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# Contents

## Idea

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

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

for fixed $p$ but all $n \geq 1$.

Effectively the conjecture says that the generating function for the number of points as $n$ 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 19:11:38. See the history of this page for a list of all contributions to it.