On the local-to-global principle for zero-cycles on products of CM elliptic curves 41 views

Let X be a nice algebraic variety over a ``global'' number field K, and suppose we wish to understand X(K), the set of K-rational points of X. As a first step, we may try to understand the rational points over all ``local'' completions of K, where the robust topological structure facilitates computations. If the geometry of X is sufficiently simple, this can provide extensive information about X(K), a phenomenon known as a local-to-global principle. However, this approach generally fails. The reciprocity laws on K can block the transfer of solutions via the Brauer--Manin obstruction, and a full list of further reasons such principles may fail remains elusive. As an alternative, we may ``linearize" the points of X, giving the Chow group of zero-cycles. While the Brauer--Manin obstruction still plays a role here, conjectures of Colliot-Thelene--Sansuc and Kato--Saito predict that this is the only obstruction to several local-to-global principles. Unfortunately, in dimension at least 2 these conjectures are widely open. The purpose of this thesis is to offer some evidence for their truth in the case that X is given as a product of elliptic curves with complex multiplication (CM). We do this by developing computational methods for better understanding zero-cycles defined over local fields on these varieties, then applying these methods to construct the required zero-cycles globally. Using this approach, we are able to establish unconditional results supporting these conjectures for infinite families of varieties and seven of the nine CM discriminants of class number 1.

PHD (Doctor of Philosophy)

Arithmetic geometry; Local-to-global principles; Elliptic curves; Algebraic cycles

Corrected title per LS-30378

English

2026-04-30