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The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves
Abstract
We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on o-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.
Related
- Coset representatives corresponding to Yetter-Drinfeld modules of modular group and continued fraction
- Rational Points on Drinfeld Modular Curves of Prime Level
- Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
- Skew cyclic codes through Drinfeld modules
- Rank-Two Drinfeld Modules over Elliptic Coordinate Rings
BibTeX
@misc{binyamini2026pilazannier,
title = {{The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves}},
author = {Gal Binyamini and Dmitry Novikov and Francesco Maria Saettone},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.19952},
eprint = {2607.19952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19952},
}
RIS
TY - UNPB
TI - The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves
AU - Gal Binyamini
AU - Dmitry Novikov
AU - Francesco Maria Saettone
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2607.19952
UR - https://arxiv.org/abs/2607.19952
AB - We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on o-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.
ER -