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$$p$$-adic interpolation of special values of Hecke L-functions. (English) Zbl 0777.11049
Let $$K$$ be an imaginary quadratic field and $$F/K$$ be an extension of degree $$n$$. Let $$\psi$$ be a so-called $$K$$-admissible Hecke character of $$F$$ and $$L(\psi,s)$$ its Hecke $$L$$-function. Let $$p>5$$ be a prime splitting in $$K$$.
The authors study the $$p$$-adic behavior of the “algebraic part” $$\Lambda(\psi)$$ of the special value $$L(\psi,0)$$. More precisely, they construct a measure on a certain $$p$$-Galois group such that essentially $$\Lambda(\psi)$$ is obtained as an integral over this measure. This generalizes results of M. M. Vishik and Yu. I. Manin [Math. USSR, Sb. 24, 345-371 (1974); translation from Mat. Sb., Nov. Ser. 95(137), 357-383 (1974; Zbl 0352.12013)] and N. M. Katz [Ann. Math., II. Ser. 104, 459-571 (1976; Zbl 0354.14007)] in the case $$n=1$$.
Reviewer: W.Kohnen (Bonn)

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 14G20 Local ground fields in algebraic geometry
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