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Higher coherent coholomogy of Shimura varieties

Project description

Research on the cohomology of Shimura varieties takes a step forward

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve. Its geometry is closely linked to the theory of automorphic forms over the corresponding reductive algebraic group. It is a central part of automorphic forms, Galois representations and motives. As a result, it makes a natural test case for investigating the conjectural relations between motives and automorphic forms, and for whether all zeta functions are automorphic. The EU-funded HiCoShiVa project will focus on understanding torsion appearing in the coherent cohomology of Shimura varieties. Compared to previous studies that explored cohomology classes of degree 0, the project will focus on higher cohomology groups. The main project innovation will be the construction of p-adic variations of higher coherent cohomology groups.


One can attach certain complex analytic functions to algebraic varieties defined over the rational numbers, called Zeta functions. They are a vast generalization of Riemann’s zeta function. The Hasse-Weil conjecture predicts that these Zeta functions satisfy a functional equation and admit a meromorphic continuation to the whole complex plane. This follows from the conjectural Langlands program, which aims in particular at proving that Zeta functions of algebraic varieties are products of automorphic L-functions.
Automorphic forms belong to the representation theory of reductive groups but certain automorphic forms actually appear in the cohomology of locally symmetric spaces, and in particular the cohomology of automorphic vector bundles over Shimura varieties. This is a bridge towards arithmetic geometry.
There has been tremendous activity in this subject and the Hasse-Weil conjecture is known for proper smooth algebraic varieties over totally real number fields with regular Hodge numbers. This covers in particular the case of genus one curves. Nevertheless, lots of basic examples fail to have this regularity property : higher genus curves, Artin motives...
The project HiCoShiVa is focused on this irregular situation. On the Shimura Variety side we will 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 will be 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.

Host institution

Net EU contribution
€ 1 288 750,00
75794 Paris

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Ile-de-France Ile-de-France Paris
Activity type
Research Organisations
Total cost
€ 1 288 750,00

Beneficiaries (1)