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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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