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Let $S$ be a finite set of prime numbers and $X$ be a proper smooth variety (reduced scheme of finite type) over $\text{Spec} \mathbb{Z} [1/S]$. Let $p$ be a prime number not in $S$. I can consider the base change $X_{\mathbb{F}_{p}}$ over $\mathbb{F}_{p}$ to which I can associate its zeta function $Z(X_{\mathbb{F}_{p}})$. I can also consider the base change $X_\mathbb{C}$ to which I can associate its Hodge numbers $h^{p,q}$.

My question is :

** Are there some S,p as above and some proper smooth variety $X$ and $Y$ over $\text{spec} \mathbb{Z}[1/S]$ such that $X$ and $Y$ have the same zeta function over $\mathbb{F}_{p}$ but different Hodge numbers ?**

In the mathoverflow discussion Can one find the hodge number by counting points over finite fields? an example is given which answers the question if I replace $\mathbb{F}_{p}$, $p$ prime, by a general finite field $\mathbb{F}_{q}$, $q$ power of a prime. The idea of this example is to consider a supersingular elliptic curve over $\mathbb{F}_{q}$ such that the frobenius acts on $H^1$ by multiplication by $\sqrt{q}$ (in order to construct by product a variety whose cohomology looks like cohomology of a projective plane ...) : of course, for the existence of such an elliptic curve, a necessary condition is $2 \sqrt{q}$ integer so $q$ must be a square. This shows that this example cannot be easily changed to answer the question. What I want is stronger : the zeta function over $\mathbb{F}_{p}$ contains more information than the zeta function over $\mathbb{F}_{p^{n}}$ for some $n$.

Finally, one remark to prevent eventual misunderstanding : the data in the question is the zeta function over $\mathbb{F}_{p}$ for one particular prime $p$. It is known (for example in How much complex geometry does the zeta-function of a variety know) that the knowledge of the zeta functions on $\mathbb{F}_{p}$ for all but finitely many primes $p$ determine the Hodge numbers.

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Sorry for the parallel editing. I hope things are as you intended know. The issue is that underscores (and some other things) in tex causes problems as they are interpreted as markdown synstax. A quick way to avoid this is to put backticks around the tex; cf the current source. –  quid Jul 24 '12 at 12:25
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One can obtain examples of the type you want by restriction of scalars, i.e., if you have such examples over the ring of $S$-integers of a number field, then the restriction of scalars to $\mathbb{Z}[1/S]$ will give what you want. The point is that after base change to $\mathbb{C}$ the restriction of scalars is just a product and if $X_1$ and $X_2$ have the same number of points over each finite extension of $\mathbb{F}_q$, then the same will hold for the restriction of scalars over $\mathbb{F}_p$. –  ulrich Jul 24 '12 at 13:19
    
Thanks for your answer. Does the zeta function changes in a simple way under restriction on scalars ? –  user25309 Jul 26 '12 at 16:43
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