On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

Abstract

For $e$ a positive integer, we find restrictions modulo $2^e$ on the coefficients of the characteristic polynomial ${\chi }_S(x)$ of a Seidel matrix $S$. We show that, for a Seidel matrix of order $n$ even (resp. odd), there are at most $2^{\left(\genfrac{}{}{0ex}{}{e-2}{2}\right)}$ (resp. $2^{\left(\genfrac{}{}{0ex}{}{e-2}{2}\right)+1}$) possibilities for the congruence class of ${\chi }_S(x)$ modulo $2^e\mathbb{Z}[x]$. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in ${\mathbb{R}}^{17}$, that is, we reduce the known upper bound from $50$ to $49$.

21 pages, fixed typo in Lemma 2.1