For a smooth projective variety
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Abstract X over an algebraic number fieldk a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofX is a torsion group. In this article we consider a product of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$X . For a product of two curves over$$X=C_1\times C_2$$ $X={C}_{1}\times {C}_{2}$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$\mathbb {Q} $$ $Q$ is finite, where$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ ${J}_{1}\left(Q\right)\otimes {J}_{2}\left(Q\right)\stackrel{\epsilon}{\to}{\phantom{\rule{0ex}{0ex}}\text{CH}\phantom{\rule{0ex}{0ex}}}_{0}({C}_{1}\times {C}_{2})$ is the Jacobian variety of$$J_i$$ ${J}_{i}$ . Our constructions include many new examples of nonisogenous pairs of elliptic curves$$C_i$$ ${C}_{i}$ 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$$E_1, E_2$$ ${E}_{1},{E}_{2}$ for which the analogous map$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$ has finite image.$$\varepsilon $$ $\epsilon $ 
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.more » « less