Modular invariance, modular identities and supersingular j-invariants

Abstract

To every $k$-dimensional modular invariant vector space we associate a modular form on $SL(2,\mathbb{Z})$ of weight $2k$. We explore number theoretic properties of this form and find a sufficient condition for its vanishing which yields modular identities (e.g., Ramanujan-Watson's modular identities). Furthermore, we focus on a family of modular invariant spaces coming from suitable two-dimensional spaces via the symmetric power construction. In particular, we consider a two-dimensional space spanned by graded dimensions of certain level one modules for the affine Kac-Moody Lie algebra of type $D_4^{(1)}$. In this case, the reduction modulo prime $p=2k+3 \geq 5$ of the modular form associated to the $k$-th symmetric power classifies supersingular elliptic curves in characteristic $p$. This construction also gives a new interpretation of certain modular forms studied by Kaneko and Zagier.