nLab Weil conjectures

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.

Related concepts

• number theory

• Riemann zeta function

• Frobenius endomorphism

• Weil conjecture on Tamagawa number

References

• James Milne, section 26 of Lectures on Étale Cohomology

• Sophie MorelThe Weil conjectures, from Abel to Deligne (IAS video). Note that the title was chosen as a joke by Morel; she clarifies that there is no known connection between Abel and the Weil conjectures.

• Wikipedia, Weil conjectures

• Dennis Gaitsgory, Jacob Lurie, Weil’s Conjecture for Function Fields (pdf)