Overconvergent cohomology of Hilbert modular varieties and $p$-adic $L$-functions

Barrera Salazar, Daniel

Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2177-2213.

Pour une forme de Hilbert de pente non critique, l’on construit une distribution $p$-adique sur le groupe de Galois de l’extension abélienne maximale du corps totalement réel, non-ramifiée en dehors de $p$ et $\infty$. On démontre que la distribution obtenue est admissible et interpole les valeurs critiques de la fonction $L$ complexe de la forme de Hilbert. Cette construction est basée sur l’étude de la cohomologie surconvergente des variétés modulaires de Hilbert et de certains cycles sur ces variétés.

For each Hilbert modular form of non-critical slope we construct a $p$-adic distribution on the Galois group of the maximal abelian extension unramified outside $p$ and $\infty$ of the totally real field. We prove that the distribution is admissible and interpolates the critical values of the complex $L$-function of the form. This construction is based on the study of the overconvergent cohomology of Hilbert modular varieties and certain cycles on these varieties.

Classification : 11F41, 11F67, 11S80
Keywords: $p$-adic $L$-functions, Hilbert modular forms
Mot clés : Fonctions $L$ $p$-adiques, Formes modulaires de Hilbert

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