mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Weil's criterion is a criterion of

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

for the

Generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...

to be true. It takes the form of an equivalent statement, to the effect that a certain

generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...

positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...

. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of prime number theory, as they apply to

Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...

s, and other more general global L-functions. A single statement thus combines statements on the complex zeroes of ''all'' Dirichlet L-functions. Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions ( Artin-Hecke L-functions); and to the

global function field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

case. Here the inclusion of

Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...

s, in particular, implicates Artin's conjecture; so that the criterion involves a Generalized Riemann Hypothesis plus Artin Conjecture. The case of function fields, of curves over finite fields, is one in which the analogue of the Riemann Hypothesis is known, by Weil's classical work begun in 1940; and Weil also proved the analogue of the Artin Conjecture. Therefore, in that setting, the criterion can be used to show the corresponding statement of positive-definiteness does hold.

References

* A. Weil, "Sur les 'formules explicites' de la théorie des nombres premiers", Comm. Lund (vol. dédié a Marcel Riesz) (1952) 252–265; Collected Papers II * A. Weil, "Sur les formules explicites de la théorie des nombres, Izvestia Akad. Nauk S.S.S.R., Ser. Math. 36 (1972) 3-18; Collected Papers III, 249-264 Zeta and L-functions