Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids
Abstract
Let M(N) be the number of similarity classes of positive rational perfect cuboids with ordered edges a,b,c, space diagonal g, and squarefree invariant N=sf(abcg/2). We prove that M(N) is finite for every positive squarefree N, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, M(N)≤ C1+rN for an absolute constant C>1, where rN is the rank of y2=x3−N2x. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove limX→∞1/X ∑substackN≤ X N positive squarefreeM(N)q=0 for every fixed q>0. Thus the result counts cuboid classes with their full multiplicities at each N. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.
Related
BibTeX
@misc{asiryan2026finiteness,
title = {{Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids}},
author = {Valery Asiryan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38650},
eprint = {2609.38650},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38650},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids
AU - Valery Asiryan
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38650
UR - https://arxiv.org/abs/2609.38650
AB - Let M(N) be the number of similarity classes of positive rational perfect cuboids with ordered edges a,b,c, space diagonal g, and squarefree invariant N=sf(abcg/2). We prove that M(N) is finite for every positive squarefree N, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, M(N)≤ C^1+r_N for an absolute constant C>1, where r_N is the rank of y^2=x^3−N^2x. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove lim_X→∞1/X ∑_substackN≤ X N positive squarefreeM(N)^q=0 for every fixed q>0. Thus the result counts cuboid classes with their full multiplicities at each N. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.
ER -