The Hasse–Davenport relations, introduced by Davenport and Hasse (1935), are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory relating Gauss sums over different fields. Weil (1949) used it to calculate the zeta function of a Fermat hypersurface over a finite field, which motivated the Weil conjectures.
Gauss sums are analogues of the gamma function over finite fields, and the Hasse–Davenport product relation is the analogue of Gauss's multiplication formula
Hasse–Davenport lifting relation
Let F be a finite field with q elements, and Fs be the field such that [Fs:F] = s, that is, s is the dimension of the vector space Fs over F.
Let be an element of .
Let be the norm from to defined by
Let be the multiplicative character on which is the composition of with the norm from Fs to F, that is
Let ψ be some nontrivial additive character of F, and let be the additive character on which is the composition of with the trace from Fs to F, that is
be the Gauss sum over F, and let be the Gauss sum over .
Then the Hasse–Davenport lifting relation states that
Hasse–Davenport product relation
The Hasse–Davenport product relation states that
where ρ is a multiplicative character of exact order m dividing q–1 and χ is any multiplicative character and ψ is a non-trivial additive character.
- Davenport, Harold; Hasse, Helmut (1935), "Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. (On the zeros of the congruence zeta-functions in some cyclic cases)", Journal für Reine und Angewandte Mathematik (in German) 172: 151–182, ISSN 0075-4102, Zbl 0010.33803
- Gross, Benedict H.; Koblitz, Neal (1979), "Gauss sums and the p-adic Γ-function", Annals of Mathematics. Second Series 109 (3): 569–581, doi:10.2307/1971226, ISSN 0003-486X, MR 534763
- Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory. Springer. pp. 158–162. ISBN 0-387-97329-X.
- Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5