Ramanujan and the Modular j-Invariant

Abstract

AbstractA new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between themodular jinvariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of . This shows that tn is a new class invariant according to H. Weber’s definition of class invariants.