Gaußsche Summen über endlichen Körpern und Gammafunktion

Anna Helversen-Pasotto

Séminaire lotharingien de combinatoire, Tome 33 (1994)

Résumé

Gaussian sums over finite fields are analogous to values of the gamma-function as can be seen via the Eulerian integral for the gamma-function and this seems to have been known to Gauss already; indeed, this seems to have been for him one of the reasons to introduce these sums. Nevertheless it has been astonishing for several authors to discover identities for Gaussian sums which bear a strong formal analogy with classical identities for the gamma function; also a p-adic gamma function has been invented and many of the well-known classical identities have been shown to admit p-adic analogs. The aim of these lectures is to give an introduction to the subject discussing some identities, giving some proofs and pointing out relations to the representation theory of the general linear group of a finite field (Hecke algebras). The analogy between binomial coefficients and Jacobi sums leading to hypergeometric functions over finite fields is mentioned.