I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the problem is in the proof of Theorem 1 (§2). At the beginning of the last paragraph of page 288 he says: "Let, then, $\Delta\subset \mathcal{H}^3$ be a tetrahedron with vertices $P_i=(z_i,r_i)\in K_1\times (K_1\cap \mathbb{R})_+^*\subset \mathbb{C}\times \mathbb{R}_+^*$ $(i=0,1,2,3)$. The geodesic through $P_0$ and $P_1$, continued in the direction from $P_0$ to $P_1$, meets the ideal boundary $\mathbb{P}^1(\mathbb{C})=\mathbb{C}\cup\{\infty\}$ of $\mathcal{H}^3$ in a point of $\mathbb{P}^1(K_1)$ [...]".

Here $\mathcal{H}^3$ is the half-upper space model for the 3-dimensional hyperbolic space. For the notation please read the pdf in the link above.

My question is: why does the geodesic meets the boundary in $\mathbb{P}^1(K_1)=K_1\cup\{\infty\}$?

I tried to do some obvious calculations and I considered a semicircle going from $P_0$ to $P_1$ and then to the boundary. The coordinates for the centre $C$ of the semicircle are $$C=\left(z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right],0\right).$$ The intersections of the geodesic with $\mathbb{C}\times\{0\}$ are $$\left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right] \pm (z_1-z_0)\sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2},0 \right)=\\ \left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right],0 \right). $$

So, if Zagier is right, these two points should be in $K_1\times \{0\}$ and this means that $$\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right]\in K_1$$

but $K_1$ is not supposed to be closed under complex conjugation and, even if we make this hypothesis, how could we know that those numbers are in $K_1$?

For these reasons I think there is a lack in the proof (or a mistake!). Can anyone help me? Thank you in advance.

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the problem is in the proof of Theorem 1 (§2). In this proof we have a quaternion division algebra $B/K$ where $K$ is a number field with just one complex place. $B$ is ramified at all real places (i.e. $B\otimes_K \mathbb{R}\simeq \mathbb{H}$ for al real places) and $K_1$ is a quadratic field extension of $K$ such that $B$ splits on $K_1$ (i.e. $B\otimes_K K_1\simeq M(2,K_1)$).

At the beginning of the last paragraph of page 288 Zagier says: "Let, then, $\Delta\subset \mathcal{H}^3$ be a tetrahedron with vertices $P_i=(z_i,r_i)\in K_1\times (K_1\cap \mathbb{R})_+^*\subset \mathbb{C}\times \mathbb{R}_+^*$ $(i=0,1,2,3)$. The geodesic through $P_0$ and $P_1$, continued in the direction from $P_0$ to $P_1$, meets the ideal boundary $\mathbb{P}^1(\mathbb{C})=\mathbb{C}\cup\{\infty\}$ of $\mathcal{H}^3$ in a point of $\mathbb{P}^1(K_1)$ [...]". Here $\mathcal{H}^3$ is the half-upper space model for the 3-dimensional hyperbolic space.

My question is: why does the geodesic meets the boundary in $\mathbb{P}^1(K_1)=K_1\cup\{\infty\}$?

I tried to do some obvious calculations and I considered a semicircle going from $P_0$ to $P_1$ and then to the boundary. The coordinates for the centre $C$ of the semicircle are $$C=\left(z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right],0\right).$$ The intersections of the geodesic with $\mathbb{C}\times\{0\}$ are $$\left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right] \pm (z_1-z_0)\sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2},0 \right)=\\ \left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right],0 \right). $$

So, if Zagier is right, these two points should be in $K_1\times \{0\}$ and this means that $$\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right]\in K_1$$

but $K_1$ is not supposed to be closed under complex conjugation and, even if we make this hypothesis, how could we know that those numbers are in $K_1$?

For these reasons I think there is a lack in the proof (or a mistake!). Can anyone help me? Thank you in advance.

