Counting modular forms by rationality field

TL;DR

This paper studies the distribution of degrees and rationality fields of weight 2 newforms, providing heuristic bounds, predictions, and conjectures about their frequency and finiteness for various degrees and specific quadratic fields.

Contribution

It offers new heuristic bounds, predictions, and conjectures on the distribution and frequency of rationality fields of weight 2 newforms, especially for degrees up to 7.

Findings

Heuristic upper bounds on the occurrence of degree d rationality fields for squarefree levels.

Predictions of finiteness of such fields when d ≥ 7.

Conjecture that Q(√5) is the most common quadratic rationality field.

Abstract

We investigate the distribution of degrees and rationality fields of weight 2 newforms. In particular, we give heuristic upper bounds on how often degree $d$ rationality fields occur for squarefree levels, and predict finiteness if $d \ge 7$. When $d=2$, we make predictions about how frequently specific quadratic fields occur, prove lower bounds, and conjecture that $\mathbb{Q}(\sqrt 5)$ is the most common quadratic rationality field.