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Weil conjectures

Contents

Idea

The Weil conjectures are a sequence of conjectures about counting the number of points of algebraic varieties XX over finite fields; 𝔽 p\mathbb{F}_p and extensions thereof. That is, the number of homomorphisms

Spec𝔽 p nX \operatorname{Spec} \mathbb{F}_{p^n} \longrightarrow X \,

for fixed pp but all n1n \geq 1.

Effectively the conjecture says that the generating function for the number of points as nn varies – the Weil zeta function – is a rational function with some nice properties.

It was realized that the all except one of the conjectures (the Riemann hypothesis) would follow formally from the existence of a suitable cohomology theory on algebraic varieties which behaves in essential aspects like ordinary cohomology of topological spaces and which in particular satisfies a Lefschetz fixed point theorem - what is now called a Weil cohomology theory.

Later Alexander Grothendieck found that the relevant cohomology theory is étale cohomology of schemes.

References

Last revised on July 15, 2016 at 15:11:38. See the history of this page for a list of all contributions to it.