Higher coherent coholomogy of Shimura varieties

Informazioni relative al progetto

HiCoShiVa

ID dell’accordo di sovvenzione: 818856

DOI

10.3030/818856

Progetto chiuso

Data della firma CE 16 Gennaio 2019

Data di avvio 1 Febbraio 2019

Data di completamento 31 Luglio 2024

Finanziato da

• EXCELLENT SCIENCE - European Research Council (ERC)

Costo totale

€ 1 288 750,00

Contributo UE

€ 1 288 750,00

Coordinato da

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

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

Periodo di rendicontazione: 2023-08-01 al 2024-07-31

Sintesi del contesto e degli obiettivi generali del progetto

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.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

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).

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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