Hasse–Davenport relation

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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).

Hasse–Davenport lifting relation[edit]

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 a multiplicative character from F to the complex numbers.

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

Let

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[edit]

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.

References[edit]