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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.