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


\Gamma(z) \; \Gamma\left(z + \frac{1}{k}\right) \; \Gamma\left(z + \frac{2}{k}\right) \cdots
\Gamma\left(z + \frac{k-1}{k}\right) =
(2 \pi)^{ \frac{k-1}{2}} \; k^{1/2 - kz} \; \Gamma(kz). \,\!

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 \alpha be an element of F_s.

Let \chi be a multiplicative character from F to the complex numbers.

Let N_{F_s/F}(\alpha) be the norm from F_s to F defined by

N_{F_s/F}(\alpha):=\alpha\cdot\alpha^q\cdots\alpha^{q^{s-1}}.\,

Let \chi' be the multiplicative character on F_s which is the composition of \chi with the norm from Fs to F, that is

\chi'(\alpha):=\chi(N_{F_s/F}(\alpha))

Let ψ be some nontrivial additive character of F, and let \psi' be the additive character on F_s which is the composition of \psi with the trace from Fs to F, that is

\psi'(\alpha):=\psi(Tr_{F_s/F}(\alpha))

Let

\tau(\chi,\psi)=\sum_{x\in F}\chi(x)\psi(x)

be the Gauss sum over F, and let \tau(\chi',\psi') be the Gauss sum over F_s.

Then the Hasse–Davenport lifting relation states that

(-1)^s\cdot \tau(\chi,\psi)^s=-\tau(\chi',\psi').

Hasse–Davenport product relation[edit]

The Hasse–Davenport product relation states that

\prod_{a\bmod m} \tau(\chi\rho^a,\psi) = -\chi^{-m}(m)\tau(\chi^m,\psi)\prod_{a\bmod m} \tau(\rho^a,\psi)

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]