Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups

Alexei Panchishkin

doi:10.5802/afst.1504

Arithmetical modular forms and new constructions of

p

-adic

L

-functions on classical groups

Alexei Panchishkin ¹

¹ Institut Fourier, Université Grenoble Alpes 100, rue des mathématiques, 38610, Gières, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 543-568.

Abstract
Abstract

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 $s \in ℤ_{p}$ , $χ mod p^{r}$ 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 $s \in ℤ_{p}$ , $χ mod p^{r}$ 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: 2016-07-11

MR Zbl

DOI: 10.5802/afst.1504

Author's affiliations:

Alexei Panchishkin ¹

¹ Institut Fourier, Université Grenoble Alpes 100, rue des mathématiques, 38610, Gières, France

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     title = {Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {543--568},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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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, 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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