Jacobi sum

Last updated January 03, 2025 • 2 min readFrom Wikipedia, The Free Encyclopedia

In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by

J(\chi ,\psi )=\sum \chi (a)\psi (1-a)\,,

where the summation runs over all residues a = 2, 3, ..., p − 1 mod p (for which neither a nor 1 − a is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. Jacobi sums J can be factored generically into products of powers of Gauss sums g. For example, when the character χψ is nontrivial,

J(\chi ,\psi )={\frac {g(\chi )g(\psi )}{g(\chi \psi )}}\,,

analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums g have absolute value p^1⁄2, it follows that J(χ, ψ) also has absolute value p^1⁄2 when the characters χψ, χ, ψ are nontrivial. Jacobi sums J lie in smaller cyclotomic fields than do the nontrivial Gauss sums g. The summands of J(χ, ψ) for example involve no pth root of unity, but rather involve just values which lie in the cyclotomic field of (p − 1)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,

J(\chi ,\chi )=-\chi (-1)=(-1)^{\frac {p+1}{2}}\,.

In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the formula p + 1 for the number of points on a conic section that is a projective line 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 characters. 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 curves, for example. The exact conductors of these characters, a question Weil had left open, were determined in later work.

Related Research Articles

<span class="mw-page-title-main">Quadratic reciprocity</span> Gives conditions for the solvability of quadratic equations modulo prime numbers

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter $ζ$ (zeta), is a mathematical function of a complex variable defined as $for, and its analytic continuation elsewhere.$

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function $:\mathbb {Z} \rightarrow \mathbb {C} }$ is a Dirichlet character of modulus if for all integers $and :$

$that is, is completely multiplicative.$
$; that is, is periodic with period .$

<span class="mw-page-title-main">Hurwitz zeta function</span> Special function in mathematics

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables $s$ with $Re(s) > 1$ and $a \neq 0, -1, -2, \dots$ by

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis. They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.

In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.

In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, around 1951.

In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ_K(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζ_K(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.

In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

In mathematics, a character sum is a sum $of values of a Dirichlet character χ modulo N, taken over a given range of values of n . Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N . Character sums are often closely linked to exponential sums by the Gauss sums.$

In number theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function.

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 certain others which have functional equations analogous to that of the Riemann zeta-function.

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.

The Hasse–Davenport relations, introduced by Davenport and Hasse, 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.

In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic. For example, the domain could be the p-adic integersZ_p, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbersQ_p or its algebraic closure.

In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight 2k, discovered by Goro Shimura. It has the property that the eigenvalue of a Hecke operator T_n² on F is equal to the eigenvalue of T_n on f.

References

Berndt, B. C.; Evans, R. J.; Williams, K. S. (1998). Gauss and Jacobi Sums. Wiley. ISBN 978-0-471-12807-6.
Lang, S. (1978). Cyclotomic fields . Graduate Texts in Mathematics. Vol. 59. Springer Verlag. ch. 1. ISBN 0-387-90307-0.
Weil, André (1949). "Numbers of solutions of equations in finite fields". Bull. Amer. Math. Soc. 55 (5): 497–508. doi: 10.1090/s0002-9904-1949-09219-4 .
Weil, André (1952). "Jacobi sums as Grössencharaktere". Trans. Amer. Math. Soc. 73 (3): 487–495. doi: 10.1090/s0002-9947-1952-0051263-0 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.