Certain Set of Smooth Conics is Finite

Question

This question comes from Darmon's ''Rational Points on Curves," page 16.

Let $S$ be a finite set of primes. Let $\mathcal O$ denote the ring of $S$-integers of a number field $K$. Let $\mathcal M_0(\mathcal O)$ denote the set of smooth genus $0$ curves over $\operatorname{Spec} \mathcal O$, which we know is the set of smooth conics over $K$ with good reduction outside $S$. Darmon says that the results of Class Field Theory can be used to show that this set is finite, of cardinality $2^{\#S + r - 1}$, where $r$ is the number of real places of $K$. Why is this true/where can I read more about it?

Answer 1

This follows from 3 facts:

1. Conics over $\newcommand{\O}{\mathcal{O}} \O$ are in bijection with quaternion algebras over $\O$.

2. The Brauer group $\newcommand{\Br}{\operatorname{Br}} \Br(F) \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}}$ of a field $F$ classifies central simple algebras over $F$ up to similarlity, and when $F$ is a local or global field, its $2$-torsion $\Br(F)[2]$ consists exactly of the similarity classes of quaternion algebras over $F$.

3. The fundamental exact sequence of global class field theory states, that, given any nonempty set of finite places $S$ of $K$, the following sequence is exact

\begin{align} \label{fundamental} 0 \to \Br(\O_{K,S}) \to \bigoplus_{v \ \in\ S\ \cup\ \Sigma_K^\infty} \operatorname{Br}(K_v) \overset{\operatorname{inv}}\longrightarrow \mathbb{Q}/\mathbb{Z} \to 0 \, , \tag{$*$} \end{align} where $\Sigma_K^\infty$ is the set of infinite places of $K$.

We will obtain the desired result by taking the $2$-torsion of the above exact sequence. Note that $$ \frac{\Q}{\Z}[2] = \frac{\frac{1}{2} \Z}{\Z} \cong \Z/2\Z \, . $$ Now, over an algebraically closed field $L$, every quaternion algebra splits, i.e., is isomorphic to the matrix algebra $\newcommand{\M}{\operatorname{M}} \M_2(L)$. Over $\newcommand{\R}{\mathbb{R}} \R$, there are exactly two isomorphism classes of quaternion algebras: the matrix algebra $\M_2(\R)$ and the Hamiltonian quaternions $\newcommand{\H}{\mathbb{H}} \H$. Similarly, over a nonarchimedean local field $F$, there is a unique division quaternion algebra, so together with the matrix algebra $\M_2(F)$ there are again exactly 2 possibilities. (Cf., Voight's Quaternion Algebras, Theorems 12.3.2 and 13.3.11.) Thus

\begin{align*} \Br(K_v)[2] \cong \begin{cases} 0 & \text{if $v \in \Sigma_K^\infty$ is complex}\\ \mathbb{F}_2 & \text{if $v \in \Sigma_K^\infty$ is real}\\ \mathbb{F}_2 & \text{if $v$ is finite}. \end{cases} \end{align*}

Thus, taking $2 \newcommand{\F}{\mathbb{F}}$-torsion of (\ref{fundamental}), we obtain the exact sequence \begin{align*} 0 \to \Br(\O_{K,S})[2] \to \mathbb{F}_2^{r + \#S} \to \mathbb{F}_2 \to 0 \, , \end{align*} where $r$ is the number of real places of $K$. (The sequence remains exact on the right because the division quaternion algebra over $K_v$ for $v \in S$ has nonzero Brauer invariant.) Then \begin{align*} 0 &= \dim_{\F_2}(\Br(\O_{K,S})[2]) - \dim_{\F_2}(\mathbb{F}_2^{r + \#S}) + \dim_{\F_2}(\mathbb{F}_2) = \dim_{\F_2}(\Br(\O_{K,S})[2]) - (r + \#S) + 1 \end{align*} so $$ \dim_{\F_2}(\Br(\O_{K,S})[2]) = r + \#S - 1 \, , $$ as desired.

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Some comments and references about the 3 facts above.

1. This is given (over a field, anyway) in Corollary 5.5.2 of Voight’s book. One direction of the bijection is easily stated. Given a quaternion algebra $B$ over a field $F$ defined by $i^2 = a$ and $j^2 = b$ for some $a,b \in F$, the reduced norm of a generic pure quaternion $xi + yj + zij$ is \begin{align*} \operatorname{nrd}(xi + yj + zij) &= (xi + yj + zij)(-xi - yj - zij) = -(xi + yj + zij)^2\\ &= -a x^2 - b y^2 + ab z^2 \, . \end{align*} The vanishing locus $-a x^2 - b y^2 + ab z^2 = 0$ of this quadratic form in $\mathbb{P}^2$ is a plane conic.

2. See Chapter 4 of Milne’s Class Field Theory, especially section IV.4. The Brauer group of a field $F$ can also be defined in terms of Galois cohomology. Letting $\overline{F}$ be a separable closure of $F$ and $G_F = \operatorname{Gal}(\overline{F}/F)$, then $\Br(F) \cong H^2(G_F, \overline{F}^\times)$.

3. See Theorem VIII.4.2 of Milne or Theorem 8.1.17 of Neukirch, Schmidt, and Wingberg’s Cohomology of Number Fields. These technically only state the result for a global field, but you should be able to get the result for a ring of $S$-integers by specializing the more general result stated using the idèles in Corollary 8.1.16 of Neukirch et al.