Reduction mod. 2 of del Pezzo lattices

TL;DR

This paper investigates the properties of del Pezzo lattices under mod 2 reduction, establishing bijections and isomorphisms related to their root systems and automorphism groups.

Contribution

It demonstrates that for certain root system lattices, mod 2 reduction creates a bijection between roots and vectors with q=1, and an isomorphism of automorphism groups.

Findings

Mod 2 reduction induces a bijection between roots and vectors with q=1.

Automorphism group O(L)/<-1> is isomorphic to O(L/2L).

Results apply to lattices associated with specific root systems.

Abstract

For an even lattice L , the form v --> (v.v)/2 induces a quadratic form q on the (Z/2)-vector space L/2L . For the lattices associated to some particular root systems, we show that reduction mod. 2 induces a bijection between the roots of L and the vectors of L/2L with q=1 , and an isomorphism of O(L)/<-1> onto O(L/2L).