Counting Unimodular Lattices in $\R^{r,s}$

Details Counting Unimodular Lattices in $\R^{r,s}$ Counting Unimodular Lattices in $\R^{r,s}$

2003-01-10

arxiv.org/abs/math/0301095v1

Mathematics

Quantum Algebra

TeX, 10pp; uncomment \listtoc and \writetoc to get table of contents

Scientific paper

Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects -- rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice $U^2$, and isomorphisms of discriminant groups of rank 2 lattices.

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