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The \(p\)-adic Shintani cocycle. (English) Zbl 1317.11116
Let \(V\) be a vector space over the rationals of finite dimension, and let \(C\) be a Shintani set in \(V_R=V\otimes_QR\), as defined by S. Dasgupta [Duke Math. J. 143, No. 2, 225–279 (2008; Zbl 1235.11102)]. The author considers the Shintani zeta functions defined by \[ \zeta_{Sh}(f,C;s)=\sum_{v\in C\cap V}{f(v)\over N(v)^s}, \] (where \(N(v)\) is the product of coordinates, and \(f\) is test function), shows how to attach with these functions certain \(p\)-adic pseudo-measures, and proves (Theorem 4.19) that under certain conditions on \(f\) these pseudo-measures are measures. This is applied to a construction of \(p\)-adic \(L\)-functions for totally real number fields (Theorem 6.1).

11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions
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