On non-congruent numbers with $8a\pm1$ type odd prime factors and tame kernels

TL;DR

This paper characterizes when certain square-free integers with primes of form 8a±1 are non-congruent with specific Shafarevich-Tate groups, linking it to class group ranks and tame kernels, extending prior results.

Contribution

It provides a new criterion for non-congruence based on 4-ranks of class groups and tame kernels for integers with primes of the form 8a±1.

Findings

Determines conditions for non-congruence involving 2-primary Shafarevich-Tate groups.

Links the vanishing of 4-rank of tame kernels to prime factor forms.

Generalizes previous results on class groups and tame kernels.

Abstract

Let $n$ be a positive square-free integer, where every odd prime factor of $n$ has form $8a\pm 1$. We determine when $n$ is non-congruent with second minimal $2$-primary Shafarevich-Tate group, in terms of the $4$-ranks of class groups and a Jacobi symbol. In particular, when every odd prime factor of $n$ has form $8a+1$, this condition is equivalent to the vanishing of the $4$-rank of the tame kernel of $\mathbb Q(\sqrt{n})$ for odd $n$, or $\mathbb Q(\sqrt{-n})$ for even $n$. This generalizes previous results.