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 $J\left(X\right)$. In many cases, each component of the group algebra decomposition of $J\left(X\right)$ is isogenous to a Prym varieties of an intermediate covering of the Galois covering ${\pi }_G:X\to X/G$; 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 $J\left(X\right)$ is not affordable by Prym varieties; namely, affine groups $\mathrm{Aff}\left(F_q\right)$ 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 $J\left(X\right)$ by Prym varieties.