Prym-Tyurin varieties via Hecke algebras

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.

24 pages. Accepted in J. Reine Angew. Math. Minor changes

Cited by in corpus (2)

• Prym-Tyurin varieties using self-products of groups
• Products of Jacobians as Prym-Tyurin varieties