Prym-Tyurin varieties using self-products of groups

TL;DR

This paper introduces a method to construct Prym-Tyurin varieties of higher exponent using subgroup and representation data, exemplified with dihedral groups, and analyzes the Jacobian decompositions of related curves.

Contribution

It generalizes previous constructions of Prym-Tyurin varieties by incorporating subgroup and representation structures, extending to dihedral groups of any odd prime.

Findings

Constructed Prym-Tyurin varieties with exponent [G:H]q

Provided explicit example with dihedral group D_p for odd prime p

Computed isogeny decomposition of Jacobians in the example

Abstract

Given Prym-Tyurin varieties of exponent $q$ with respect to a finite group $G$, a subgroup $H$ and a set of rational irreducible representations of $G$ satisfying some additional properties, we construct a Prym-Tyurin variety of exponent $[G:H]q$ in a natural way. We study an example of this result, starting from the dihedral group $\mathbf{D}_p$ for any odd prime $p$. This generalizes the construction of arXiv:math/0412103v2[math.AG] for $p=3$. Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example.