Finiteness theorems for some representations of $\mathrm{GL}_3$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916920129748992 |
|---|---|
| author | Janbazi, Fatemehzahra Shankar, Arul |
| author_facet | Janbazi, Fatemehzahra Shankar, Arul |
| contents | Let $n \geq 2$ be an integer and let $K$ be a number field with ring of integers $\mathcal{O}_K$. We prove that the set of ternary $n$-ic forms with coefficients in $\mathcal{O}_K$ and fixed nonzero discriminant, breaks up into finitely many $\mathrm{GL}_3(\mathcal{O}_K)$-orbits. This generalizes a result of Birch--Merriman in the binary forms case. We also prove a similar finiteness result on the $\mathrm{GL}_3(\mathcal{O}_K)$-orbits of the 27-dimensional representation of $\mathrm{GL}_3$ with highest weight $(4, 2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness theorems for some representations of $\mathrm{GL}_3$ Janbazi, Fatemehzahra Shankar, Arul Number Theory Let $n \geq 2$ be an integer and let $K$ be a number field with ring of integers $\mathcal{O}_K$. We prove that the set of ternary $n$-ic forms with coefficients in $\mathcal{O}_K$ and fixed nonzero discriminant, breaks up into finitely many $\mathrm{GL}_3(\mathcal{O}_K)$-orbits. This generalizes a result of Birch--Merriman in the binary forms case. We also prove a similar finiteness result on the $\mathrm{GL}_3(\mathcal{O}_K)$-orbits of the 27-dimensional representation of $\mathrm{GL}_3$ with highest weight $(4, 2)$. |
| title | Finiteness theorems for some representations of $\mathrm{GL}_3$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.05641 |