p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound

TL;DR

This paper introduces p-adic equiangular lines and establishes a fundamental relation between their common angle, the space dimension, and the number of lines, extending classical bounds to the p-adic setting.

Contribution

It defines p-adic equiangular lines and derives the first p-adic van Lint-Seidel relative bound relating lines, angles, and dimension.

Findings

Established the p-adic van Lint-Seidel relative bound.

Connected p-adic equiangular lines with classical bounds.

Provided a new framework for p-adic geometric configurations.

Abstract

We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if $\{\tau_j\}_{j=1}^n$ is p-adic $\gamma$-equiangular lines in $\mathbb{Q}^d_p$, then \begin{align*} (1) \quad\quad \quad \quad |n|^2\leq |d|\max\{|n|, \gamma^2 \}. \end{align*} We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel \textit{[Indag. Math., 1966]} relative bound for equiangular lines in the p-adic case.