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Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups. (English) Zbl 1424.11096

Summary: We wish to use Krasner graded and Krasner-Vuković paragraded structures [M. Krasner, in: Algebra Colloquium. Rennes: Univ. Rennes. 209–308 (1980; Zbl 0521.16001); M. Krasner and M. Vuković, Structures paragraduées (groupes, anneaux, modules). (Paragraded structures (groups, rings, modules)). Ontario, Canada: Queen’s University Press (1987; Zbl 0624.08001); M. Vuković, “Structures graduées et paragraduées”, Prépublication de l’Institut Fourier, No. 536 (2001)] on differential 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 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 \(L(s, \pi, r, \chi)\).
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 quasi-modular forms and their Fourier coefficients.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11F33 Congruences for modular and \(p\)-adic modular forms
14G20 Local ground fields in algebraic geometry
22E50 Representations of Lie and linear algebraic groups over local fields
16W50 Graded rings and modules (associative rings and algebras)
16E45 Differential graded algebras and applications (associative algebraic aspects)
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