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Non-Archimedean Rankin convolution of unbounded growth. (Russian) Zbl 0732.11025

The Rankin convolution of two holomorphic Hecke forms f and g for \(\Gamma_ 0(N)\) not only gives a complex valued L-function L(,f,g), but also an L-function \(\Psi_ p(,f,g)\) with values in \({\mathbb{C}}_ p\) (the completion of the algebraic closure of \({\mathbb{Q}}_ p)\). The domain of \(\Psi_ p\) is the group of continuous homomorphisms \((\prod_{q\in S}{\mathbb{Z}}^*_ q)\to {\mathbb{C}}_ p\), with S a finite set of primes containing the fixed prime p.
The p-adic L-function \(\Psi_ p\) interpolates the values of the complex L-functions L(,f,g\({}_ 1)\) at special points, where \(g_ 1\) runs through the twists of g by Dirichlet characters. To make this interpolation possible an embedding \(i_ p\) of the algebraic closure of \({\mathbb{Q}}\) into \({\mathbb{C}}_ p\) has to be fixed. The case that the Fourier coefficient \(a_ p\) of f satisfies \(| i_ p(a_ p)|_ p=1\) has been considered by A. A. Panchishkin [Izv. Akad. Nauk SSSR, Ser. Mat. 52, 336-354 (1988; Zbl 0656.10020); English translation Math. USSR, Izv. 32, 339-358 (1989)]. He represents \(\Psi_ p\) as the Mellin transform of a measure \(\mu\) on \(\prod_{q\in S}{\mathbb{Z}}^*_ q\), proves its analyticity and functional equation.
The author considers the case \(| i_ p(a_ p)|_ p<1\), following Panchishkin closely. (Some parts are taken word by word from A. A. Panchishkin, Non-Archimedean automorphic zeta-functions, (1988; Zbl 0667.10017). Now \(\mu\) is no longer a measure, but a distribution satisfying a certain growth condition. To handle this the author uses results on this type of distributions from M. M. Vishik [Mat. Sb., Nov. Ser. 99, (141), 248-260 (1976; Zbl 0358.14014); translation in Math. USSR, Sb.].
In Panchishkin’s approach the weight of g should be less than that of f. I think that on top of page 167 the author implicitly introduces an additional assumption on the weight of g.

MSC:

11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11S20 Galois theory
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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