Year: 2026
Type: preprint
Abstract: 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 $\mathbb{Q}$, which is $U\oplus\langle-4\rangle^{\oplus4}$. Every two-cube representation by rational functions over $\mathbb{Q}$ 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 $\frac{4}{3}|n|\log(8+\sqrt{65})+O(1)$, 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.
Source: arXiv (Cornell University)
Author: Raghava, K. Srinivasa
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168408786