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

Shinobu Hosono University of Tokyo Bong H. Lian Brandeis University Keiji Oguiso University of Tokyo Shing-Tung Yau Harvard University

Quantum Algebra mathscidoc:1608.29001

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
Unimodular Lattices
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  title={Counting Unimodular Lattices in $\R^{r,s}$},
  author={Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau},
Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau. Counting Unimodular Lattices in $\R^{r,s}$. 2003.
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