\(p\)-adic Hecke series of imaginary quadratic fields. (English) Zbl 0329.12016

Math. USSR, Sb. 24 (1974), 345-371 (1976); translation from Mat. Sb., N. Ser. 95(137), 357-383 (1974).
The work of Kubota-Leopoldt and Iwasawa on \(p\)-adic analogues of the Riemann zeta-function and Dirichlet L-series and that of Serre on the \(p\)-adic analogue of the Dedekind zeta function for totally real algebraic number fields are well-known. From Mazur and Manin, one knows how to introduce \(p\)-adic Mellin transforms of modular forms of one variable belonging to congruence subgroups of the elliptic modular group. In this article, the authors study a \(p\)-adic


11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G20 Local ground fields in algebraic geometry
11R11 Quadratic extensions
11S40 Zeta functions and \(L\)-functions
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