Congruent numbers and lower bounds on class numbers of real quadratic fields

Kim, Jigu; Lee, Yoonjin

doi:10.1090/proc/15993

Congruent numbers and lower bounds on class numbers of real quadratic fields
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Proc. Amer. Math. Soc. 150 (2022), 4671-4684 Request permission

Abstract:

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields $\mathbb {Q}(\sqrt {t^4-n^2})$ as $t$ varies over positive integers for a congruent number $n$. Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form $\mathbb {Q}(\sqrt {n^2k^4-1})$ for positive integers $k$ and congruent numbers $n$ whose elliptic curves have algebraic rank greater than $2$.

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