Categoricity of the two sorted j-function

TL;DR

This paper proves that a specific two-sorted theory involving the modular j-function is categorical in all uncountable sizes, linking it to deep conjectures in arithmetic geometry.

Contribution

It establishes the categoricity of a natural two-sorted theory with the j-function and relates it to the adelic Mumford-Tate conjecture for elliptic curves.

Findings

The theory is categorical in all uncountable cardinals.

A weakened Mumford-Tate conjecture is necessary for categoricity.

Connections between model theory and arithmetic geometry are demonstrated.

Abstract

We show that a natural, two sorted $\cL_{\omega_1,\omega}$ theory involving the modular $j$-function is categorical in all uncountable cardinaities. It is also shown that a slight weakening of the adelic Mumford-Tate conjecture for products of elliptic curves is necessary and (along with a couple of other results from arithmetic geometry) sufficient for categoricity.