Shapes of unit lattices in $D_p$-number fields
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916986090422272 |
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| author | Harron, Robert Holmes, Erik Vemulapalli, Sameera |
| author_facet | Harron, Robert Holmes, Erik Vemulapalli, Sameera |
| contents | The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the unit shape, within the family of prime degree $p$ number fields whose Galois closure has dihedral Galois group $D_p$ and a unique real embedding. In the case $p = 5$, we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank $2$ lattices). For general $p$, we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shapes of unit lattices in $D_p$-number fields Harron, Robert Holmes, Erik Vemulapalli, Sameera Number Theory 11R27, 11R18, 11R33 The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the unit shape, within the family of prime degree $p$ number fields whose Galois closure has dihedral Galois group $D_p$ and a unique real embedding. In the case $p = 5$, we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank $2$ lattices). For general $p$, we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes. |
| title | Shapes of unit lattices in $D_p$-number fields |
| topic | Number Theory 11R27, 11R18, 11R33 |
| url | https://arxiv.org/abs/2501.12504 |