A functional equation of the non-Archimedian Rankin convolution
Let S be a finite set of prime numbers containing a fixed prime p. The author describes a functional equation satisfied by the S-adic L- functions obtained by the non-Archimedean interpolation of special values of the Rankin convolution of two cusp forms of different weights. The ideas follow those used by \textit{H. Hida} [Invent. Math. 79, 159-195 (1985; Zbl 0573.10020)] for his p-adic version of Rankin's method. The author first describes the spaces of S-adic modular forms, and S-adic ordinary forms, and then an S-adic version of the Petersson inner product with an S-adic ordinary form. Using this he produces an S-adic Rankin formula, and ultimately an S-adic version of the Rankin convolution. Finally, using results of \textit{W.-C. W. Li} [Math. Ann. 244, 135-166 (1979; Zbl 0396.10017)] he deduces the desired functional equation.
scientific article; zbMATH DE number 4029680
scientific article