Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

TL;DR

This paper explores how graded structures on differential operators and quasimodular forms can be used to construct p-adic measures and L-functions on classical groups, advancing automorphic L-function theory in the p-adic setting.

Contribution

It introduces a novel approach using graded structures to build p-adic L-functions and measures, extending automorphic L-function constructions to non-archimedean contexts.

Findings

Constructs p-adic measures and L-functions on classical groups.

Extends the doubling method to p-adic L-functions.

Provides a framework for p-adic automorphic L-function construction.

Abstract

We wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular 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 uni-tary groups and their p-adic avatars is presented. For an algebraic group G over a number eld K these L functions are certain Euler products L(s, $\pi$, r, $\chi$). In particular, our constructions cover the L-functions in [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 , $\chi$ mod p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof. Marc Krasner on Friday, September 23, 2016, International University Centre (IUC), Dubrovnik, Croatia.