Prym-Tyurin varieties via Hecke algebras

Abstract

Let G denote a finite group and π : Z → 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 ℚ[H\G/H] on the Jacobian of the curve X = Z/H. To each rational irreducible representation 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 𝒲, H, and the action of G on Z for P to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.