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Theorem 4.4 (Cayley's theorem). Let $C$ and $D$ two non-degenerate conics of $\mathbb{P}_{\mathbb{C}}^2$ meeting at four different points, and let
\[
\sqrt{\operatorname{det}(t C+D)}=A_0+A_1 t+A_2 t^2+\ldots
\]
be the Taylor expansion, at the point $t=0$, of the function $\sqrt{\operatorname{det}(t C+D)}$. Then, there exists a $n$-sided polygon inscribed in $C$ and circumscribed about $D$ if and only if,
\[
\begin{aligned}
& \left|\begin{array}{ccc}
A_2 & \ldots & A_{m+1} \\
\vdots & & \vdots \\
A_{m+1} & \ldots & A_{2 m}
\end{array}\right|=0, \quad \text { when } n \text { is odd and } n=2 m+1 \text {, for some } m \geq 1 \\
& \left|\begin{array}{ccc}
A_3 & \ldots & A_{m+1} \\
\vdots & & \vdots \\
A_{m+1} & \ldots & A_{2 m-1}
\end{array}\right|=0, \quad \text { when } n \text { is even and } n=2 m, \text { for } \text { some } m \geq 2
\end{aligned}
\]
Proof. Recall that we have the curves $E=\left\{(r, u): u_0^2 r_1^3=u_1^2 \cdot \operatorname{det}\left(r_0 C+r_1 D\right)\right\}$ and $\gamma=\{(r, s): H(r, s)=0\}$ in $\mathbb{P}_{\mathbb{C}}^1 \times \mathbb{P}_{\mathbb{C}}^1$, as well as the isomorphisms
\[
G: E \longrightarrow \gamma, \quad G\left(\left(r_0: r_1\right),\left(u_0: u_1\right)\right)=\left(\left(r_0: r_1\right),\left(2 T_2(r) \cdot u_1:-T_1(r) \cdot u_1+u_0 \cdot r_1^2\right)\right)
\]
\[
F: \gamma \longrightarrow \mathfrak{M}, \quad F(r, s)=(p(r), l(s))
\]
Let's consider the isomorphism $\psi=G^{-1} \circ F^{-1}: \mathfrak{M} \longrightarrow E$.
If we take $\theta=\left(p_0, l_0\right)=\left(p((1: 0)), l_0\right)$ as the neutral element on $\mathfrak{M}$ (where $l_0$ is the tangent line to $D$ through $p_0$ ),
\[
\psi(\theta)=G^{-1}\left(F^1\left(p((1: 0)), l_0\right)\right)=G^{-1}\left((1: 0), l^{-1}\left(l_0\right)\right)=\left((1: 0),\left(u_0: u_1\right)\right)
\]
for some $\left(u_0: u_1\right)$ such that $\left((1: 0),\left(u_0: u_1\right)\right) \in E$. That is, $\psi(\theta)=((1: 0),(1: 0))$.
Thus choosing $((1: 0),(1: 0))$ as the neutral element on $E$, by corollary $2.18 \psi$ is also a group isomorphism.
Note that $\eta(\theta)=(\widetilde{p}, \widetilde{l})$, where $\widetilde{p}$ satisfies $C \cap l_0=\left\{p_0, \widetilde{p}\right\}$ and $\widetilde{l}$ is the tangent line to $D$ through $\widetilde{p}$. But $\widetilde{p}=p((0: 1))$ (since $l_0$ is the tangent line to $D=C_{(0: 1)}$ through $p_0$ ), so
\[
\psi(\eta(\theta))=G^{-1}\left(F^1(p((0: 1)), \widetilde{l})\right)=G^{-1}\left((0: 1), l^{-1}(\widetilde{l})\right)=\left((0: 1),\left(u_0: u_1\right)\right)
\]
for some $\left(u_0: u_1\right)$ such that $\left((0: 1),\left(u_0: u_1\right)\right) \in E$. From the equation for $E$, it follows that $\left(u_0: u_1\right)=( \pm \sqrt{\operatorname{det} D}: 1)$.
Using that $\psi$ is a group isomorphism and Lemma 4.3, we deduce that
\[
\begin{aligned}
& \eta^n=I d_{\mathfrak{M}} \Longleftrightarrow \eta(\theta) \text { is a torsion point of } \mathfrak{M} \text { of order } n \Longleftrightarrow \\
& \Longleftrightarrow \psi(\eta(\theta))=((0: 1),( \pm \sqrt{\operatorname{det} D}: 1)) \text { is a } n \text {-torsion point of } E
\end{aligned}
\]
Now, let's study the restriction of the elliptic curve $E$ to the affine chart $A_1=\left\{((x: 1),(y: 1)) \in \mathbb{P}_{\mathbb{C}}^1 \times \mathbb{P}_{\mathbb{C}}^1: x, y \in \mathbb{C}\right\}$ of $\mathbb{P}_{\mathbb{C}}^1 \times \mathbb{P}_{\mathbb{C}}^1$. Namely, consider the plane affine curve
\[
E^{\prime}=\left\{(x, y) \in \mathbb{C}^2:((x: 1),(y: 1)) \in E\right\}=\left\{(x, y) \in \mathbb{C}^2: y^2=\operatorname{det}(x C+D)\right\}
\]
and its projective closure
\[
E^{\prime \prime}=\left\{(x: y: z) \in \mathbb{P}_{\mathbb{C}}^2: y^2 z=\operatorname{det}(x C+D z)\right\}
\]
Since the pair of points at infinity $((1: 0),(1: 0))$ is the neutral element on $E$, the neutral element on $E^{\prime \prime}$ must be on the line at infinity $z=0$ : it's the point $(0: 1: 0)$. Then,
$((0: 1),( \pm \sqrt{\operatorname{det} D}: 1))$ is a $n$-torsion point of $E \Longleftrightarrow(0, \pm \sqrt{\operatorname{det} D})$ is a $n$-torsion point of $E^{\prime} \Longleftrightarrow$
$\Longleftrightarrow(0: \pm \sqrt{\operatorname{det} D}: 1)$ is a $n$-torsion point of $E^{\prime \prime}$
and Cayley's theorem becomes a consequence of theorem 2.19 . |
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