On non-congruent numbers with 8a±1 type odd prime factors and tame kernels

2021/11/23 by Shenxing Zhang, Zhang, Shenxing
Mathematics · #11R29 #11R70 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #Primary 11G05 #Secondary 11D25

paper · pdf · doi:10.48550/arxiv.2111.11618

openalex publication_date 2021/11/23 · openalex created_date 2021/12/06 · openalex updated_date 2026/07/28

Abstract

Let n be a positive square-free integer, where every odd prime factor of n has form 8a± 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(√(n)) for odd n, or \mathbb Q(√(-n)) for even n. This generalizes previous results.

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