Higher coherent coholomogy of Shimura varieties

Project Information

HiCoShiVa

Grant agreement ID: 818856

DOI

10.3030/818856

Project closed

EC signature date 16 January 2019

Start date 1 February 2019

End date 31 July 2024

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 288 750,00

EU contribution

€ 1 288 750,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

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

Reporting period: 2023-08-01 to 2024-07-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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