Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem
Abstract
We study the arithmetic of the elliptic curves Am:y2=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≡11pmod40 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.
Related
- Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
- Tunnell-type criteria for variants of the congruent number problem
- A proof of Alexander’s Conjecture on an Inequality of Cassels
- The Bernstein problem in higher dimensional Heisenberg groups
- Visibility problem in higher dimensions
BibTeX
@misc{lei2026higher,
title = {{Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem}},
author = {Kaisheng Lei and Shisong Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23419},
eprint = {2609.23419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23419},
}
RIS
TY - UNPB
TI - Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem
AU - Kaisheng Lei
AU - Shisong Xu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23419
UR - https://arxiv.org/abs/2609.23419
AB - 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≡11pmod40 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.
ER -