On the Local-to-Global Principle for Zero-Cycles on Self Products of Elliptic Curves with CM

Abstract

For a smooth projective variety $X$ defined over a global field $K$, one can form a notion of Weak Approximation for the Chow group of zero-cycles of $X$. There exists a Brauer-Manin obstruction to Weak Approximation here akin to that for rational points. However, unlike for rational points, it is conjectured that this obstruction is the only one; early versions of this conjecture date back to work of Colliot-Th\'el\`ene and Sansuc (1981) and of Kato and Saito (1986). In this paper, we provide evidence for this when $X$ is the self-product of an elliptic curve with complex multiplication. For some varieties of this form, we construct infinitely many extensions $L/K$ for which the base change $X\times_K \mathrm{Spec}\ L$ satisfies a local-to-global principle for a fixed prime $p$. We do this via explicitly constructing global zero-cycles, and our results have applications over all but two of the complex multiplication fields.