CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem

Jingxuan Geng

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arXiv · 2610.16953

CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem

Jingxuan Geng

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Abstract

We determine thirteen entirely CM identity fibers of Shimura parametrizations attached to elliptic curves of rank two and conductor below $2000$. Each yields a proof of the imaginary quadratic class number one theorem. The argument combines the Gross--Zagier formula with the Ribet--Takahashi degree comparison to show that the fiber is completely explained by known CM points.

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Jingxuan Geng. 2026-10-12. CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem. https://arxiv.org/abs/2610.16953

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