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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2509.13641 |
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| _version_ | 1866908543606587392 |
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| author | Wills, Michael |
| author_facet | Wills, Michael |
| contents | 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-Thélène 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Local-to-Global Principle for Zero-Cycles on Self Products of Elliptic Curves with CM Wills, Michael Algebraic Geometry 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-Thélène 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. |
| title | On the Local-to-Global Principle for Zero-Cycles on Self Products of Elliptic Curves with CM |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2509.13641 |