On the 2-part of the class number of \(\mathbb{Q}(\sqrt{\pm D})\) for a congruent number \(D\)
When \(D\) is a congruent number which is a product of two primes \(p\) and \(q\) both congruent to 5 modulo 8, and which satisfy the Legendre symbol \((q/p)=-1\), the authors, assuming some condition on the Tate-Shafarevich group, proved that the class number of \(\mathbb{Q}(\sqrt{D})\) is divisible by 4. Assuming some extra hypotheses involving the representations of \(p\) band \(q\) by certain binary quadratic forms, they also proved that the class number of \(\mathbb{Q}(\sqrt{- D})\) is congruent to 8 modulo 16. Some other properties and comments are made. This is a nice contribution to the problem of characterizing congruent numbers.
scientific article; zbMATH DE number 7600670