Torsion phenomena for zero-cycles on a product of curves over a number field

Abstract For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product X=C₁ Cd X = C 1 × ⋯ × C d of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product X=C₁ C₂ X = C 1 × C 2 of two curves over Q Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map J₁ (Q) J₂ (Q) {\, CH\, }₀ (C₁ C₂) J 1 (Q) ⊗ J 2 (Q) → ε CH 0 (C 1 × C 2) is finite, where Jᵢ J i is the Jacobian variety of Cᵢ C i. Our constructions include many new examples of non-isogenous pairs of elliptic curves E₁, E₂ E 1, E 2 with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products X=C₁ Cd X = C 1 × ⋯ × C d for which the analogous map ε has finite image.