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June 2021 An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields
Yoshichika IIZUKA, Yutaka KONOMI, Shin NAKANO
Tokyo J. Math. 44(1): 33-47 (June 2021). DOI: 10.3836/tjm/1502179314

Abstract

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.

Citation

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Yoshichika IIZUKA. Yutaka KONOMI. Shin NAKANO. "An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields." Tokyo J. Math. 44 (1) 33 - 47, June 2021. https://doi.org/10.3836/tjm/1502179314

Information

Published: June 2021
First available in Project Euclid: 18 June 2020

Digital Object Identifier: 10.3836/tjm/1502179314

Subjects:
Primary: 11R29
Secondary: 11G05, 11R11

Rights: Copyright © 2021 Publication Committee for the Tokyo Journal of Mathematics

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Vol.44 • No. 1 • June 2021
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