# Tagged Questions

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### Reference for counting points over finite fields

The following fact is extremely well known: Fact. Let $Y$ be a geometrically irreducible variety (not necessarily smooth or proper) over a finite field $k$. Then there is a constant $B$, ...
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### How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes. Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...
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### Elementary examples of the Weil conjectures

I'm looking for examples of the Weil conjectures---specifically rationality of the zeta function---that can be appreciated with minimal background in algebraic geometry. Are there varieties for which ...
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### Hodge numbers of reduction mod $p$

Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$) ...
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### On the Hasse-Weil L-function of $P^n$

So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...
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### Can one find the hodge number by counting points over finite fields?

Given a proper smooth variety $X$ of dimension $n$ over $\mathbb{C}$, assume it has a model over a DVR of mixed characteristic $(0,p)$ with residue field $\mathbb{F}_q$, and assume the closed fiber ...
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### Is the integrality of the zeta function easy?

I'm trying to get the gist of the proof of the Weil conjectures. Let $X$ be a variety over $\mathbb{F}_{p^n}$. A priori $Z(X,t)\in \mathbb{Q}[[t]]$. Due to the Grothendieck-Lefschetz fixed point ...
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### How would a motivic proof of the Riemann hypothesis over finite fields go?

It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...
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### Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula. My first question is based on the ...