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

2025/09/17 by Michael S. Wills, Wills, Michael
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.13641

openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-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×_K 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.

Citations

• Compatibility of weak approximation for zero-cycles on products of varieties
• The Brauer-Manin obstruction for zero-cycles on K3 surfaces
• Zero-cycles on a product of elliptic curves over a p-adic field
• Milnor K-groups attached to elliptic curves over a p-adic field
• On the cycle map for products of elliptic curves over a p-adic field
• Beyond the Manin obstruction

Related

• Compatibility of weak approximation for zero-cycles on products of varieties
• Zero cycles on products of elliptic curves over local fields with supersingular reduction
• Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields
• Local-global principle for 0-cycles on fibrations over rationally connected bases
• Local and local-to-global Principles for zero-cycles on geometrically Kummer K3 surfaces