Hirzebruch Functional Equation and Elliptic Functions of Level $d$

A function $f(x)$ of a complex variable $x$ regular in a neighborhood of $x=0$ and such that $f(0)=0$ and $f'(0)=1$ is said to be $n$-rigid if the sum of residues of the function $\prod_{i=0}^n1/f(x-x_i)$ does not depend on the choice of different points $x_0,\dots,x_n$ in a small neighborhood of $x=0$. The power series expansion of an $n$-rigid function is determined by a functional equation. We refer to this equation as the Hirzebruch $n$-equation. If $d$ is a divisor of $n+1$, then any elliptic function of level $d$ is $n$-rigid. A description of the manifold of all $2$-rigid functions has been obtained very recently. The main result of this work is a description of the manifold of all $3$-rigid functions.