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CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem
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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BibTeX
@misc{geng2026fibers,
title = {{CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem}},
author = {Jingxuan Geng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16953},
eprint = {2610.16953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16953},
}
RIS
TY - UNPB
TI - CM fibers of Shimura parametrizations and thirteen proofs of the class number one theorem
AU - Jingxuan Geng
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16953
UR - https://arxiv.org/abs/2610.16953
AB - 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.
ER -