This article is about the theory of algebraic varieties over finite fields. For other uses, see Weil conjecture.
On generating functions from counting points on algebraic varieties over finite fields
In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements.
Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by Bernard Dwork (1960), the functional equation by Alexander Grothendieck (1965), and the analogue of the Riemann hypothesis by Pierre Deligne (1974).
In mathematics, the Weilconjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them...
The term Weilconjecture may refer to: The Weilconjectures about zeta functions of varieties over finite fields, proved by Dwork, Grothendieck, Deligne...
Poincaré conjecture), Fermat's Last Theorem, and others. Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf....
theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Shimura-Weilconjecture or modularity conjecture for elliptic curves) states that elliptic...
1949, André Weil posed the landmark Weilconjectures about the local zeta-functions of algebraic varieties over finite fields. These conjectures offered a...
mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories...
1944) is a Belgian mathematician. He is best known for work on the Weilconjectures, leading to a complete proof in 1973. He is the winner of the 2013...
geometry), motivic cohomology. Weilconjectures The Weilconjectures were three highly influential conjectures of André Weil, made public around 1949, on...
This is a list of notable mathematical conjectures. The following conjectures remain open. The (incomplete) column "cites" lists the number of results...
étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weilconjectures, ultimately completed in the 1970s by...
with André and Simone Weil, translated from the French by Benjamin Ivry. Math Intelligencer *34, *76–78 (2012) "The WeilConjectures by Karen Olsson review...
Tamagawa number τ(G) is defined to be the Tamagawa measure of G(A)/G(k). Weil'sconjecture on Tamagawa numbers states that the Tamagawa number τ(G) of a simply...
one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weilconjectures (specifically, he deduced it...
Ramanujan conjecture, one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic...
l-adic cohomology) Kleiman, S. L. (1968), "Algebraic cycles and the Weilconjectures", Dix exposés sur la cohomologie des schémas, Amsterdam: North-Holland...
functions, and in particular for a proof of the first part of the Weilconjectures: the rationality of the zeta-function of a variety over a finite field...
writing up the algebraic geometry involved. This led him to the general Weilconjectures. Alexander Grothendieck developed scheme theory for the purpose of...
completes the proof. A proof is also possible assuming Weil'sconjecture on Tamagawa numbers. The conjecture asserts for the case of the algebraic group SL2(R)...
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