Research: Higher coherent coholomogy of Shimura varieties
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.
Max ERC Funding
1 288 750 €
Start date: 2019-02-01, End date: 2024-01-31
ERC Consolidator Grant
ERC Consolidator Grants support promising researchers who are at the beginning of an independent research career. The consolidation of their own research team will be funded with the grant. For the evaluation of the researcher profile several benchmarks according to research domain and career development are taken into account. For instance, applicants should have several important publications without their PhD supervisor. Other benchmarks include publications as first author in high-ranking international journals, (translated) monographs, patents, conference presentations or (inter)national prizes and awards.