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Counting modular forms by rationality field

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arxiv 2301.10357 v2 pith:345SOEIP submitted 2023-01-25 math.NT

classification math.NT
keywords rationalityfieldsboundsfieldoccurquadraticcommonconjecture
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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.

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Cited by 1 Pith paper

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  1. Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

    math.NT 2025-05 conditional novelty 7.0 of 10

    Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.

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