questions, which we collected in \[quest:opology1\]. The paper also ends in this spirit: with a conjecture (\[conj:liegeneration\]) and a question (\[quest:arithmetic2\]).

We thank Ben Moonen for drawing our attention to a paper by Masa-Hiko Saito and we thank Tyakal Venkataramana for helpful consultations.

*Conventions*. If a group $G$ acts properly discretely on a smooth object $S$ in a category where this makes sense, then $S_G$ denotes the associated orbifold object.

When $G$ is a finite group and $k$ is a field of characteristic zero, then we denote by $X(k G)$ the set of irreducible characters of $k G$. For every $\chi\in X(k G)$, we let $V_\chi $ be a finitely generated (irreducible) $k G$-module with that character. Is $H$ is a $k G$-module of finite rank, then we write $H[\chi]$ for the $V_\chi$ isogeny space ${\operatorname{Hom}}_{kG}(V_\chi,H)$. The *$\chi$-isotypical summand* of $H$ is the image of the evaluation map $V_\chi\otimes_k H[\chi]\to H$ and denoted by $H_\chi$. We indeed have $H=\oplus_{\chi\in X(k G)} H_\chi$.

We write ${{\mathbb K}}$ for the Hamilton quaternions, considered as a division ring containing ${{\mathbb R}}$.

Endomorphisms of Jacobians of $G$-curves {#endomorphisms}
========================================

A central role in this paper is played by a moduli stack whose definition we 