Model Complete Expansions of the Real Field by Modular Functions and Forms

We prove a strong form of model completenes for expansions of the field of real numbers by (the real and imaginary parts of) the modular function J, by the modular forms $E_4$ and $E_6$ and quasimodular form $E_2$ defined in the usual fundamental domain, and the restricted sine function and the (unrestricted) exponential function. This is done using ideas of Peterzil and Starchenko’s paper \cite{peterzil-starchenko-wp2004} on the uniform definability of $\mathrm{\wp }$ function in ${\mathbb{R}}_{an}$ (and of the modular function $J$). In the conclusion we pose some open problems related to this work.