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arXiv · 2310.12519

The unimodular equivalence of sublattices in an $n$-dimensional lattice

Shikui Shang
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Abstract

In this paper, we study the unimodular equivalence of sublattices in an $n$-dimensional lattice. A recursive procedure is given to compute the cardinalities of the unimodular equivalent classes with the indices which are powers of a prime $p$. We also show that these are integral polynomials in $p$. When $n=2$, the explicit formulae of the cardinalities are presented depending on the prime decomposition of the index $m$. We also give an explicit formula on the number of co-cyclic sublattices with a fixed index $m$, which consist a unimodular equivalent class of sublattices.

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Shikui Shang. 2023-10-19. The unimodular equivalence of sublattices in an $n$-dimensional lattice. https://arxiv.org/abs/2310.12519

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