PreprintCC BY 4.0arXiv
Faltings' Isogeny Theorem via Equidistribution
Abstract
We give a new proof of Faltings' isogeny theorem. More precisely, we show that Yuan's non-archimedean equidistribution theorem can be used to "pump" homomorphisms and semisimplicity from finite fields to number fields, thereby reducing Faltings' theorem directly to Tate's theorem. In the appendix joint with Yap, we explain how an ultraproduct variant of our method gives another proof of the Shafarevich conjecture, and hence a new route to the Mordell conjecture via Parshin's trick.
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BibTeX
@misc{xie2026faltings,
title = {{Faltings' Isogeny Theorem via Equidistribution}},
author = {Junyi Xie and Ziquan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10302},
eprint = {2609.10302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10302},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Faltings' Isogeny Theorem via Equidistribution
AU - Junyi Xie
AU - Ziquan Yang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.10302
UR - https://arxiv.org/abs/2609.10302
AB - We give a new proof of Faltings' isogeny theorem. More precisely, we show that Yuan's non-archimedean equidistribution theorem can be used to "pump" homomorphisms and semisimplicity from finite fields to number fields, thereby reducing Faltings' theorem directly to Tate's theorem. In the appendix joint with Yap, we explain how an ultraproduct variant of our method gives another proof of the Shafarevich conjecture, and hence a new route to the Mordell conjecture via Parshin's trick.
ER -