I'm currently trying to familiarize myself with reducible Jacobians of hyperelliptic curves. A construction which I recently saw in a paper was the quotient $C/\tau$ of a curve $C$ of genus $2$ by a non-hyperelliptic involution $\tau$.
I can get the general idea, but the details of the construction and the following proofs escape me. Almost all of the literature seems to assume a familiarity with properties of such involutions and quotients, but they don't show up at all in the books I read on elliptic curves, abelian varieties and functions fields (there is, as far as I know, very few explicit introductory literature on hyperelliptic curves).
Can someone recommend a good review or some book source which devotes not only two lines to involutions of curves? The genus does not have to be $2$, in fact a more general exposition would be preferable.
Thank you in advance, I would appreciate any guidance and hints.