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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
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.
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)
Last revised on July 15, 2016 at 19:11:38. See the history of this page for a list of all contributions to it.