following more concrete (and equivalent) formulation of the conclusion of this theorem: if ${{\mathscr C}}/Y$ is a $G$-curve over a complex variety $Y$ of type $\phi$ such that the associated morphism $Y\to{{\mathscr M}}^\phi$ is dominant, then ${{\mathbb Q}}G\to {\operatorname{End}}_{{\mathbb Q}}(J({{\mathscr C}}/Y))$ is onto. This is essentially what we shall prove and it is also the formulation through which we will use it.

\[notinjective\]If the genus of the quotient curve is zero, then the natural homomorphism ${{\mathbb Q}}G\to{\operatorname{End}}_{{\mathbb Q}}(J(C))$ is not injective, as the trivial $G$-module is not represented in $H^0(C,\Omega_C)$. But as we will see, the Chevalley-Weil formula implies that this map is injective when the genus of $C_G$ is at least $2$. This formula will also show that the conditions we imposed on the $G$-action on $H^0(C, \Omega_C)$ are topological, in the sense that they only depend on how $G$ acts on $\phi$ (as we should expect).

The assumption that ${{\mathscr M}}^\phi$ has positive dimension translates into asking that the moduli space of the quotient orbifold curve (so endowed with the finite subset representing irregular $G$-orbits) is not rigid. This simply amounts to demanding that the regular $G$-orbit space of $C$ has negative Euler characteristic, but is not a 