Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that we have arithmetic equivalence. Does this imply that $K_1$ and $K_2$ have the same genus as well?
The strongest relationship I know between genus and the zeta function is a theorem from Rosen's book on function fields in number theory: if $K$ is a global function field over a finite field of genus $g$, then there is a polynomial $L_K \in \mathbb{Z}[u]$ of degree $2g$ such that
$$ \zeta_K (s) = \frac{L_K(q^{-s})}{(1-q^{-s})(1-q^{1-s})}$$
Is there a stronger relationship? Is the genus completely determined ?