Decomposing Jacobians via Galois covers

Abstract

Let $\varphi :X\to Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $\varphi$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $\varphi$ with a well-chosen map $Y\to {\mathbb{P}}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genus $1$ and $\varphi$ is a degree $3$ map totally ramified over $2$ points.