February 12, 2024Open Access

The Siegel Modular Group is the Lattice of Minimal Covolume in the Symplectic Group

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The Siegel modular group represents the lattice with the smallest covolume in the symplectic group.
It was shown that up to conjugation, this lattice is unique among other groups.
The proof focuses on the relation between the symplectic groups over rational and real numbers and their respective lattices involved in geometry and algebra contexts. The implication highlights the structural significance of these groups in mathematical systems.

Implication

Proves the symplectic group has the smallest covolume lattice, indicating its unique structure.

Let n 2. We prove that, up to conjugation, Sp₂ₙ (Z) is the lattice in Sp₂ₙ (R) which has the smallest covolume.

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Džambić et al. (2024) studied this question.

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Proves the symplectic group has the smallest covolume lattice, indicating its unique structure.

Let n 2. We prove that, up to conjugation, Sp₂ₙ (Z) is the lattice in Sp₂ₙ (R) which has the smallest covolume.