on $\phi$, so that this character may be regarded as a topological invariant. We are therefore justified in denoting it by $\chi^\phi$. It is clear that for every $\rho\in {\operatorname{Aut}}(G)$, $\chi^{\phi\sigma}=\rho^*\chi^{\phi}$.

Let us say that a geometric point $\{x\}\to {{\mathscr M}}^\phi$ is a *very general point* of ${{\mathscr M}}^\phi$ if it is not contained in a given countable union of closed proper substacks of ${{\mathscr M}}^\phi$.

\[endo\]Let $G$ be a finite group and $\phi : G\to {\operatorname{Mod}}(S)$ a monomorphism such that the centralizer of the image of $\phi$ does not contain a hyperelliptic involution. Assume that the associated moduli stack ${{\mathscr M}}^\phi$ has positive dimension and that for a $G$-curve $C$ representing a point of ${{\mathscr M}}^\phi$ the action of $G$ on $H^0(C, \Omega_C)$ satisfies the following two properties:

1.  every irreducible symplectic representation occurring in $H^0(C, \Omega_C)$ occurs with multiplicity $\ge 3$ and

2.  every non-self-dual representation occurring in $H^0(C, \Omega_C)$ occurs along with its dual and both do so with multiplicity $\ge 2$.

Then the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is surjective, provided the $G$-curve $C$ represents a very general member of ${{\mathscr M}}^\phi$.

\[rem:verygeneral=dominant\] We can avoid the notion of a very general point by using the 