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Ax-Schanuel for the Drinfeld j-function
Abstract
We prove an analog of the Ax-Schanuel theorem for the Drinfeld j-function in odd characteristic. Roughly speaking, if the graph of j:Ωn→𝔸nℂ∞ and its derivatives has an atypical intersection 𝒱 with an algebraic variety, then 𝒱 projects to a weakly-special subvariety in 𝔸nℂ∞. More generally, we prove a positive characteristic analog of the differential Galois theoretic Ax-Schanuel theorem of Blázquez-Sanz, Casale, Freitag and Nagloo. Our main theorem for j follows by applying this result to a suitable foliation. A theorem of Pink allows us to conclude that the special varieties in the sense of Blázquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.
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BibTeX
@misc{binyamini2026axschanuel,
title = {{Ax-Schanuel for the Drinfeld $j$-function}},
author = {Gal Binyamini and Dmitry Novikov and Francesco Maria Saettone},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37595},
eprint = {2609.37595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37595},
}
RIS
TY - UNPB
TI - Ax-Schanuel for the Drinfeld j-function
AU - Gal Binyamini
AU - Dmitry Novikov
AU - Francesco Maria Saettone
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37595
UR - https://arxiv.org/abs/2609.37595
AB - We prove an analog of the Ax-Schanuel theorem for the Drinfeld j-function in odd characteristic. Roughly speaking, if the graph of j:Ω^n→𝔸^n_ℂ_∞ and its derivatives has an atypical intersection 𝒱 with an algebraic variety, then 𝒱 projects to a weakly-special subvariety in 𝔸^n_ℂ_∞. More generally, we prove a positive characteristic analog of the differential Galois theoretic Ax-Schanuel theorem of Blázquez-Sanz, Casale, Freitag and Nagloo. Our main theorem for j follows by applying this result to a suitable foliation. A theorem of Pink allows us to conclude that the special varieties in the sense of Blázquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.
ER -