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Modular invariance, modular identities and supersingular

To every k-dimensional modular invariant vector space we associate a modular form on SL(2, Z) of weight 2k. We explore number theoretic properties of this form and found 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 level one modules for the affine Kac-Moody Lie algebra of type D (1) 4 . In this case the reduction modulo prime p = 2k + 3 ≥ 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.