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Class number divisibility for imaginary quadratic fields

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Abstract

In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let \(A,B,g \ge 3\) be positive integers such that \(\gcd (A,B)\) is square-free. We refine Soundararajan’s result to show that if \(4 \not \mid g\) or if A and B satisfy certain conditions, then the number of negative square-free \(D \equiv A \pmod {B}\) down to \(-X\) such that the ideal class group of \({\mathbb {Q}} (\sqrt{D})\) contains an element of order g is bounded below by \(X^{\frac{1}{2} + \epsilon (g) - \epsilon }\), where the exponent is the same as in Soundararajan’s theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with p-Selmer group of rank at least 2, where \(p \in \{3,5,7\}\).

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Acknowledgements

The author thanks Ken Ono for suggesting this project, the two reviewers for some very helpful comments, and Professor Soundararajan for a helpful conversation.

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Correspondence to Olivia Beckwith.

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Beckwith, O. Class number divisibility for imaginary quadratic fields. Res. number theory 6, 13 (2020). https://doi.org/10.1007/s40993-020-0188-4

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