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The mean number of 3-torsion elements in ray class groups of quadratic fields

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Abstract

We determine the average number of 3-torsion elements in the ray class groups of a fixed (integral) conductor c of quadratic fields ordered by their absolute discriminant, generalizing Davenport and Heilbronn’s theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor c is taken to be a squarefree integer having very few prime factors none of which are congruent to 1 mod 3. Additionally, we compute the second main term for the number of 3-torsion elements in ray class groups with a fixed conductor of quadratic fields ordered by their discriminant.

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Acknowledgements

The author would like to thank Manjul Bhargava for suggesting this problem and answering many questions. She would also like to thank Djordjo Milovic, Carlo Pagano, Arul Shankar, and Jacob Tsimerman for helpful discussions. The author was supported by a National Defense Science & Engineering Fellowship and NSF Grant DMS-1502834.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Ila Varma

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  1. Ila Varma
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Varma, I. The mean number of 3-torsion elements in ray class groups of quadratic fields. Isr. J. Math. 252, 149–185 (2022). https://doi.org/10.1007/s11856-022-2346-y

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  • DOI: https://doi.org/10.1007/s11856-022-2346-y

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