A differential approach to Ax-Schanuel, I

TL;DR

This paper establishes new Ax-Schanuel type results for uniformizers of geometric structures, including derivatives, using differential algebra, geometry, and model theory, advancing understanding of differential relations in complex uniformization.

Contribution

It proves the full Ax-Schanuel theorem with derivatives for uniformizers of simple projective structures and extends results to Shimura varieties, combining multiple mathematical techniques.

Findings

Proved Ax-Schanuel with derivatives for simple projective structures.

Established Ax-Schanuel for derivatives of the j-function and exponential.

Extended Ax-Schanuel results to uniformizing maps of Shimura varieties.

Abstract

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curves including unifomizers of any Fuchsian group of the first kind and any genus. Combining our techniques with those of Ax, we give a strong Ax-Schanuel result for the combination of the derivatives of the j-function and the exponential function. In the general setting of Shimura varieties, we obtain an Ax-Schanuel theorem for the derivatives of uniformizing maps. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields.