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