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 In fact the Hasse–Davenport product relation follows from the analogous multiplication formula for p-adic gamma functions together with the Gross–Koblitz formula of Gross & Koblitz (1979).
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 a multiplicative character from F to the complex numbers.
be the multiplicative character on
with the norm from Fs to F, that is Let ψ be some nontrivial additive character of F, and let
with the trace from Fs to F, that is Let be the Gauss sum over F, and let
be the Gauss sum over
Then the Hasse–Davenport lifting relation states that 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.