PreprintCC BY 4.0arXiv
Finiteness of an Isogeny Class via Equidistribution
Abstract
In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings' isogeny theorem over number fields by reducing to Tate's theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.
Related
- Faltings' Isogeny Theorem via Equidistribution
- Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
- Recovering curves from isogeny classes of Prym varieties
- Isogeny rigidity under algebraic group actions and finite translation orbits
- Models of Elliptic Curves with Maximal 2-adic Image Subject to a Rational 2-Isogeny
BibTeX
@misc{yap2026finiteness,
title = {{Finiteness of an Isogeny Class via Equidistribution}},
author = {Jit Wu Yap},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13735},
eprint = {2609.13735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13735},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finiteness of an Isogeny Class via Equidistribution
AU - Jit Wu Yap
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13735
UR - https://arxiv.org/abs/2609.13735
AB - In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings' isogeny theorem over number fields by reducing to Tate's theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.
ER -