A Note on the Degree of Field Extensions Involving Classical and Nonholomorphic Singular Moduli

TL;DR

This paper establishes bounds on the degree of field extensions generated by almost holomorphic modular functions at quadratic points, using o-minimality techniques, extending known results for classical modular functions.

Contribution

It provides new bounds on the degree of fields generated by a broad class of level 1 almost holomorphic modular functions at quadratic points, generalizing prior results.

Findings

Bounds relate [Q(f(τ)):Q] to [Q(j(τ)):Q] with a constant M

Techniques of o-minimality are used to prove the bounds

Results are uniform and depend only on the degree of f

Abstract

In their 2015 paper, Mertens and Rolen prove that for a certain level 6 "almost holomorphic" modular function $P$, the degree of $P(\tau)$ over $\mathbb{Q}$ for quadratic $\tau$ is as large as expected, settling a conjecture of Bruinier and Ono. Analogously for level 1 modular functions $f$, we expect $\mathbb{Q}(f(\tau))$ to have similar degree to $\mathbb{Q}(j(\tau))$. In this paper, I show for a wide class of level 1 almost holomorphic modular functions that \[\dfrac{1}{M}[\mathbb{Q}(j(\tau)):\mathbb{Q}]\leq [\mathbb{Q}(f(\tau)):\mathbb{Q}]\leq[\mathbb{Q}(j(\tau)):\mathbb{Q}]\] for all quadratic $\tau$ and some constant $M$. This is proven using techniques of o-minimality, and hence can easily be made uniform; the constant $M$ depends only upon the "degree" of $f$ (in a certain well-defined sense).