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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).

##### MSC:
 11R42 Zeta functions and $$L$$-functions of number fields 11S40 Zeta functions and $$L$$-functions
##### Keywords:
Shintani zeta functions; $$p$$-adic $$L$$-functions
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