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PariTorMod SIGNED

P-adic Arithmetic Geometry, Torsion Classes, and Modularity

Total Cost €

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EC-Contrib. €

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Project "PariTorMod" data sheet

The following table provides information about the project.

Coordinator
IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE 

Organization address
address: SOUTH KENSINGTON CAMPUS EXHIBITION ROAD
city: LONDON
postcode: SW7 2AZ
website: http://www.imperial.ac.uk/

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Total cost 1˙469˙805 €
 EC max contribution 1˙469˙805 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2018-STG
 Funding Scheme ERC-STG
 Starting year 2019
 Duration (year-month-day) from 2019-01-01   to  2023-12-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE UK (LONDON) coordinator 1˙469˙805.00

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 Project objective

The overall theme of the proposal is the interplay between p-adic arithmetic geometry and the Langlands correspondence for number fields. At the heart of the Langlands program lies reciprocity, which connects Galois representations to automorphic forms. Recently, new developments in p-adic arithmetic geometry, such as the theory of perfectoid spaces, have had a transformative effect on the field. This proposal would establish a research group that will develop and exploit novel techniques, that will allow us to move significantly beyond the state of art. I intend to make fundamental progress on three major interlinked problems.

Torsion in the cohomology of Shimura varieties: in joint work with Scholze, I proved a strong vanishing result for torsion in the cohomology of compact unitary Shimura varieties. In work in progress, we have extended this to many non-compact cases. To obtain a complete picture, I propose to develop new techniques using point-counting and the trace formula and combine them with ingredients from arithmetic geometry.

Local-global compatibility is essential for establishing new instances of Langlands reciprocity. I will use the results on Shimura varieties described above to prove local-global compatibility for torsion in the cohomology of locally symmetric spaces for general linear groups over CM fields. This is one of the fundamental questions in the field. Solving it will require progress on a diverse set of problems in representation theory and integral p-adic Hodge theory.

The Fontaine–Mazur conjecture is the most general reciprocity conjecture. Very little is known outside the case of two-dimensional representations of the absolute Galois group of the rational numbers, which relies crucially on a connection to p-adic local Langlands. I will attack the Fontaine–Mazur conjecture for imaginary quadratic fields. Some crucial inputs will come from the first two projects above.

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The information about "PARITORMOD" are provided by the European Opendata Portal: CORDIS opendata.

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