Hirzebruch sums and Hecke operators
Let p be a rational prime congruent to 3 modulo 4. \textit{F. Hirzebruch} [Hilbert Modular Surfaces, Enseign. Math., II. Ser. 19, 183-281 (1973; Zbl 0285.14007)] showed that, under certain conditions, the class number of \({\mathbb{Q}}(\sqrt{-p})\) is related to a sum over the partial quotients of the continued fraction of \(\sqrt{p}\). This sum, defined in general for a real quadratic irrational and dubbed the Hirzebruch sum, is the subject of this paper. The author shows (Theorem 7) that this sum satisfies a simple, but notationally cumbersome, identity which on one side resembles a weighted form of the action of the Hecke operators from modular form theory. In those cases involving irrational numbers from fields of discriminant p or 4p, p prime, this identity reduces to the result that this Hirzebruch sum is an eigenfunction for the Hecke operators themselves. The proof of the identity itself depends on an earlier result of the author on the transformation formula for the Dedekind \(\eta\)- function [Acta Math. Sci. 8, 249-262 (1988; Zbl 0665.10016)] and an identity of \textit{M. Knopp} involving Dedekind sums [J. Number Theory 12, 2-9 (1980; Zbl 0423.10015)].
scientific article; zbMATH DE number 4214119