Higher coherent coholomogy of Shimura varieties

Projektinformationen

HiCoShiVa

ID Finanzhilfevereinbarung: 818856

DOI

10.3030/818856

Projekt abgeschlossen

EK-Unterschriftsdatum 16 Januar 2019

Startdatum 1 Februar 2019

Enddatum 31 Juli 2024

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 288 750,00

EU-Beitrag

€ 1 288 750,00

Koordiniert durch

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

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

Berichtszeitraum: 2023-08-01 bis 2024-07-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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).

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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