Question

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

Answer 1

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.