Jacobian varieties with group algebra decomposition not affordable by Prym varieties

Abstract

The action of a finite group $G$ on a compact Riemann surface $X$ naturally induces another action of $G$ on its Jacobian variety $\operatorname{J}(X)$. In many cases, each component of the group algebra decomposition of $\operatorname{J}(X)$ is isogenous to a Prym varieties of an intermediate covering of the Galois covering $\pi_G\colon X \to X/G$; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of $\operatorname{J}(X)$ is not affordable by Prym varieties; namely, affine groups $\operatorname{Aff}(\mathbb{F}_q)$ with some exceptions: $q = 2$, $q = 9$, $q$ a Fermat prime, $q = 2^n$ with $2^n-1$ a Mersenne prime and some particular cases when $X/G$ has genus $0$ or $1$. In each one of this exceptional cases, we give the group algebra decomposition of $\operatorname{J}(X)$ by Prym varieties.