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

TL;DR

This paper proves the irreducibility of a polynomial related to singular moduli of a non-holomorphic modular function, addressing a question posed by Bruinier and Ono, using analytic estimates and numerical calculations.

Contribution

It establishes the irreducibility of a key polynomial linked to singular moduli, advancing understanding of non-holomorphic modular functions.

Findings

The polynomial with singular moduli as zeros is essentially irreducible.

Analytic estimates support the irreducibility proof.

Numerical calculations corroborate theoretical results.

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 singuar 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.