\(p\)-adic limits of Shintani generating functions for a real quadratic field
In the paper [\textit{T. Shintani}, J. Fac. Sci. Univ. Tokyo, Sect. I A 23, No. 2, 393-417 (1976; Zbl 0349.12007)], special values were evaluated at non-positive integers of zeta functions for a totally real algebraic number field \(F\) by introducing a kind of generating functions. These evaluations are associated to ``cone decompositions'' in \(F\otimes \mathbb{R}\) which, however, seem rather noncanonical for \(F\neq\mathbb{Q}\). The author of this paper introduces new generating functions by passing to a certain \(p\)-adic limit in the case of a real quadratic field \(F\) with \(h_+ (F) =1\) motivated by Iwasawa Theory and \(p\)-adic \(L\)-functions. In this paper, the author investigates these limiting functions, showing that they possess several interesting properties not shared by those of Shintani.
scientific article; zbMATH DE number 913834