André Weil proved the Artin conjecture in the case of function fields.

The Weil conjecture on Tamagawa numbers proved resistant for many years.

He picked up another credited Weil conjecture, around 1967, which later under pressure from Serge Lang ( resp.

Taniyama – Weil conjecture ) based on a roughly formulated question of Taniyama at the 1955 Nikkō conference.

* Weil conjecture disambiguation page

* Weil conjecture on Tamagawa numbers

Because of the Mordell – Weil theorem, Faltings ' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell – Lang conjecture, which has been proved.

In mathematics the modularity theorem ( formerly called the Taniyama – Shimura – Weil conjecture and several related names ) states that elliptic curves over the field of rational numbers are related to modular forms.

The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform ( an eigenvector of all Hecke operators ); this is the Hasse – Weil conjecture, which follows from the modularity theorem.

The conjecture attracted considerable interest when suggested that the Taniyama – Shimura – Weil conjecture implies Fermat's Last Theorem.

In the summer of 1986, proved the epsilon conjecture, thereby proving that the Taniyama – Shimura – Weil conjecture implied Fermat's Last Theorem.

, with some help from Richard Taylor, proved the Taniyama – Shimura – Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

The full Taniyama – Shimura – Weil conjecture was finally proved by,, and who, building on Wiles ' work, incrementally chipped away at the remaining cases until the full result was proved.

and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Artin for attacking the Weil conjectures, as outlined in.

* had previously shown that the Ramanujan-Petersson conjecture follows from the Weil conjectures.

The term Weil conjecture may refer to:

* The Weil conjecture on Tamagawa numbers about the Tamagawa number of an algebraic group, proved by Kottwitz and others.

* The Hasse – Weil conjecture about zeta functions.

For example, the Künneth standard conjecture, which states the existence of algebraic cycles π < sup > i </ sup > ⊂ X × X inducing the canonical projectors H < sup >∗</ sup >( X ) ↠ H < sup > i </ sup >( X ) ↣ H < sup >∗</ sup >( X ) ( for any Weil cohomology H ) implies that every pure motive M decomposes in graded pieces of weight n: M = ⊕ Gr < sub > n </ sub > M.

The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse – Weil L-function L ( E, s ) of E at s = 1.

The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.

The conjecture attracted considerable interest when Frey ( 1986 ) suggested that the Taniyama – Shimura – Weil conjecture implies Fermat's Last Theorem.

One of the more dramatic successes of his theory was his prediction of the existence of secondary and tertiary alcohols, a conjecture that was soon confirmed by the synthesis of these substances.

His attitude towards conjectures was that one should not dignify a guess as a conjecture lightly, and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s.

‘‘ The most prevalent conjecture was that they were some of the German peoples which extended as far as the northern ocean ,</ br >

Although Tiberius was 77 and on his death bed, some ancient historians still conjecture that he was murdered.

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for k

* In the 1960s, EDSAC was used to gather numerical evidence about solutions to elliptic curves, which led to the Birch and Swinnerton-Dyer conjecture.

The conjecture was first proposed in 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed.

This formula, the Heawood conjecture, was conjectured by P. J.

In 1973 the number theorist Hugh Montgomery was visiting the Institute for Advanced Study and had just made his pair correlation conjecture concerning the distribution of the zeros of the Riemann zeta function.

Gin, though, was blamed for various social problems, and it may have been a factor in the higher death rates which stabilized London's previously growing population, although there is no evidence for this and it is merely conjecture.

But Steinschneider will not admit the possibility of this conjecture, while Renan scarcely strengthens it by regarding " Andreas " as a possible northern corruption of " En Duran ," which, he says, may have been the Provençal surname of Anatoli, since Anatoli, in reality, was but the name of his great-grandfather.

In fact, whether one can smooth certain higher dimensional spheres was, until recently, an open problem — known as the smooth Poincaré conjecture.

Yusuf Ali ’ s translation reads " That they said ( in boast ), " We killed Christ Jesus the son of Mary, the Messenger of Allah ";― but they killed him not, nor crucified him, but so it was made to appear to them and those who differ therein are full of doubts, with no ( certain ) knowledge, but only conjecture to follow, for of a surety they killed him not .― ( 157 ) Nay, Allah raised him up unto Himself ; and Allah is Exalted in Power, Wise.

" And while the conjecture may one day be solved, the argument applies to similar unsolved problems ; to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution.

He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper " On a conjecture by Littlewood and idempotent measures ", and lends his name to the Cohen-Hewitt factorization theorem.

The Poincaré conjecture, before being proven, was one of the most important open questions in topology.

The influence of the Tamagawa number idea was felt in the theory of arithmetic of abelian varieties through its use in the statement of the Birch and Swinnerton-Dyer conjecture, and through the Tamagawa number conjecture developed by Spencer Bloch, Kazuya Kato and many other mathematicians.