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.

A new non-arithmetic lattice in PU(3,1)

We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1)

Perfect forms over totally real number fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and later generalized by Koecher to arbitrary number fields. One knows that up to a natural "change of variables'' equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main result explains how to find an initial perfect form for any such field. We also compute the inequivalent binary perfect forms over real quadratic fields Q(\sqrt{d}) with d \leq 66.Comment: 11 pages, 2 figures, 1 tabl

Abelian Spiders

If G is a finite graph, then the largest eigenvalue L of the adjacency matrix of G is a totally real algebraic integer (L is the Perron-Frobenius eigenvalue of G). We say that G is abelian if the field generated by L^2 is abelian. Given a fixed graph G and a fixed set of vertices of G, we define a spider graph to be a graph obtained by attaching to each of the chosen vertices of G some 2-valent trees of finite length. The main result is that only finitely many of the corresponding spider graphs are both abelian and not Dynkin diagrams, and that all such spiders can be effectively enumerated; this generalizes a previous result of Calegari, Morrison, and Snyder. The main theorem has applications to the classification of finite index subfactors. We also prove that the set of Salem numbers of "abelian type" is discrete.Comment: This work represents, in part, the PhD thesis of the second autho