The Torelli theorem holds for curves over an arbitrary ground field $k$ (in particular, $k$ need not be perfect). A very nice treatment of the "strong" Torelli theorem may be found in the appendix by J.-P. Serre to Kristin Lauter's 2001 Journal of Algebraic Geometry paper *Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields*. It is available on the arxiv:

http://arxiv.org/abs/math/0104247

Here are the statements (translated into English):

Let $k$ be a field, and let $X_{/k}$ be a nice (= smooth, projective and geometrically integral) curve over $k$ of genus $g > 1$. Let $(\operatorname{Jac}(X),\theta_X)$ denote the Jacobian of $X$ together with its canonical principal polarization. Let $X'_{/k}$ be another nice curve.

Theorem 1: Suppose $X$ is hyperelliptic. Then for every isomorphism of polarized abelian varieties $(\operatorname{Jax}(X),\theta_X) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),\theta_{X'})$, there exists a *unique* isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ such that $F = \operatorname{Jac} f$.

Theorem 2: Suppose $X$ is not hyperelliptic. Then, for every isomorphism $F: (\operatorname{Jax}(X),\theta_X) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),\theta_{X'})$ there exists an isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ and $e \in \{ \pm 1\}$ such that $F = e \cdot \operatorname{Jac} f$. Moreover, the pair $(f,e)$ is uniquely determined by $F$.