Reflective Lorentzian lattices of signature (5,1)

In this paper we give a complete classification of strongly square-free reflective Z-lattices of signature (5,1). This is done by reducing the classification of Lorentzian lattices to those of positive-definite lattices. The classification of totally-reflective genera breaks up into two parts. The first part consists of classifying the square free, totally-reflective, primitive genera by calculating strong bounds on the prime factors of the determinant of positive-definite quadratic forms (lattices) with this property. We achieve these bounds by combining the Minkowski-Siegel mass formula with the combinatorial classification of reflective lattices accomplished by Scharlau & Blaschke. In a second part, we use a lattice transformation that goes back to Watson, to generate all totally-reflective, primitive genera when starting from the square free case.