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

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 $2^{\binom{e-2}{2}}$ (resp. $2^{\binom{e-2}{2}+1}$) 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 $49$.Comment: 21 pages, fixed typo in Lemma 2.

Real quadratic fields with a universal form of given rank have density zero

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.Comment: 18 pages, minor change

Equiangular lines in low dimensional Euclidean spaces

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively.Comment: 23 page