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

