Computing quaternionic representations via twisted forms of Bruhat-Tits trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914276556406784 |
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| author | Arenas-Carmona, Luis Bravo, Claudio |
| author_facet | Arenas-Carmona, Luis Bravo, Claudio |
| contents | This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing quaternionic representations via twisted forms of Bruhat-Tits trees Arenas-Carmona, Luis Bravo, Claudio Representation Theory Number Theory Rings and Algebras This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups. |
| title | Computing quaternionic representations via twisted forms of Bruhat-Tits trees |
| topic | Representation Theory Number Theory Rings and Algebras |
| url | https://arxiv.org/abs/2512.22713 |