meeting (Paris, 13/11/2006)

Second meeting (London, 02/05/2007)

Third meeting (Paris, 12/11/2007)

Fourth meeting (London, 7,8/05/2008)

Fifth meeting (Paris, 17/11/2008)

Sixth meeting (London, 03/06-04/06/2009)

Seventh meeting (Paris, 16/11/2009)

Eighth meeting (London, 02/06/2010)

Ninth meeting (Paris, 08/11/2010)

Tenth meeting (London, 01/06/2011)

Eleventh meeting (Paris, 21/11/2011)

Twelfth meeting (London, 30/05/2012)

13th meeting (Paris, 22/10/2012)

14th meeting (London, 3,4/6/2013)

This page is maintained by Kevin Buzzard.