Unit lattices of $D_4$-quartic number fields with signature $(2,1)$
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912624819568640 |
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| author | Ceballos, Sergio Ricardo Zapata Chari, Sara Holmes, Erik Jalalvand, Fatemeh Nchiwo, Rahinatou Yuh Njah O'Connor, Kelly Ramirez, Fabian Vemulapalli, Sameera |
| author_facet | Ceballos, Sergio Ricardo Zapata Chari, Sara Holmes, Erik Jalalvand, Fatemeh Nchiwo, Rahinatou Yuh Njah O'Connor, Kelly Ramirez, Fabian Vemulapalli, Sameera |
| contents | There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results.
In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unit lattices of $D_4$-quartic number fields with signature $(2,1)$ Ceballos, Sergio Ricardo Zapata Chari, Sara Holmes, Erik Jalalvand, Fatemeh Nchiwo, Rahinatou Yuh Njah O'Connor, Kelly Ramirez, Fabian Vemulapalli, Sameera Number Theory Primary 11R27, Secondary 11R33 There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$. |
| title | Unit lattices of $D_4$-quartic number fields with signature $(2,1)$ |
| topic | Number Theory Primary 11R27, Secondary 11R33 |
| url | https://arxiv.org/abs/2507.06130 |