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I am looking for a reference for the following fact.

Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such that $C(k_v) = \emptyset$ if and only if $v \in S$.

I need this fact in a paper I am writing. I know how to prove it using class field theory, and it is very well-known so I would not like to have to prove it again in my paper. If anybody knows a reference I could use, I would be most obligued. I tried looking in the "usual" texts on class field theory, but I could not find the statement I needed.

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  • $\begingroup$ You might try looking in a quadratic forms book. Saying that $C(k_{v}) = \emptyset$ is equivalent to saying that the ternary quadratic form $Q$ whose zero set is $C$ is anisotropic at $v$. The statement that you want is that the only restriction to constructing a quadratic form $Q/k$ given a choice of local reductions $Q_{v}$ is the product formula. $\endgroup$
    – Jeremy Rouse
    Commented Sep 24, 2014 at 13:42
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    $\begingroup$ It is surprising that no standard references on class field theory prove period = index over global fields. It is proved as Theorem 1.2.4.4 in the book Complex multiplication and lifting problems by Chai, Conrad, & Oort. So you can say "Since conics over a field $F$ are $F_s/F$-forms of $\mathbf{P}^1_F$, the set of isomorphism classes is in natural bijection with ${\rm{H}}^1(F, {\rm{PGL}}_2)$, which is the set of isomorphism classes of quaternion division algebras over $F$. For a global field $F$ these are the elements of order 2 in ${\rm{Br}}(F)$ (see [CCO, Thm. 1.2.4.4] for a proof)." $\endgroup$
    – user27920
    Commented Sep 24, 2014 at 17:12
  • $\begingroup$ @user52824: I agree, it is suprising. I will check out the reference, thanks! $\endgroup$
    – Daniel Loughran
    Commented Sep 24, 2014 at 18:41
  • $\begingroup$ I should have written "quaternion algebras" rather than "quaternion division algebras", and "2-torsion elements" rather than "elements of order 2" (corresponding to allowing a conic with a rational point, as one does when applying functoriality relative to the map from a global field to a completion). $\endgroup$
    – user27920
    Commented Sep 24, 2014 at 21:08
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    $\begingroup$ It's a special case of 72.1 in O'Meara's 1963 book (see also 71.19) and of VIII 6.12 in Milne's notes CFT. $\endgroup$
    – anon
    Commented Sep 27, 2014 at 23:51

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