AF AND MSOGR

Automorphic Forms and Moduli Spaces of Galois Representations

Coordinatore	IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE    more  Organization address address: SOUTH KENSINGTON CAMPUS EXHIBITION ROAD city: LONDON postcode: SW7 2AZ contact info Titolo: Ms. Nome: Brooke Cognome: Alasya Email: send email Telefono: +44 20 7594 1181 Fax: +44 20 7594 1418
Nazionalità Coordinatore	United Kingdom [UK]
Totale costo	100˙000 €
EC contributo	100˙000 €
Programma	FP7-PEOPLE Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)
Code Call	FP7-PEOPLE-2011-CIG
Funding Scheme	MC-CIG
Anno di inizio	2012
Periodo (anno-mese-giorno)	2012-04-02   -   2016-04-01

 Partecipanti

#	participant	country	role	EC contrib. [€]
1	IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE  Organization address address: SOUTH KENSINGTON CAMPUS EXHIBITION ROAD city: LONDON postcode: SW7 2AZ contact info Titolo: Ms. Nome: Brooke Cognome: Alasya Email: send email Telefono: +44 20 7594 1181 Fax: +44 20 7594 1418	UK (LONDON)	coordinator	100˙000.00

Mappa

 Word cloud

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 Obiettivo del progetto (Objective)

'I propose to solve three problems. The first is to prove Serre’s conjecture for real quadratic fields. I will do this by using automorphic induction to transfer the problem to U(4) over the rational numbers, where I will use automorphy lifting theorems and results on the weight part of Serre's conjecture that I established in my earlier work to reduce the problem to improving results in small weight and level. I will prove these base cases via integral p-adic Hodge theory and discriminant bounds.

The second problem is to develop a geometric theory of moduli spaces of mod p and p-adic Galois representations, and to use it to establish the Breuil–Mezard conjecture in arbitrary dimension, by reinterpreting the conjecture in geometric terms, independently of any fixed mod p Galois representation. This will transform the subject by building the first connections between the p-adic Langlands program and the geometric Langlands program, providing an entirely new world of techniques for number theorists. As a consequence of the Breuil-Mezard conjecture, I will be able to deduce far stronger automorphy lifting theorems (in arbitrary dimension) than those currently available.

The third problem is to prove a strengthened version of the Gouvea–Mazur conjecture, by completely determining the reduction mod p of certain two-dimensional crystalline representations. I will do this by means of explicit computations with the p-adic local Langlands correspondence for GL_2(Q_p), as well as by improving existing arguments which prove multiplicity one theorems via automorphy lifting theorems. This work will show that the existence of counterexamples to the Gouvea-Mazur conjecture is due to a purely local phenomenon, and that when this local obstruction vanishes, far stronger conjectures of Buzzard on the slopes of the U_p operator hold.'