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Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis 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 forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$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$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$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. 

Abstract We consider a product $$X=E_1\times \cdots \times E_d$$ of elliptic curves over a finite extension $$K$$ of $${\mathbb{Q}}_p$$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $$X$$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $$CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$$. We give specific criteria that guarantee this map is injective for every $$n\geq 1$$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $$L$$ of $$K$$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi.