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Gauss Sums and the Classical Γ‐Function

Gauss Sums and the Classical Γ‐Function GAUSS SUMS AND THE CLASSICAL T-FUNCTION C.-G. SCHMIDT 1. Introduction {N) Let N be a positive integer (N £ 2 mod 4), and let K = Q be the cyclotomic field of N-th roots of unity. Using the notation of [2] we consider, for some prime ^B/iV of K and for some element a = £m(a)<5 of the free abelian group with basis T = (l/iV)Z/Z-{0}, the generalized Gauss sum: and the product of the values of the F-function: mia) » . (2) Deligne has shown (see [2]) that, for all a belonging to the relation module of the principal ideals generated by the Gauss sums A = {a;(g(<x,yj) = l for all the quantities Q are algebraic numbers, and that the Frobenius automorphism <7< p e Gal (Q/K) operates like ^(QJ^a,^)" ^. (3) The proof involves the Hodge theory of Fermat hypersurfaces. A weaker version of Deligne's theorem was recently shown by Gross and Koblitz [2]. They use Morita's p-adic F-function and a result of Katz on the p-adic cohomology of the Fermat curve to prove among other things that (3) is valid for the submodule B of A generated by the elements a(d,a) = S - t^ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Gauss Sums and the Classical Γ‐Function

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.5.344
Publisher site
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Abstract

GAUSS SUMS AND THE CLASSICAL T-FUNCTION C.-G. SCHMIDT 1. Introduction {N) Let N be a positive integer (N £ 2 mod 4), and let K = Q be the cyclotomic field of N-th roots of unity. Using the notation of [2] we consider, for some prime ^B/iV of K and for some element a = £m(a)<5 of the free abelian group with basis T = (l/iV)Z/Z-{0}, the generalized Gauss sum: and the product of the values of the F-function: mia) » . (2) Deligne has shown (see [2]) that, for all a belonging to the relation module of the principal ideals generated by the Gauss sums A = {a;(g(<x,yj) = l for all the quantities Q are algebraic numbers, and that the Frobenius automorphism <7< p e Gal (Q/K) operates like ^(QJ^a,^)" ^. (3) The proof involves the Hodge theory of Fermat hypersurfaces. A weaker version of Deligne's theorem was recently shown by Gross and Koblitz [2]. They use Morita's p-adic F-function and a result of Katz on the p-adic cohomology of the Fermat curve to prove among other things that (3) is valid for the submodule B of A generated by the elements a(d,a) = S - t^

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1980

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