Arithmetical modular forms and new constructions of p-adic L-functions on classical groups
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, parties 1 et 2, Volume 25 (2016) no. 2-3, pp. 543-568.

An approach to constructions of automorphic L-functions and their p-adic avatars is presented as a work in progress with Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). For an algebraic group G over a number field K these L functions are certain Euler products L(s,π,r,χ). In particular, our constructions cover the L-functions in [52] via the doubling method of Piatetski-Shapiro and Rallis.

A p-adic avatar of L(s,π,r,χ) is a p-adic analytic function L p (s,π,r,χ) of p-adic arguments s p , χmodp r which interpolates algebraic numbers defined through the normalized critical values L * (s,π,r,χ) of the corresponding complex analytic L-function. We present a method using arithmetic nearly-holomorphic forms and general quasi-modular forms, related to algebraic automorphic forms. It gives new technique of constructing p-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.

Une nouvelle approche pour construire des fonctions L p-adiques pour les groupes classiques est présentée comme un projet en cours avec Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). Pour un groupe algébrique G sur un corps de nombres K les fonctions L complexes sont certains produits d’Euler L(s,π,r,χ). En particulier, notre construction couvre les fonctions L étudiées par Shimura dans [52] via la méthode de doublement de Piatetski-Shapiro et Rallis. Un avatar p-adique L(s,π,r,χ) est une fonction p-adique analytique L p (s,π,r,χ) de s p , χmodp r interpolant les valeurs spéciales normalisées algébriques L * (s,π,r,χ) de la fonction L complexe analytique attachée. Nous utilisons les formes presque-holomorphes et quasi-modulaires générales pour calculer et pour interpoler les valeurs spéciales normalisées.

Published online:
DOI: 10.5802/afst.1504

Alexei Panchishkin 1

1 Institut Fourier, Université Grenoble Alpes 100, rue des mathématiques, 38610, Gières, France
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Alexei Panchishkin. Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, parties 1 et 2, Volume 25 (2016) no. 2-3, pp. 543-568. doi : 10.5802/afst.1504. https://afst.centre-mersenne.org/articles/10.5802/afst.1504/

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