Let $R$ be a commutative ring. Given a free rank $2$ étale $R$-algebra $A$ equipped with a basis, any $a \in A^\times$ defines an $R$-linear multiplication-by-$a$ map $A \to A$, so we get an injective homomorphism $A^\times \to \Aut_{\text{$R$-module}}(A) \simeq \GL_2(R)$, and the image is called a **Cartan subgroup** of $\GL_2(R)$. The canonical involution of the $R$-algebra $A$ gives another element of $\Aut_{\text{$R$-module}}(A)$; we call the group generated by it and the Cartan subgroup $A^\times$ the **extended Cartan subgroup**. The Cartan subgroup has index $2$ in the extended Cartan subgroup.

If $R=\F_p$, there are two possibilities for $A$: the split algebra $\F_p \times \F_p$ and the nonsplit algebra $\F_{p^2}$; the resulting Cartan subgroups are called split and nonsplit. The extended Cartan subgroup equals the normalizer of the Cartan subgroup in $\GL_2(\F_p)$ except when $p=2$ and $A$ is split. In the split case, if we use the standard basis of $\F_p \times \F_p$, the Cartan subgroup is the subgroup of diagonal matrices in $\GL_2(\F_p)$, and the extended Cartan subgroup is this together with the coset of antidiagonal matrices in $\GL_2(\F_p)$.

If $R=\Z/p^e\Z$, again there are two possibilities for $A$: the split algebra $R \times R$, or the nonsplit algebra. The nonsplit algebra can be described as $\mathcal{O}/p^e \mathcal{O}$ where $\mathcal{O}$ is either the degree $2$ unramified extension of $\Z_p$ or a quadratic order in which $p$ is inert. The nonsplit algebra can also be described as the ring of length $e$ Witt vectors $W_e(\F_{p^2})$.

If $R=\Z/N\Z$ for some $N \ge 1$, then $A$ can be split or nonsplit independently at each prime dividing $N$.

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- Review status: reviewed
- Last edited by Andrew Sutherland on 2022-03-26 13:01:45

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**History:**(expand/hide all)

- 2022-03-26 13:01:45 by Andrew Sutherland (Reviewed)
- 2022-03-25 22:01:14 by Bjorn Poonen
- 2022-03-25 21:54:35 by Bjorn Poonen
- 2022-03-25 21:37:08 by Bjorn Poonen
- 2022-03-25 21:30:51 by Bjorn Poonen
- 2022-03-20 19:48:56 by Andrew Sutherland
- 2021-09-18 13:56:46 by Andrew Sutherland
- 2021-09-18 13:56:21 by Andrew Sutherland
- 2021-09-18 13:53:50 by Andrew Sutherland
- 2021-09-18 10:14:24 by Andrew Sutherland
- 2021-09-18 10:13:13 by Andrew Sutherland
- 2021-09-18 09:58:16 by Andrew Sutherland
- 2021-09-18 09:57:06 by Andrew Sutherland
- 2020-10-29 12:32:02 by Eran Assaf
- 2017-03-16 13:51:31 by Andrew Sutherland

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