Open Access

On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono

Contributed equally
Research in Number Theory20151:4

DOI: 10.1007/s40993-015-0007-5

Received: 17 January 2015

Accepted: 3 February 2015

Published: 30 November 2015

Abstract

Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this paper we prove that the rational polynomial having these singular moduli as zeros is (essentially) irreducible, settling a question of Bruinier and Ono. The proof uses careful analytic estimates together with some related work of Dewar and Murty, as well as extensive numerical calculations of Sutherland.