Higher coherent coholomogy of Shimura varieties

Informacje na temat projektu

HiCoShiVa

Identyfikator umowy o grant: 818856

DOI

10.3030/818856

Projekt został zamknięty

Data podpisania przez KE 16 Stycznia 2019

Data rozpoczęcia 1 Lutego 2019

Data zakończenia 31 Lipca 2024

Finansowanie w ramach

• EXCELLENT SCIENCE - European Research Council (ERC)

Koszt całkowity

€ 1 288 750,00

Wkład UE

€ 1 288 750,00

Koordynowany przez

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

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

Okres sprawozdawczy: 2023-08-01 do 2024-07-31

Podsumowanie kontekstu i ogólnych celów projektu

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.

Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty

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).

Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)

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