Totally-Reflective Genera of Integral Lattices

TL;DR

This paper provides a complete classification of totally-reflective primitive genera of integral lattices in dimensions 3 and 4, using bounds on determinants and lattice transformations.

Contribution

It introduces a method combining mass formulas and lattice transformations to classify all totally-reflective primitive genera in low dimensions.

Findings

Classified all totally-reflective primitive genera in dimensions 3 and 4.

Established bounds on prime factors of determinants for such genera.

Developed a method to generate all such genera from square-free cases.

Abstract

In this paper we give a complete classification of totally-reflective, primitive genera in dimension 3 and 4. Our method 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 genera 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 with the square-free case.