Prym-Tyurin varieties via Hecke algebras

TL;DR

This paper constructs new Prym-Tyurin varieties from Galois covers of curves using Hecke algebra actions, providing explicit conditions for their existence and generating numerous examples of arbitrary exponent.

Contribution

It introduces a method to produce Prym-Tyurin varieties via Hecke algebra actions on Jacobians, expanding the known families with explicit criteria.

Findings

New families of Prym-Tyurin varieties of arbitrary exponent are constructed.

Sufficient conditions for a subvariety to be Prym-Tyurin are established.

The approach links group representations, Hecke algebras, and algebraic geometry.

Abstract

Let $G$ denote a finite group and $\pi: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash G/H]$ on the Jacobian of the curve $X = Z/H$. To each rational irreducible representation $\mathcal{W}$ of $G$ we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve $X$ and thus an abelian subvariety $P$ of the Jacobian $JX$. We give sufficient conditions on $\mathcal{W}$, $H$, and the action of $G$ on $Z$, which imply $P$ to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.