Riemann hypothesis (function fields)
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| Riemann hypothesis (function fields) | |
|---|---|
| Name | Riemann hypothesis (function fields) |
| Field | Number theory, Algebraic geometry |
| Introduced | 1940s |
| Proved by | André Weil |
| Proof date | 1940s |
Riemann hypothesis (function fields) The Riemann hypothesis for function fields is an analogue of the classical Riemann hypothesis formulated for zeta functions of algebraic curves over finite fields, connecting zeros of L-functions to geometric invariants in the style of Bernhard Riemann, André Weil, and Alexander Grothendieck. It inspired developments in algebraic geometry, arithmetic geometry, and representation theory through links to the Weil conjectures, Hasse–Weil zeta function, and the work of Pierre Deligne, Maxwell Rosenlicht, and Jacob Tsimerman.
Introduction
The problem concerns zeta functions attached to function fields over finite fields such as F_q and was shaped by contributions of Helmut Hasse, André Weil, Emil Artin, Emmy Noether, and Oscar Zariski in analogy with the classical work of Bernhard Riemann on the Riemann zeta function, the investigations of Ernst Steinitz and the formulation of the Weil conjectures later proved by Pierre Deligne and influenced by Alexander Grothendieck and the Bourbaki circle.
Zeta functions of function fields
For a smooth projective curve C over a finite field F_q one defines the zeta function Z(C, T) following the formalism introduced by Emil Artin and refined by Helmut Hasse and André Weil; this zeta function encodes point counts over extensions F_{q^n} and relates to the Hasse–Weil zeta function and L-series considered by Hasse, Weil, and later studied by Jean-Pierre Serre and John Tate. The rationality of Z(C, T) was conjectured in the context of the Weil conjectures and proven using techniques developed by André Weil and axiomatized by Alexander Grothendieck in the theory of étale cohomology, later formalized by Pierre Deligne and linked to traces of Frobenius elements studied by Emil Artin and Richard Brauer.
Statement of the Riemann hypothesis for function fields
The statement asserts that the reciprocal zeros of Z(C, T) lie on the circle |T| = q^{-1/2} in the complex plane, a formulation originating from comparisons to Bernhard Riemann's original hypothesis and articulated by André Weil as part of his list of conjectures for varieties over finite fields; this mirrors the spectral interpretations pursued by Atle Selberg and conceptual connections to Harmonic analysis made by Harish-Chandra and Roger Godement. Equivalent formulations involve bounds for Frobenius eigenvalues acting on étale cohomology groups as developed by Pierre Deligne, Jean-Pierre Serre, and Alexander Grothendieck in the language of sheaves and monodromy.
Weil's proof and methods
Weil's proof for curves used ideas from Algebraic topology-style intersection theory, correspondences, and the Jacobian variety, building on prior work of Hasse on elliptic curves, ideas from Emmy Noether and Oscar Zariski in algebraic geometry, and later framed by the cohomological machinery of Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne. Weil related the zeta function to the characteristic polynomial of the Frobenius endomorphism acting on the Jacobian, invoking the theory of correspondences and the Riemann–Roch theorem which had predecessors in the work of Bernhard Riemann and Gustav Roch; subsequent conceptual proofs used étale cohomology, weights, and purity introduced by Grothendieck and completed by Deligne in his proofs of the remaining Weil conjectures.
Consequences and applications
The resolution for function fields yielded sharp point-counting estimates such as the Hasse bound for curves and influenced the study of error-correcting codes via algebraic-geometric codes introduced by Vladimir Goppa, applications in cryptography developed by Neal Koblitz and Victor Miller, and links to random matrix models explored by Freeman Dyson and Hugh Montgomery. It also provided tools used in the proofs of results about L-functions of varieties appearing in the work of Pierre Deligne and informed the Langlands program articulated by Robert Langlands and developed by Michael Harris and Richard Taylor.
Examples and explicit computations
For elliptic curves over F_q the zeta function can be written explicitly in terms of the trace of Frobenius pioneered by Helmut Hasse and computed in examples studied by André Weil; Hasse's bound and the Deuring–Waterhouse classification link to computations of endomorphism rings as considered by Max Deuring and John Waterhouse. For hyperelliptic curves and modular curves the explicit characteristic polynomials of Frobenius have been computed in work influenced by Gautam Chinta, Andrew Sutherland, and computational projects connected to the L-functions and Modular Forms Database and to algorithms developed by François Morain and Robert Schoof.