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Galois representations modulo \(p\) and cohomology of Hilbert modular varieties. (English) Zbl 1160.11325

Summary: The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:
– control of the image of Galois representations modulo \(p\),
– Hida’s congruence criterion outside an explicit set of primes,
– freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.
We study the arithmetic properties of Hilbert modular forms by studying their modulo \(p\) Galois representations and our main tool is the action of inertia groups at primes above \(p\). In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the \(p\)-adic (resp. modulo \(p\)) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.

MSC:

11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14G35 Modular and Shimura varieties
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