Extremal modular lattices, McKay Thompson series, quadratic iterations, and new series for $\pi$

Abstract

We give 20 new Ramanujan-type formulae and 20 quadratic approximations to {\small $\pi$}, parameterized by extremal modular lattices of minimal square norm 2 and 4, at the ten special levels corresponding to square-free orders of the Mathieu group {\small $M_{23}$}. An algorithm for uncovering rational relations between two given power series that we used to discover some of the explicit relations is also given. Explicit relations parameterized by modular lattices between McKay Thompson series for the group {\small $\Gamma_0(\ell)^+$} and {\small $\Gamma_0(2\ell)+p_j$}, where {\small $p_j$} range over all odd primes dividing the special level {\small $\ell$}, are uncovered.