MODULAR EQUATIONS OF ORDER p AND THETA FUNCTIONS

Abstract

Let $p$ be a prime integer and ${\mathbf{H}}_{\mathbf{g}}$ be a collection of complex positive definite symmetric $g\times g$ matrices $\tau$. Denote by $p\tau$ the multiplication of $\tau$ by $p$. In this note we describe an explicit process to obtain algebraic identities between theta functions with integral characteristics evaluated at $\tau$ and $p\tau$. For $g=1$ this produces modular equations between $\lambda \left(\tau \right)$, $\lambda \left(p\tau \right)$ where $\lambda \left(\tau \right)$ is the invariant associated with elliptic curve generated by $\tau$, described by the equation: $y^2=x(x-1)(x-\lambda ({\tau }_1\left)\right)$. Consequently, if the algebraic identities we obtain might serve as a higher dimensional generalization for the one dimensional modular equations.