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

Abstract

AbstractFor 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 $$X=C_1\times \cdots \times C_d$$ 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 $$X=C_1\times C_2$$ X = C 1 × C 2 of two curves over $$\mathbb {Q} $$ Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map $$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ J 1 ( Q ) ⊗ J 2 ( Q ) → ε CH 0 ( C 1 × C 2 ) is finite, where $$J_i$$ J i is the Jacobian variety of $$C_i$$ C i . Our constructions include many new examples of non-isogenous pairs of elliptic curves $$E_1, E_2$$ 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 $$X=C_1\times \cdots \times C_d$$ X = C 1 × ⋯ × C d for which the analogous map $$\varepsilon $$ ε has finite image.