Weil conjectures

The Weil conjectures describe zeta functions of smooth projective varieties over finite fields through rationality, functional equations, and constraints on their zeros.

Finite field: A field with finitely many elements, whose size is a power of a prime. The conjectures concern varieties defined over fields of this kind.
Point counting over finite fields: The enumeration of solutions to equations over finite fields and their finite extensions. The sequence of these counts determines the variety's zeta function.
André Weil: A French mathematician whose work shaped number theory, algebraic geometry, and the theory of automorphic forms. He formulated the conjectures after studying analogies between curves and number fields.
Langlands program: A web of conjectures connecting representations of reductive groups with arithmetic and geometric objects. The Weil conjectures helped establish cohomological methods later used in related arithmetic correspondences.
Algebraic variety: A geometric space defined by polynomial equations over a field. The zeta function in the conjectures counts points on varieties over finite fields.
Rationality of zeta functions over finite fields: The theorem that a variety's zeta function over a finite field is a rational function. This is the first of the conjectures, proved by Dwork.
Weil's 1949 paper on the Riemann hypothesis for function fields: A paper in which André Weil proved the Riemann-hypothesis analogue for curves over finite fields. Its proof provided the model and evidence for the broader conjectures.
Modular forms: Highly symmetric holomorphic functions on the upper half-plane with transformation and growth conditions. Their arithmetic became linked to Frobenius eigenvalues through later developments in number theory.
Zeta function of a variety over a finite field: A generating function that encodes point counts of a variety over finite-field extensions. Its structure is the direct subject of the conjectures.
Functional equation of a zeta function over a finite field: A symmetry relating a variety's zeta function at a variable to its value at a transformed variable. The conjectures predict a functional equation shaped by the variety's dimension and degree.

Linked from 6 pages