Further Development on Krasner-Vuković Paragraded Structures and $p$-adic Interpolation of Yubo Jin $L$-values

Abstract

This paper is a joint project with Siegfried Bocherer (Mannheim), developing a recent preprint of  Yubo Jin (Durham UK) previous works of Anh Tuan Do (Vietnam)  and Dubrovnik, IUC-2016 papers from \textit{Sarajevo Journal of Mathematics} (Vol.12, No.2-Suppl., 2016). We wish to use paragraded structures on {differential operators and arithmetical automorphic forms on classical groups and show that these structures provide a tool to construct $p$-adic measures and $p$-adic $L$-functions on the corresponding non-archimedean weight spaces.}   An approach to constructions of automorphic $L$-functions on unitary groups and their $p$-adic analogues is presented. 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 \cite{Shi00} via the doubling method of Piatetski-Shapiro and Rallis.  A $p$-adic analogue of $L(s,\pi, r, \chi)$ is a $p$-adic analytic function  $L_p(s,\pi, r, \chi)$ of $p$-adic arguments $s  \in \Z_p$, $\chi \bmod p^r$ 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 a technique of constructing $p$-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.