Modular curves and their pseudoanalytic cover
Abstract
We find a natural $L_{\omega_1,\omega}$axiomatisation $\Sigma$ of a structure on the upper halfplane $\mathbb{H}$ 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 CMpoints of the modular curves while our results here are over $\mathbb{Q}.$
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 DOI:
 10.48550/arXiv.2107.11110
 arXiv:
 arXiv:2107.11110
 Bibcode:
 2021arXiv210711110Z
 Keywords:

 Mathematics  Logic;
 03C45;
 11G18