Algebraic cycles on Jacobian varieties

Abstract

Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J, tensored with ℚ. We study in this paper the smallest ℚ-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J): intersection and Pontryagin products, pull back and push down under multiplication by integers. We prove that this 'tautological subring' is generated (over ℚ) by the classes of the subvarieties W1=C, W2=C + C,...,Wg-1. If C admits a morphism of degree d onto ℙ1, we prove that the last d - 1 classes suffice. © Foundation Compositio Mathematica 2004.