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Rationality of the zeta function mod p December 12, 2011

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
5 comments

Here’s a neat argument about counting points that you could present at the end of a second course in number theory. I’m sure it’s not original, but, hey, that’s what blogs are for!

Let X be a smooth hypersurface in \mathbb{P}^{n}, over the field \mathbb{F}_p with p elements. The Weil conjectures are conjectures about the number of points of X over \mathbb{F}_{p^k}. Specifically, they say that there should be some matrix A such that

\displaystyle{ \# X(\mathbb{F}_{p^k}) = 1+p^k+p^{2k} + \cdots + p^{(n-1)k} + (-1)^{n-1} \mathrm{Tr} (A^k),}

and that the eigenvalues of A should be algebraic integers of norm p^{(n-1)/2}.
Here I am using the Lefschetz hyperplane theorem to know what H_{et}^i(X) is for i \neq n-1.

This is, of course, a famously hard theorem. The claim about the eigenvalues is the hardest part, but simply the existence of a matrix for which this formula holds is already quite hard; the first proof was due to Dwork.

What I am going to show you is that there is a much easier proof of the above formula modulo p; a proof of the sort that could be appear in Ireland and Rosen. Many of the terms above disappear mod p, so our goal is just to show that there is some matrix B such that

\displaystyle{ \# X(\mathbb{F}_{p^k}) \equiv 1 + (-1)^{n-1} \mathrm{Tr} (B^k) \mod p.}

(more…)

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