Classification of the sublattices of a lattice

In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $Λ$ of determinant ${\rm det}(Λ)$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(Λ)$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. This paper presents a systematic treatment for counting the sublattices and deduces a formula for the number of the sublattice classes of determinant $m\cdot {\rm det}(Λ)$.

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