Weil Conjectures

In mathematics, the Weil conjectures were some highly-influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with qk elements containing that field. The generating function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements.

Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation by Grothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974)

Read more about Weil Conjectures:  Background and History, Statement of The Weil Conjectures, Weil Cohomology, Grothendieck's Formula For The Zeta Function, Deligne's First Proof, Deligne's Second Proof, Applications

Other articles related to "weil, weil conjectures, conjecture, conjectures, weil conjecture":

Timeline Of Category Theory And Related Mathematics - 1945–1970
1949 André Weil Formulates the Weil conjectures on remarkable relations between the cohomological structure of algebraic varieties over C and the diophantine structure of algebraic. 1959 Bernard Dwork Proves the rationality part of the Weil conjectures (the first conjecture) ... development in SGA4 of the long-anticipated Weil cohomology ...
Hilbert's Problems - Sequels
... One of the exceptions is furnished by three conjectures made by André Weil during the late 1940s (the Weil conjectures) ... In the fields of algebraic geometry, number theory and the links between the two, the Weil conjectures were very important ... The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via l-adic cohomology was given by Alexander ...
Weil Conjectures - Applications
... (1980) was able to prove the hard Lefschetz theorem (part of Grothendieck's standard conjectures) using his second proof of the Weil conjectures ... Deligne (1971) had previously shown that the Ramanujan-Petersson conjecture follows from the Weil conjectures ... Deligne (1974, section 8) used the Weil conjectures to prove estimates for exponential sums ...
Wiles's Proof Of Fermat's Last Theorem - Progress of The Previous Decades
... it became well known through a 1967 paper by André Weil ... With Weil giving conceptual evidence for it, it is sometimes called the Taniyama–Shimura–Weil conjecture ... Again, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its ...
Glossary Of Arithmetic And Diophantine Geometry - W
... Weil cohomology The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a cohomology theory applying to ... Weil conjectures The Weil conjectures were three highly-influential conjectures of André Weil, made public around 1949, on local zeta-functions ... which comes from an elementary method, and improvements of Weil bounds, e.g ...

Famous quotes containing the words conjectures and/or weil:

    After all, it is putting a very high price on one’s conjectures to have a man roasted alive because of them.
    Michel de Montaigne (1533–1592)

    A test of what is real is that it is hard and rough. Joys are found in it, not pleasure. What is pleasant belongs to dreams.
    —Simone Weil (1909–1943)