On some cubic modular identities

Abstract:

Let $4K$ and $2iK’$ be the periods of $\operatorname {sn} z$. By evaluating the functional equations ${\operatorname {sn} ^2}z + {\operatorname {cn} ^2}z = 1$ and ${k^2}{\operatorname {sn} ^2}z + {\operatorname {dn} ^2}z = 1$ at $z = iK’/3$, we deduce a set of cubic modular identities from which the familiar modular equation of degree $3$ follows directly as a corollary.