Quasi-inner automorphisms of Drinfeld modular groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916207041445888 |
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| author | Mason, A. W. Schweizer, Andreas |
| author_facet | Mason, A. W. Schweizer, Andreas |
| contents | Let $A$ be the set of elements in an algebraic function field $K$ over ${\mathbb F}_q$ which are integral outside a fixed place $\infty$. Let $G=GL_2(A)$ be a {\it Drinfeld modular group}. The normalizer of $G$ in $GL_2(K)$, where $K$ is the quotient field of $A$, gives rise to automorphisms of $G$, which we refer to as {\it quasi-inner}. Modulo the inner automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to ${\mathrm Cl}(A)_2$, the $2$-torsion in the ideal class group ${\mathrm Cl}(A)$.
The group $Quinn(G)$ acts on all kinds of objects associated with $G$. For example, it acts freely on the cusps and elliptic points of $G$. If ${\mathcal T}$ is the associated Bruhat-Tits tree the elements of $Quinn(G)$ induce non-trivial automorphisms of the quotient graph $G\setminus{\mathcal T}$, generalizing an earlier result of Serre. It is known that the ends of $G\setminus{\mathcal T}$ are in one-one correspondence with the cusps of $G$. Consequently $Quinn(G)$ acts freely on the ends. In addition $Quinn(G)$ acts transitively on those ends which are in one-one correspondence with the vertices of $G\setminus{\mathcal T}$ whose stabilizers are isomorphic to $GL_2({\mathbb F}_q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_03261 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quasi-inner automorphisms of Drinfeld modular groups Mason, A. W. Schweizer, Andreas Number Theory Group Theory 11F06, 20E08, 20E36, 20G30 Let $A$ be the set of elements in an algebraic function field $K$ over ${\mathbb F}_q$ which are integral outside a fixed place $\infty$. Let $G=GL_2(A)$ be a {\it Drinfeld modular group}. The normalizer of $G$ in $GL_2(K)$, where $K$ is the quotient field of $A$, gives rise to automorphisms of $G$, which we refer to as {\it quasi-inner}. Modulo the inner automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to ${\mathrm Cl}(A)_2$, the $2$-torsion in the ideal class group ${\mathrm Cl}(A)$. The group $Quinn(G)$ acts on all kinds of objects associated with $G$. For example, it acts freely on the cusps and elliptic points of $G$. If ${\mathcal T}$ is the associated Bruhat-Tits tree the elements of $Quinn(G)$ induce non-trivial automorphisms of the quotient graph $G\setminus{\mathcal T}$, generalizing an earlier result of Serre. It is known that the ends of $G\setminus{\mathcal T}$ are in one-one correspondence with the cusps of $G$. Consequently $Quinn(G)$ acts freely on the ends. In addition $Quinn(G)$ acts transitively on those ends which are in one-one correspondence with the vertices of $G\setminus{\mathcal T}$ whose stabilizers are isomorphic to $GL_2({\mathbb F}_q)$. |
| title | Quasi-inner automorphisms of Drinfeld modular groups |
| topic | Number Theory Group Theory 11F06, 20E08, 20E36, 20G30 |
| url | https://arxiv.org/abs/2108.03261 |