AN EXTENSION OF ROHRLICH’S THEOREM TO THE $j$-FUNCTION

Abstract

We start by recalling the following theorem of Rohrlich [17]. To state it, let $\unicode[STIX]{x1D714}_{\mathfrak{z}}$ denote half of the size of the stabilizer $\unicode[STIX]{x1D6E4}_{\mathfrak{z}}$ of $\mathfrak{z}\in \mathbb{H}$ in $\text{SL}_{2}(\mathbb{Z})$ and for a meromorphic function $f:\mathbb{H}\rightarrow \mathbb{C}$ let $\text{ord}_{\mathfrak{z}}(f)$ be the order of vanishing of $f$ at $\mathfrak{z}$. Moreover, define $\unicode[STIX]{x1D6E5}(z):=q\prod _{n\geqslant 1}(1-q^{n})^{24}$, where $q:=e^{2\unicode[STIX]{x1D70B}iz}$, and set $\unicode[STIX]{x1D55B}(z):=\frac{1}{6}\log (y^{6}|\unicode[STIX]{x1D6E5}(z)|)+1$, where $z=x+iy$. Rohrlich’s theorem may be stated in terms of the Petersson inner product, denoted by $\langle ~,\,\rangle$.