Higher coherent coholomogy of Shimura varieties

Informations projet

HiCoShiVa

N° de convention de subvention: 818856

DOI

10.3030/818856

Projet clôturé

Date de signature de la CE 16 Janvier 2019

Date de début 1 Février 2019

Date de fin 31 Juillet 2024

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 288 750,00

Contribution de l’UE

€ 1 288 750,00

Coordonné par

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 4 - HiCoShiVa (Higher coherent coholomogy of Shimura varieties)

Période du rapport: 2023-08-01 au 2024-07-31

Résumé du contexte et des objectifs généraux du projet

The project is concerned with the problem of understanding the coherent cohomology of Shimura varieties. We have to deal with higher cohomology groups and torsion. The main innovation of the project is to construct p-adic variations of the coherent cohomology. We are able to consider higher coherent cohomology classes, while previous works in this area have been concerned with degree 0 cohomology.
The applications are to the construction of automorphic Galois representations, the modularity of irregular motives and new cases of the Hasse-Weil conjecture, and the construction of p-adic L-functions.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

We develop local cohomology techniques to study the finite slope part of the cohomology of Shimura varieties. The local cohomology groups we consider are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. Overconvergent modular forms are a particular case of these local cohomologies. We construct a spectral sequence from local cohomology to cohomology. We are able to obtain vanishing theorems for the cohomology, as well as clas- sicality theorem comparing local and classical cohomology. We also develop eigenvarieties by p-adic deformation of the local cohomology groups. As an application, we prove some new properties of Galois representations arising from certain non-regular algebraic cuspidal automorphic forms.
The main outcomes are the proof of the potential modularity of abelian surfaces (with Boxer, Calegari, Gee) (>300 pages manuscript published in Publ. Math. IHES) and Higher Coleman theory (with Boxer) (175 pages preprint).

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

We have inverstigated an integral generalisation of Higher Coleman theory, called Higher Hida theory. We also have obtained new arithmetic applications to the modularity of abelian surfaces.

Pierre de Fermat