The Hasse–Davenport relations, introduced by , are two related identities for
Gauss sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
:G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r)
where the sum is over elements of some finite commutative ring , is a ...
s, 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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
relating Gauss sums over different fields. used it to calculate the zeta function of a
Fermat hypersurface over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, which motivated the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
.
Gauss sums are analogues of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
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
In mathematics, the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenpor ...
of .
Hasse–Davenport lifting relation
Let ''F'' be a finite field with ''q'' elements, and ''F''
s be the field such that
s:''F''">'F''s:''F''= ''s'', that is, ''s'' is the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''F''
s over ''F''.
Let
be an element of
.
Let
be a
multiplicative character
Character or Characters may refer to:
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* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
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
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
from ''F''
s 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
Trace may refer to:
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* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
from ''F''
s 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
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
*
*
*
* Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil
{{DEFAULTSORT:Hasse-Davenport Relation
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