In
mathematics, a Jacobi sum is a type of
character sum
In mathematics, a character sum is a sum \sum \chi(n) of values of a Dirichlet character χ '' modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic ...
formed with
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
s. Simple examples would be Jacobi sums ''J''(''χ'', ''ψ'') for Dirichlet characters ''χ'', ''ψ'' modulo a prime number ''p'', defined by
:
where the summation runs over all
residues (for which neither ''a'' nor is 0). Jacobi sums are the analogues for
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 ...
s of the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
. Such sums were introduced by
C. G. J. Jacobi early in the nineteenth century in connection with the theory of
cyclotomy
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
. Jacobi sums ''J'' can be factored generically into products of powers of
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 ''g''. For example, when the character ''χψ'' is nontrivial,
:
analogous to the formula for the beta function in terms of
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 ...
s. Since the nontrivial Gauss sums ''g'' have absolute value ''p''
, it follows that also has absolute value ''p''
when the characters ''χψ'', ''χ'', ''ψ'' are nontrivial. Jacobi sums ''J'' lie in smaller
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
s than do the nontrivial Gauss sums ''g''. The summands of for example involve no ''p''th
root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
, but rather involve just values which lie in the cyclotomic field of th roots of unity. Like Gauss sums, Jacobi sums have known
prime ideal factorisations in their cyclotomic fields; see
Stickelberger's theorem.
When ''χ'' is the
Legendre symbol,
:
In general the values of Jacobi sums occur in relation with the
local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective alg ...
s of
diagonal form
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is
:\sum_^n a_i ^m\
for some given degree ''m''.
Such forms ''F'', and the hypersurfaces ''F'' = ...
s. The result on the Legendre symbol amounts to the formula for the number of points on a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
that is a
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
over the field of ''p'' elements. A paper of
André Weil from 1949 very much revived the subject. Indeed, through the
Hasse–Davenport relation of the late 20th century, the formal properties of powers of Gauss sums had become current once more.
As well as pointing out the possibility of writing down local zeta-functions for diagonal hypersurfaces by means of general Jacobi sums, Weil (1952) demonstrated the properties of Jacobi sums as
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
s. This was to become important once the
complex multiplication of abelian varieties became established. The Hecke characters in question were exactly those one needs to express the
Hasse–Weil ''L''-functions of the
Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation
:X^n + Y^n = Z^n.\
Therefore, in terms of the affine plane its equation is
:x^ ...
s, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.
References
*
*
*
*{{cite journal, first=André, last=Weil, authorlink=André Weil, title=Jacobi sums as Grössencharaktere, journal=Trans. Amer. Math. Soc., volume=73, date=1952, issue=3, pages=487–495, doi=10.1090/s0002-9947-1952-0051263-0, doi-access=free
Cyclotomic fields