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

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 \(X=C_1\times \cdots \times C_d\) $X=C_1\times \cdots \times C_d$ 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 \(X=C_1\times C_2\) $X=C_1\times C_2$ of two curves over \(\mathbb {Q} \) $\mathbb{Q}$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map \(J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)\) $J_1(\mathbb{Q})\otimes J_2(\mathbb{Q})\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$ 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 \(X=C_1\times \cdots \times C_d\) $X=C_1\times \cdots \times C_d$ for which the analogous map \(\varepsilon \) $\epsilon$ has finite image.