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# Jacobi sum

In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. [1]

## Abelian variety of CM-type

In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A).

## André Weil

André Weil (6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century, known for his foundational work in number theory, algebraic geometry.

## Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by for.

## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

## Character sum

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 (this is like a finite Mellin transform).

## Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

## Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.

## Diagonal form

In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates.

## Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.

## 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 Therefore, in terms of the affine plane its equation is An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa.

## 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.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Gauss sum

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically where the sum is over elements of some finite commutative ring, is a group homomorphism of the additive group into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit where it takes the value 0.

## Hasse–Davenport relation

The Hasse–Davenport relations, introduced by, are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation.

## Hasse–Weil zeta function

In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.

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

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## Local zeta-function

In number theory, the local zeta function Z(V,s) (sometimes called the congruent zeta function) is defined as where N_m is the number of points of V defined over the degree m extension \mathbf_ of \mathbf_q, and V is a non-singular n-dimensional projective algebraic variety over the field \mathbf_q with q elements.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

## Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

## Remainder

In mathematics, the remainder is the amount "left over" after performing some computation.

## Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

## Stickelberger's theorem

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields.

## References

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