  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 $\mathbb{Q}.$
