3
votes
3answers
206 views

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 $B$, ...
8
votes
1answer
882 views

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 ...
24
votes
5answers
1k views

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 ...
4
votes
4answers
600 views

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 $p$) ...
15
votes
1answer
696 views

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 ...
8
votes
2answers
694 views

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 ...
2
votes
2answers
579 views

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 ...
10
votes
2answers
921 views

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 ...
17
votes
0answers
764 views

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 ...
8
votes
0answers
866 views

Do the Standard Conjectures imply parts of the “Weil II” Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
8
votes
3answers
1k views

Are there “motivic” proofs of Weil conjectures in special cases?

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...