Document Zbl 1261.19004

Certain values of Hecke \(L\)-functions and generalized hypergeometric functions. (English) Zbl 1261.19004

J. Number Theory 131, No. 4, 648-660 (2011).

For a smooth projective curve \(X\), one has the regular map from the motivic cohomology group of \(X\), which is given by Milnor \(K\)-groups, to the real Deligne cohomology group. The regular map is closely related to the Beilinson’s conjecture on special values of \(L\)-functions of motives.
Now let \(X\) be a Fermat curve of degree \(N\). For the cases \(N=3,4\), in this paper the author reformulates the regular maps (after Bloch) and then obtains the special values of \(L\)-functions at \(s=0\) for the Jacobi-sum Hecke characters of the \(N\)th cyclotomic fields over \(\mathbb Q\) by means of generalized hypergeometric functions, respectively.

Reviewer: Feng-Wen An (Hubei)

Cited in 7 Documents

MSC:

19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
33C20	Generalized hypergeometric series, \({}_pF_q\)
33C65	Appell, Horn and Lauricella functions
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Keywords:

Fermat curve; \(L\)-function; Milnor \(K\)-group; real Deligne cohomology; regulator map; generalized hypergeometric function; Beilinson conjecture

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References:

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