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It may be that I am missing something very simple. In S. Katok's book "Fuchsian Groups", Lemma 5.3.3, we have the following.

Lemma: Let $\Gamma$ be a Fuchsian group of finite covolume, $k_0=\mathbb{Q}(\mathrm{tr}(\Gamma))$. Assume that $[k_0:\mathbb{Q}]<\infty$ and $\mathrm{tr}(\Gamma)\subset \mathcal{O}_{k_0}$ (the ring of integers of $k_0$). Then, $$\mathcal{O}_{k_0}[\Gamma]=\left\{\sum_i a_i \gamma_i\,:\, a_i\in\mathcal{O}_{k_0},\,\gamma_0\in\Gamma\right\}$$ is an order of the quaternion algebra $$ k_0[\Gamma]=\left\{\sum_i a_i\gamma_i\,:\, a_i\in k_0,\,\gamma_0\in\Gamma\right\}. $$

In the proof, we have the following argument.


We may assume that $\gamma_0=\begin{pmatrix}\lambda\\&\lambda^{-1}\end{pmatrix}$, $\lambda\neq 1$ and $\Gamma\subseteq \mathrm{PSL}(2,K_0)$ where $K_0=k_0(\lambda)$. If $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathcal{O}_{k_0}[\Gamma]$, then $a+d$ and $\lambda a+\lambda^{-1}d$ are in $\mathcal{O}_{k_0}$. Notice that $\lambda$ and $\lambda^{-1}$ are units in $K_0$ and $\mathcal{O}_{k_0}$ is a subring of the ring of integers of $K_0$, so $a$ and $d$ are in the fractional ideal $\frac{1}{\lambda^2-1} \mathcal{O}_{k_0}$ of $K_0$.


I don't understand why $a$ is in that fractional ideal. It seemed to me that $$ (\lambda^2-1)a=\lambda(\lambda a+\lambda^{-1}d)-(a+d)\in (\lambda-1)\mathcal{O}_{k_0}-\mathcal{O}_{k_0}. $$ But I don't see why that should lie in $\mathcal{O}_{k_0}$. Also, why is that ideal fractional? Wouldn't that imply that $(\lambda^2-1)\in \mathcal{O}_{k_0}$?

On the author's website, there is a mention of this Lemma in the errata: http://www.personal.psu.edu/sxk37/errata.pdf but it just says: "The end of the proof of Lemma 5.3.3 was modified according to M.Katz’s suggestion."

Thanks for any clarification!

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