Modular curves and their pseudo-analytic cover

TL;DR

This paper develops a logical framework for the structure of the upper half-plane as a covering space of modular curves, proving its uniqueness across uncountable models using complex multiplication and Shimura varieties.

Contribution

It introduces a new $L_{,}$-axiomatisation of the modular curves' covering space with a uniqueness theorem in uncountable models, extending previous work to a $$-rational setting.

Findings

Established a unique model of the structure in every uncountable cardinal.

Connected the logical framework with deep results in complex multiplication and Shimura varieties.

Extended prior work by removing the need to name CM-points, working over $$.

Abstract

We find a natural $L_{\omega_1,\omega}$-axiomatisation $\Sigma$ of a structure on the upper half-plane $\mathbb{H}$ as the covering space of modular curves. The main theorem states that $\Sigma$ has a unique model in every uncountable cardinal. The proof relies heavily on the theory of complex multiplication and the work on Langland's conjecture on the conjugation of Shimura varieties. We also use the earlier work on a related problem by C.Daw and A.Harris. The essential difference between the setting of this work and that of the current paper is that the former was in the language which named the CM-points of the modular curves while our results here are over $\mathbb{Q}.$