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.

Categories: math.CO math.FA math.NT math.SP

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