Overconvergent cohomology, $p$-adic $L$-functions and families for $\mathrm{GL}\left(2\right)$ over CM fields

Barrera Salazar, Daniel; Williams, Chris

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 659-701.

The study of overconvergent cohomology, initiated by Pollack and Stevens in the setting of classical modular forms, has now been used to construct $p$-adic $L$-functions in a number of settings. The method is conceptual and is very closely related to the recent constructions of eigenvarieties by Ash–Stevens, Urban and Hansen. In this note, we give an exposition of the ideas behind the use of overconvergent cohomology in constructing $p$-adic $L$-functions, and use it to construct $p$-adic $L$-functions attached to base-change families of automorphic representations for $\mathrm{GL}\left(2\right)$ over CM fields. As a corollary, we prove a $p$-adic Artin formalism result for base-change $p$-adic $L$-functions.

L’étude de la cohomologie surconvergente, initiée par Pollack et Stevens dans le cadre des formes modulaires classiques, a été utilisée pour construire des fonctions $L$ $p$-adiques dans un certain nombre de contextes. La méthode est conceptuelle et très étroitement liée aux constructions récentes des variétés de Hecke par Ash–Stevens, Urban et Hansen. Dans cette note, nous exposons des idées qui sont derrière l’utilisation de la cohomologie surconvergente dans la construction de fonctions L $p$-adiques, et nous les utilisons pour construire des fonctions $L$ $p$-adiques attachées aux familles de représentations automorphes de $\mathrm{GL}\left(2\right)$ sur un corps de type CM provenant par changement de base. Comme corollaire, nous établissons une version du formalisme d’Artin $p$-adique pour ces fonctions $L$ $p$-adiques.

Classification: 11F41, 11F67, 11F85, 11S40
Keywords: Overconvergent cohomology, $p$-adic $L$-functions, automorphic representations

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