An approach to constructions of automorphic -functions and their -adic avatars is presented as a work in progress with Thanh Hung Dang and Anh Tuan Do (Hanoi, Vietnam). For an algebraic group over a number field these functions are certain Euler products . In particular, our constructions cover the -functions in [52] via the doubling method of Piatetski-Shapiro and Rallis.
A -adic avatar of is a -adic analytic function of -adic arguments , which interpolates algebraic numbers defined through the normalized critical values of the corresponding complex analytic -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 -adic zeta-functions via general quasi-modular forms and their Fourier coefficients.
Une nouvelle approche pour construire des fonctions -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 sur un corps de nombres les fonctions complexes sont certains produits d’Euler . En particulier, notre construction couvre les fonctions étudiées par Shimura dans [52] via la méthode de doublement de Piatetski-Shapiro et Rallis. Un avatar -adique est une fonction -adique analytique de , interpolant les valeurs spéciales normalisées algébriques de la fonction complexe analytique attachée. Nous utilisons les formes presque-holomorphes et quasi-modulaires générales pour calculer et pour interpoler les valeurs spéciales normalisées.
@article{AFST_2016_6_25_2-3_543_0, author = {Alexei Panchishkin}, 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}, volume = {Ser. 6, 25}, number = {2-3}, year = {2016}, doi = {10.5802/afst.1504}, zbl = {1410.11044}, mrnumber = {3530168}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1504/} }
TY - JOUR AU - Alexei Panchishkin TI - Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 543 EP - 568 VL - 25 IS - 2-3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1504/ DO - 10.5802/afst.1504 LA - en ID - AFST_2016_6_25_2-3_543_0 ER -
%0 Journal Article %A Alexei Panchishkin %T Arithmetical modular forms and new constructions of $p$-adic $L$-functions on classical groups %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 543-568 %V 25 %N 2-3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1504/ %R 10.5802/afst.1504 %G en %F AFST_2016_6_25_2-3_543_0
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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