# Classification of the sublattices of a lattice

## TL;DR

This paper systematically analyzes and derives formulas for counting sublattices of a given determinant in a lattice, building on historical results and providing exact counts for sublattice classes.

## Contribution

It offers a comprehensive method for counting sublattices and formulates the number of sublattice classes of a specified determinant, extending previous work.

## Key findings

- Derived a formula for the number of sublattice classes of a given determinant.
- Extended previous bounds and exact counts for sublattices of a lattice.
- Provided a systematic approach to classify sublattices by determinant.

## Abstract

In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $\Lambda $ of determinant ${\rm det}(\Lambda )$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(\Lambda )$. 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 $m\cdot {\rm det}(\Lambda )$.