How to define zeta function for a curve over $\mathbb{Z}$ or $\mathbb{Q}$?



How to define zeta function for curves over a number field
For any scheme $X$ of finite type over $\mathbf{Z}$ as a product of Euler factors:


$$\zeta(X,s) = \prod\_{x \in |X|}\frac{1}{1-|\kappa(x)|^{-s}}$$


One might also include the Euler factors at infinity.


For smooth projective varieties $X$ over $\mathbf{Q}$: Put $\bar{X} = X \times\_\mathbf{Q} \bar{\mathbf{Q}}$. For every prime $p$ choose a prime $\bar{p}$ of $\bar{\mathbf{Q}}$ over $p$ and let $D\_p$ and $I\_p$ be the decomposition resp. inertia subgroup, with Frobenius $\phi\_p \in D\_p/I\_p$, and consider for each $\ell \neq p$ the characteristic polynomial


$$P\_p(T) = \mathrm{det}(1-\phi\_p^{-1}T| \mathrm{H}^i(\bar{X},\mathbf{Q}\_\ell)^{I\_p}$$


*Conjecture:* $P\_p(T)$ has integral coefficients independent of $\ell$.


This is known to be true for primes of good reduction.


