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arXiv · 2611.07259

Fields Generated by a Single Point on Elliptic Curves

Minseok Kim·Myungjun Yu
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Abstract

We study the finite extensions of a number field generated by a single point on an elliptic curve. We construct geometrically non-isomorphic elliptic curves with complex multiplication that generate the same collection of extensions, giving a negative answer to a question of Mazur and Rubin. For torsion-generated fields, we prove that inclusion of the collections up to finitely many exceptions implies isogeny over the ground field, and in the non-CM case even isomorphism. When all geometric endomorphisms are defined over the ground field, we characterize equality of the torsion-generated collections by the existence, for every prime $p$, of an isogeny over the ground field of degree prime to $p$. This condition also implies equality of the full collections of fields generated by a single point.

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Minseok Kim, Myungjun Yu. 2026-11-05. Fields Generated by a Single Point on Elliptic Curves. https://arxiv.org/abs/2611.07259

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