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

Shinobu Hosono Bong H Lian Keiji Oguiso Shing-Tung Yau

Quantum Algebra mathscidoc:1912.43706

arXiv preprint math/0301095, 2003.1
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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  title={Counting Unimodular Lattices in $\R^{r, s} $},
  author={Shinobu Hosono, Bong H Lian, Keiji Oguiso, and Shing-Tung Yau},
  booktitle={arXiv preprint math/0301095},
Shinobu Hosono, Bong H Lian, Keiji Oguiso, and Shing-Tung Yau. Counting Unimodular Lattices in $\R^{r, s} $. 2003. In arXiv preprint math/0301095.
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