# Hasse-Weil zeta function of smooth projective toric varieties

Let $$X$$ be a smooth projective toric variety over a number field $$K$$ (assume the tori is split). As $$X$$ is rational, maybe the related Hasse-Weil zeta function can be well-understand, so how much do we know about the zeta function $$L(X,s)$$ from the fan (at least for some twist forms of $$X$$)? How about its special values?

• There are results on the motive associated to a toric variety, see Theorem 3.5 in arxiv.org/abs/math/0407305 See also Example 7 of math.mit.edu/~guozhen/motiv.pdf for a more precise result. – François Brunault Apr 6 at 22:44
• The cycle map from the Chow group of $X$ to $\ell$-adic cohomology would be surjective (I can dig up a reference later). Therefore $H^{2i}(X,\mathbb{Q}_\ell)= Q(-i)^{b_{2i}}$, for some $b_{2i}$, and the odd groups are zero. Therefore the Weil zeta function of the closed fibres is the product $\prod 1/(1-q^it)^{b_{2i}}$. This is should be enough to assemble the Hasse-Weil, I think. – Donu Arapura Apr 6 at 22:45