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 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 _p, χ 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.