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 $\boldsymbol{j}\colon\Omega^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ 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 Bl\'azquez-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 Bl\'azquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.