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at the link

https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01388964/fulltext.pdf,

  and I think there might be a mistake. In particular the problem is in the proof of Theorem 1 (§2). At the beginning of the last paragraph of page 288 he says: "Let, then, $\Delta\subset \mathcal{H}^3$ be a tetrahedron with vertices $P_i=(z_i,r_i)\in K_1\times (K_1\cap \mathbb{R})_+^*\subset \mathbb{C}\times \mathbb{R}_+^*$ $(i=0,1,2,3)$. The geodesic through $P_0$ and $P_1$, continued in the direction from $P_0$ to $P_1$, meets the ideal boundary $\mathbb{P}^1(\mathbb{C})=\mathbb{C}\cup\{\infty\}$ of $\mathcal{H}^3$ in a point of $\mathbb{P}^1(K_1)$ [...]".

Here $\mathcal{H}^3$ is the half-upper space model for the 3-dimensional hyperbolic space. For the notation please read the pdf in the link above.

My question is: why does the geodesic meets the boundary in $\mathbb{P}^1(K_1)=K_1\cup\{\infty\}$?

I tried to do some obvious calculations and I considered a semicircle going from $P_0$ to $P_1$ and then to the boundary. The coordinates for the centre $C$ of the semicircle are $$C=\left(z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right],0\right).$$ The intersections of the geodesic with $\mathbb{C}\times\{0\}$ are $$\left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right] \pm (z_1-z_0)\sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2},0 \right)=\\ \left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right],0 \right). $$

So, if Zagier is right, these two points should be in $K_1\times \{0\}$ and this means that $$\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right]\in K_1$$

but $K_1$ is not supposed to be closed under complex conjugation and, even if we make this hypothesis, how could we know that those numbers are in $K_1$?

For these reasons I think there is a lack in the proof (or a mistake!). Can anyone help me? Thank you in advance.

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the problem is in the proof of Theorem 1 (§2). At the beginning of the last paragraph of page 288 he says: "Let, then, $\Delta\subset \mathcal{H}^3$ be a tetrahedron with vertices $P_i=(z_i,r_i)\in K_1\times (K_1\cap \mathbb{R})_+^*\subset \mathbb{C}\times \mathbb{R}_+^*$ $(i=0,1,2,3)$. The geodesic through $P_0$ and $P_1$, continued in the direction from $P_0$ to $P_1$, meets the ideal boundary $\mathbb{P}^1(\mathbb{C})=\mathbb{C}\cup\{\infty\}$ of $\mathcal{H}^3$ in a point of $\mathbb{P}^1(K_1)$ [...]".

Here $\mathcal{H}^3$ is the half-upper space model for the 3-dimensional hyperbolic space. For the notation please read the pdf in the link above.

My question is: why does the geodesic meets the boundary in $\mathbb{P}^1(K_1)=K_1\cup\{\infty\}$?

I tried to do some obvious calculations and I considered a semicircle going from $P_0$ to $P_1$ and then to the boundary. The coordinates for the centre $C$ of the semicircle are $$C=\left(z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right],0\right).$$ The intersections of the geodesic with $\mathbb{C}\times\{0\}$ are $$\left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\right] \pm (z_1-z_0)\sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2},0 \right)=\\ \left( z_0+(z_1-z_0)\cdot\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right],0 \right). $$

So, if Zagier is right, these two points should be in $K_1\times \{0\}$ and this means that $$\left[\frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2}\pm \sqrt{\frac{r_0^2}{|z_1-z_0|^2}+\left( \frac{r_1^2-r_0^2}{2|z_1-z_0|^2}+\frac{1}{2} \right)^2} \right]\in K_1$$

but $K_1$ is not supposed to be closed under complex conjugation and, even if we make this hypothesis, how could we know that those numbers are in $K_1$?

For these reasons I think there is a lack in the proof (or a mistake!). Can anyone help me? Thank you in advance.