It is easy to check all parts of the Weil conjectures directly.For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.…

In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.

One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups.This is one of the steps used in the proof of the Weil conjectures.Behrend's formula generalizes the formula to algebraic stacks.…

He is known for work on the Weil conjectures, leading to a complete proof in 1973.

Their work came to be seen as an introduction to one form of the theory of algebraic stacks, and recently has been applied to questions arising from string theory.Perhaps Deligne's most famous contribution was his proof of the third and last of the Weil conjectures.This proof completed a programme initiated and largely developed by Alexander Grothendieck.…

It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property.These give the first non-trivial cases of the Weil conjectures (proved by Hasse).…

This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

Deligne won a Fields Medal in 1978 for his work on Weil conjectures.While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper.…

His 'matrix divisor' (vector bundle avant la lettre) Riemann-Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.The Weil conjecture on Tamagawa numbers proved resistant for many years.Eventually the adelic approach became basic in automorphic representation theory.…

In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem.Much of the second proof is a rearrangement of the ideas of his first proof.…

During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found.At that time the Weil conjectures were an outstanding motivation to research.…

Both Grothendieck and Deligne were awarded the Fields medal.However, the Weil conjectures in their scope are more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having been important in the development of many of them.…

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.This was one of the clues leading to the Weil conjectures.…

As a first application, he proved the Weil conjecture on Tamagawa numbers for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers.Previously this had been known only in a few isolated cases and for certain classical groups where it could be shown by induction.…

The first observation does not hold for all groups: found some examples whose Tamagawa numbers are not integers.The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.Several authors checked this in many cases, and finally Kottwitz proved it for all groups in 1988.…

Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures.The problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.…

It was introduced by Pierre Deligne on November 29, 1976 in a letter to David Kazhdan as an analogue of the usual Fourier transform.It was used by to simplify Deligne's proof of the Weil conjectures.…

Description: This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of modular forms and their L-functions through the introduction of representation theory.Description: Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open Weil conjectures.Description: Faltings proves a collection of important results in this paper, the most famous of which is the first proof of the Mordell conjecture (a conjecture dating back to 1922).…

Auxiliary types of sums occur in the theory, for example character sums; going back to Harold Davenport's thesis.The Weil conjectures had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an algebraic variety over a finite field).…

As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic.The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry.They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension.…