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

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 ${\mathbb{Q}}_p^d$, then \begin{align*} (1) \quad\quad \quad \quad |n|^2\leq |d|\max{|n|, γ^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 $\mathit{\text{[Indag.}}\mathit{\text{Math.,}}\mathit{\text{1966]}}$ relative bound for equiangular lines in the p-adic case.