On Functions Satisfying Modular Equations for Infinitely Many Primes

http://dx.doi.org/10.4153/CJM-1999-045-x
Canad. J. Math. 51(1999), 1020-1034

  Published:1999-10-01
 Printed: Oct 1999

In this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are: \begin{enumerate} \item[1)] every such function is analytic in the upper half plane; \item[2)] if such function takes the same value in two different points $z_1$ and $z_2$ then there exists an $f$-preserving analytic bijection between neighbourhoods of $z_1$ and $z_2$. \end{enumerate}