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2014-02-02T10:21:47Z

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The '''Hasse–Davenport relations''', introduced by {{harvs|txt|author2-link=Helmut Hasse|last2=Hasse|author1-link=Harold Davenport|last1=Davenport|year=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 sum]]s over different fields. {{harvtxt|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
:<math>
\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). \,\!
</math>
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 {{harvtxt|Gross|Koblitz|1979}}.

== Hasse–Davenport lifting relation ==
Let ''F'' be a finite field with ''q'' elements, and ''F''s be the field such that [''F''s:''F''] = ''s'', that is, ''s'' is the [[Dimension (vector space)|dimension]] of the vector space ''F''s over ''F''.

Let <math>\alpha</math> be an element of <math>F_s</math>.

Let <math>\chi</math> be a [[Multiplicative function|multiplicative]] [[Character (mathematics)|character]] from ''F'' to the complex numbers.

Let <math>N_{F_s/F}(\alpha)</math> be the norm from <math>F_s</math> to <math>F</math> defined by
:<math>N_{F_s/F}(\alpha):=\alpha\cdot\alpha^q\cdots\alpha^{q^{s-1}}.\,</math>

Let
<math>\chi'</math> be the multiplicative character on <math>F_s</math> which is the composition of <math>\chi</math> with the [[Norm (mathematics)|norm]] from ''F''s to ''F'', that is
:<math>\chi'(\alpha):=\chi(N_{F_s/F}(\alpha))</math>

Let ψ be some nontrivial additive character of ''F'', and let
<math>\psi'</math> be the additive character on <math>F_s</math> which is the composition of <math>\psi</math> with the [[Trace (mathematics)|trace]] from ''F''s to ''F'', that is
:<math>\psi'(\alpha):=\psi(Tr_{F_s/F}(\alpha))</math>

Let
:<math>\tau(\chi,\psi)=\sum_{x\in F}\chi(x)\psi(x)</math>
be the [[Gauss sum]] over ''F'', and let
<math>\tau(\chi',\psi')</math> be the Gauss sum over <math>F_s</math>.

Then the '''Hasse–Davenport lifting relation''' states that
:<math>(-1)^s\cdot \tau(\chi,\psi)^s=-\tau(\chi',\psi').</math>

== Hasse–Davenport product relation ==
The Hasse–Davenport product relation states that
:<math>\prod_{a\bmod m} \tau(\chi\rho^a,\psi) = -\chi^{-m}(m)\tau(\chi^m,\psi)\prod_{a\bmod m} \tau(\rho^a,\psi)</math>
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 ==
*{{Citation | last1=Davenport | first1=Harold | last2=Hasse | first2=Helmut | title=Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. (On the zeros of the congruence zeta-functions in some cyclic cases) | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002173123 | language=German | id={{Zbl|0010.33803}} | year=1935 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=172 | pages=151–182}}
*{{Citation | last1=Gross | first1=Benedict H. | last2=Koblitz | first2=Neal | author2-link=Neal Koblitz | title=Gauss sums and the p-adic Γ-function | url=http://dx.doi.org/10.2307/1971226 | doi=10.2307/1971226 | id={{MR|534763}} | year=1979 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=109 | issue=3 | pages=569–581}}
*{{cite book
| last = Ireland
| first = Kenneth
| first2 = Michael |last2=Rosen
| title = A Classical Introduction to Modern Number Theory
| publisher = Springer
| year = 1990
| pages = 158–162
| isbn = 0-387-97329-X}}
*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | url=http://www.ams.org/bull/1949-55-05/S0002-9904-1949-09219-4/home.html | doi=10.1090/S0002-9904-1949-09219-4 | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5}} Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5

{{DEFAULTSORT:Hasse-Davenport Relation}}
[[Category:Cyclotomic fields]]

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