Rationality fields of CM modular forms
Number Theory
2025-09-30 v1
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.
Keywords
Cite
@article{arxiv.2509.24119,
title = {Rationality fields of CM modular forms},
author = {Kimball Martin},
journal= {arXiv preprint arXiv:2509.24119},
year = {2025}
}
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17 pages