Submission date: 13 March 2018

Abstract:

We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety G which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. In the spirit of the Zilber-Pink conjecture and unlikely intersections, we show that an irreducible curve W in G meets this set Zariski-densely only if W lies in a fiber of the family or is a translate of a Ribet curve by a multiplicative section.

Mathematics Subject Classification: 14K15 (Primary), 11G15 (Secondary)

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Full text arXiv 1803.04835: pdf, ps.