On non-congruent numbers as multiples of non-congruent numbers

Abstract

Let $n=PQ$ be a square-free positive integer, where P is a product of primes congruent to $1\mathrm{mod}8$, and Q is a non-congruent number with a trivial 2-primary Shafarevich-Tate group. Under certain conditions on the Legendre symbols $\left(\frac{q}{p}\right)$ for primes $p|P,q|Q$, we establish a criterion characterizing when n is non-congruent with a minimal or a second minimal 2-primary Shafarevich-Tate group. We also provide a sufficient condition for n to be non-congruent with a larger 2-primary Shafarevich-Tate group. These results involve the class groups and tame kernels of quadratic fields.