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

Abstract

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fieldsQ(t4−n2)\mathbb {Q}(\sqrt {t^4-n^2})asttvaries over positive integers for a congruent numbernn. Furthermore, we provide lower bounds on class numbers ofRichaud-Degert typereal quadratic fields of the formQ(n2k4−1)\mathbb {Q}(\sqrt {n^2k^4-1})for positive integerskkand congruent numbersnnwhose elliptic curves have algebraic rank greater than22.