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Counting modular forms by rationality field

Alex Cowan, Kimball Martin

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.

Counting modular forms by 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 rationality fields occur for squarefree levels, and predict finiteness if . When , we make predictions about how frequently specific quadratic fields occur, prove lower bounds, and conjecture that is the most common quadratic rationality field.
Paper Structure (14 sections, 5 theorems, 10 equations, 3 figures, 6 tables)

This paper contains 14 sections, 5 theorems, 10 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Counts by degree
  3. Random Hecke polynomial model
  4. Assessment of the model
  5. Finiteness questions
  6. Hecke fields
  7. Modular varieties
  8. Rational points on Hilbert modular surfaces
  9. Lower bounds for quadratic fields
  10. Remarks for higher degree
  11. Data
  12. Data for counts by degree
  13. Data counts by quadratic field
  14. Data counts for cubic fields

Key Result

Proposition 1.4

The number of quadratic twist classes of weight $2$ newforms with rationality field $\mathbb{Q}(\sqrt 5)$ (resp. $\mathbb{Q}(\sqrt 2)$) and minimal level $N < X$ is $\gg X^{1/3}$ (resp. $\gg X^{2/7}$).

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (14)

  • Conjecture 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Proposition 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.5
  • ...and 4 more