Rationality fields of CM modular forms

We formulate a conjecture on the finitude of rationality fields (i.e., Fourier coefficient fields) of newforms of bounded degree, and prove this for CM forms assuming a generalized Riemann hypothesis. Then we explicitly determine what quadratic and cubic rationality fields occur for weight 2 CM forms, which is related to classifications of CM abelian varieties of GL(2) type. Several of these fields do not appear in the existing tables of newforms in the LMFDB.

Publication Details

Journal
ArXiv.org
Published
2025-09-28
Primary Topic
Global Energy Security and Policy
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article

Rationality fields of CM modular forms

ArXiv.org
Global Energy Security and Policy
article

Rationality fields of CM modular forms

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Abstract

We formulate a conjecture on the finitude of rationality fields (i.e., Fourier coefficient fields) of newforms of bounded degree, and prove this for CM forms assuming a generalized Riemann hypothesis. Then we explicitly determine what quadratic and cubic rationality fields occur for weight 2 CM forms, which is related to classifications of CM abelian varieties of GL(2) type. Several of these fields do not appear in the existing tables of newforms in the LMFDB.

ArXiv.org
Openalex Percentile: Top 33%
Global Energy Security and Policy
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Rationality fields of CM modular forms · ArXiv.org (2025) | TGRS Research Map | TGRS