Further Development on Krasner-Vuković Paragraded Structures and $p$-adic Interpolation of Yubo Jin $L$-values - Panchishkin, Alexei (Sarajevo Journal of Mathematics, 2024)
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 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 $L(s,π, r, χ)$. 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 $L(s,π, r, χ)$ is a $p$-adic analytic function $L_p(s,π, r, χ)$ of $p$-adic arguments $s \in \Z_p$, $χ \bmod p^r$ 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 a technique of constructing $p$-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.