Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem
Year: 2026
Type: preprint
Abstract: We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\equiv11\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and show that the next Cassels--Tate pairing is governed by the normalized representation defect, equivalently by a factorial character, a Pell symbol, and a class number congruence. For composite parameters we compute the ordinary Cassels matrices of four twists and, in the two prime case, a degree $1024$ governing field for their joint distribution. The same higher descent extends to larger radicals: a second rational pushout determines the full $Λ'$ row of the next pairing. We also determine two explicit Selmer towers with four dimensional ordinary radical, and all finite $2$-power Selmer groups when the ordinary radical is one dimensional.
Source: arXiv (Cornell University)
Authors: Lei, Kaisheng, Xu, Shisong
Institution: Nanjing University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: 同余数问题、整二次型和代数K理论相关问题的研究, G8842371449
OpenAlex ID: W7214008951