Hirzebruch functional equation and elliptic functions of level d

Abstract

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 Пi=0n 1/f(x−xi) does not depend on the choice of different points x0,..., xn 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 variety of all 2-rigid functions has been obtained very recently. The main result of this work is a description of the variety of all 3-rigid functions.