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Weil and conjecture
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.
A conjecture of André Weil was that the Tamagawa number was always 1 for a simply connected G. This arose out of Weil's modern treatment of results in the theory of quadratic forms ; the proof was case-by-case and took decades, the final steps were taken by Robert Kottwitz in 1988 and V. I.
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.

Weil and on
André Weil (; 6 May 1906 – 6 August 1998 ) was an influential French mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition.
This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).
Indian ( Hindu ) thought had great influence on Weil.
* A 1940 Letter of André Weil on Analogy in Mathematics
He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it.
On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.
With less time to spend songwriting as she focused on a burgeoning film career, during the early 1980s Parton recorded a larger percentage of material from noted pop songwriters, such as Barry Mann and Cynthia Weil, Rupert Holmes, Gary Portnoy and Carole Bayer Sager.
Recently, a large number of cryptographic primitives based on bilinear mappings on various elliptic curve groups, such as the Weil and Tate pairings, have been introduced.
Johannes Kepler was born on December 27, 1571, at the Free Imperial City of Weil der Stadt ( now part of the Stuttgart Region in the German state of Baden-Württemberg, 30 km west of Stuttgart's center ).
* Liza Weil ( 1977 -), actress known for her portrayal of Paris Geller on the television series Gilmore Girls ; was born in New Jersey but her family has resided in the borough since 1984.
This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety.
Noteworthy for its appearance on the list of Hilbert problems, Smale's list and the list of Millennium Prize Problems — and even, in its geometric guise, in the Weil Conjectures — is the Riemann hypothesis.
The songs on the album were written by such people as Tony Romeo, Terry Cashman, Tommy West, Barry Mann, and Cynthia Weil.
#" I'm on the Road " ( Barry Mann, Cynthia Weil )
Bugeaud's writings were numerous, including his Œuvres militaires, collected by Weil ( Paris, 1883 ), many official reports on Algeria and the war there, and some works on economics and political science.
Simone Weil who fought for the French resistance ( the Maquis ) argued in her final book The Need for Roots: Prelude to a Declaration of Duties Towards Mankind that for society to become more just and protective of liberty, obligations should take precedence over rights in moral and political philosophy and a spiritual awakening should occur in the conscience of most citizens, so that social obligations are viewed as fundamentally having a transcendent origin and a beneficent impact on human character when fulfilled.
Weil was a young Marxist who had written his doctoral thesis on the practical problems of implementing socialism and was published by Karl Korsch.
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions ( known as local zeta-functions ) derived from counting the number of points on algebraic varieties over finite fields.

Weil and numbers
The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers.
* Grothendieck expressed the zeta function in terms of the trace of Frobenius on l-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith l-adic cohomology group H < sup > i </ sup >( V ) of V have absolute values | α |= q < sup > i / 2 </ sup > ( for an embedding of the algebraic elements of Q < sub > l </ sub > into the complex numbers ).
For elliptic curves over the rational numbers, the Hasse – Weil conjecture follows from the modularity theorem.

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