Year: 2026
Type: preprint
Abstract: We prove an analog of the Ax-Schanuel theorem for the Drinfeld $j$-function in odd characteristic. Roughly speaking, if the graph of $\boldsymbol{j}\colonΩ^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ and its derivatives has an atypical intersection $\mathcal{V}$ with an algebraic variety, then $\mathcal{V}$ projects to a weakly-special subvariety in $\mathbb{A}^n_{\mathbb{C}_\infty}$. 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 $\boldsymbol{j}$ 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.
Source: arXiv (Cornell University)
Institution: Weizmann Institute of Science
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214987241