A Ramanujan Pell K3 Surface and Cube Representations
Abstract
A negative Pell equation produces positive integers with an3+bn3+cn3=561Nn3+(−1)n+1. We classify all positive solutions satisfying the natural linear constraints of the construction and prove an exact pairwise coprimality law, with exceptional factor $137$ in one class modulo $68$. The associated sextic K=(1−13u+26u2)3+(6+182u2)3 defines an elliptic K3 surface with six type-IV fibres. Its genus-four cubic cover has a geometrically simple Prym threefold; exact good-reduction arithmetic establishes simplicity over every finite extension of the residue field. We determine the full geometric Mordell-Weil group, the integral Neron-Severi and transcendental lattices, and the arithmetic Picard lattice over ℚ, which is U⊕⟨−4⟩⊕4. Every two-cube representation by rational functions over ℚ is obtained from a multiple of one section, with numerator degrees 2m2 and denominator degree 2m2−2. Polynomial rigidity follows without a degree assumption. Along the specified Pell orbit all specialized points are nontorsion, their heights are 4/3|n|log(8+√(65))+O(1), and infinitely many rational cubic-twist classes occur. We also identify the local zeta function at $19$ and the generic rank and rank-jump criterion for the corresponding two-parameter base-change family.
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BibTeX
@misc{raghava2026ramanujan,
title = {{A Ramanujan Pell K3 Surface and Cube Representations}},
author = {K. Srinivasa Raghava},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11925},
eprint = {2607.11925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11925},
}
RIS
TY - UNPB
TI - A Ramanujan Pell K3 Surface and Cube Representations
AU - K. Srinivasa Raghava
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2607.11925
UR - https://arxiv.org/abs/2607.11925
AB - A negative Pell equation produces positive integers with a_n^3+b_n^3+c_n^3=561N_n^3+(−1)^n+1. We classify all positive solutions satisfying the natural linear constraints of the construction and prove an exact pairwise coprimality law, with exceptional factor $137$ in one class modulo $68$. The associated sextic K=(1−13u+26u^2)^3+(6+182u^2)^3 defines an elliptic K3 surface with six type-IV fibres. Its genus-four cubic cover has a geometrically simple Prym threefold; exact good-reduction arithmetic establishes simplicity over every finite extension of the residue field. We determine the full geometric Mordell-Weil group, the integral Neron-Severi and transcendental lattices, and the arithmetic Picard lattice over ℚ, which is U⊕⟨−4⟩^⊕4. Every two-cube representation by rational functions over ℚ is obtained from a multiple of one section, with numerator degrees 2m^2 and denominator degree 2m^2−2. Polynomial rigidity follows without a degree assumption. Along the specified Pell orbit all specialized points are nontorsion, their heights are 4/3|n|log(8+√(65))+O(1), and infinitely many rational cubic-twist classes occur. We also identify the local zeta function at $19$ and the generic rank and rank-jump criterion for the corresponding two-parameter base-change family.
ER -