Ax–Schanuel for the $j$-function

By Jonathan Pila, Jacob Tsimerman. 2014

72 citations recorded by the source corpus.

In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d48a187e54c3037a4e649e32ea9e4a3828cbb363

Citations in the corpus

• Applications of the hyperbolic Ax–Schanuel conjecture (2017, unknown): “njecture and the large Galois orbits conjecture. Subsequently, Pila and Tsimerman obtained the weak complex Ax conjecture as a corollary to their proof of the Ax-Schanuel conjecture for the jfunction [31]. Habegger and Pila had previously verified the large Galois orbits conjecture for so-called asym”
• Ax-Schanuel for Shimura varieties (2017, unknown): “ersections. For example, in the extreme case that W= Ω1×X1 where X1 is a weakly special subvariety of Xand Ω1 a connected component of its preimage in Ω, we get dimW= dimU+dimX1. As in earlier papers [38, 18, 33], the proof combines arguments from complex geometry (Hwang-To), the geometry/group theo”