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

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 $L(s,\pi ,r,\chi )$. 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 $L(s,\pi ,r,\chi )$ est une fonction $p$-adique analytique $L_p(s,\pi ,r,\chi )$ de $s\in {\mathbb{Z}}_p$, $\chi mod{p}^r$ interpolant les valeurs spéciales normalisées algébriques $L^{*}(s,\pi ,r,\chi )$ 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.

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 $L(s,\pi ,r,\chi )$. In particular, our constructions cover the $L$-functions in [52] via the doubling method of Piatetski-Shapiro and Rallis.

A $p$-adic avatar of $L(s,\pi ,r,\chi )$ is a $p$-adic analytic function $L_p(s,\pi ,r,\chi )$ of $p$-adic arguments $s\in {\mathbb{Z}}_p$, $\chi mod{p}^r$ which interpolates algebraic numbers defined through the normalized critical values $L^{*}(s,\pi ,r,\chi )$ 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.

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