Classification of positive definite lattices. 22 Feb 2000 
Richard E. Borcherds, 
Mathematics department, Evans Hall #3840, University of California 
at Berkeley, CA 94720-3840 U.S.A. 
www home page www.math.berkeley.edu/~reb 

Contents.  
1. An algorithm for classifying vectors in some Lorentzian lattices. 
2. Vectors in the lattice II1,25. 
3. Lattices with no roots. 
Table 0: Primitive norm 0 vectors in II1,25. 
Table 1: Norm 2 vectors in II1,25. 
Table 2: Norm 4 vectors in II1,25. 

1.  Classification of positive norm vectors. 

In this paper we describe an algorithm for classifying orbits of vectors 
in Lorentzian lattices. The main point of this is that isomorphism classes 
of positive definite lattices in some genus often correspond to orbits of 
vectors in some Lorentzian lattice, so we can classify some positive definite 
lattices. Section 1 gives an overview of this algorithm, and in section 2 we 
describe this algorithm more precisely for the case of II1,25, and as an 
application we give the classification of the 665 25-dimensional unimodular 
positive definite lattices and the 121 even 25 dimensional positive definite 
lattices of determinant 2 (see tables 1 and 2). In section 3 we use this 
algorithm to show that there is a unique 26 dimensional unimodular positive 
definite lattice with no roots. Most of the results of this paper are taken 
from the unpublished manuscript [B], which contains more details and examples. 
For general facts about lattices used in this paper see [C-S], especially 
chapters 15-18 and 23-28.