Which sequences of $\pm 1$ can appear as the image of a Hilbert symbol

Question

In his book A Course in Arithmetic, Serre adresses the question of which sequences $\pm 1$ can appear as the image of a Hilbert symbol. By this I mean the following: Let $V$ denote the set of places of $\mathbb{Q}$ and let $(a,b)_v$ denote the Hilbert symbol of two numbers $a,b \in \mathbb{Q}^{\times}$ at a place $v \in V$:

Theorem. Let $(a_i)_{i \in I}$ be a finite family of elements in $\mathbb{Q}^{\times}$ and let $(\varepsilon_{i,v})_{i\in I, v \in V}$ be a family of numbers equal to $\pm 1$. In order that there exists $x \in \mathbb{Q}^{\times}$ such that $(a_i,x)_v = \varepsilon_{i,v}$ for all $i \in I$ and all $v \in V$, it is necessary and sufficient that the following conditions be satisfied:

1. Almost all the $\varepsilon_{i,v}$ are equal to $1$.
2. For all $i \in I$ we have $\prod_{v \in V} \varepsilon_{i,v}=1$ (i.e. the sequence always satisfies Hilbert reciprocity).
3. For all $v \in V$ there exists $x_v \in \mathbb{Q}_{\nu}^{\times}$ such that $(a_i,x_v)_v = \varepsilon_{i,v}$ for all $i \in I$.

The above is theorem 4 from section 2.2 of chapter 3 of ibid.

I have two questions concerning this:

1. Though I have searched quite extensively, I have not been able to find another resource where the question of which binary sequences can appear as the image of a Hilbert symbol is treated. Is anyone aware of any other place where a similar theorem is proved?
2. We know that property (2.) above (Hilbert reciprocity) applies to any number field (in fact to any global field). Is there a theorem analogous to the above that applies to all number fields / global fields?

Many thanks.

Answer 1

While I can't answer your questions off the top of my head right now, I can reformulate the theorem from Serre in a more conceptual way. It is about whether a finite set of quaternion algebras over $\mathbf Q$ have a certain property.

For every field $K$ we have the concept of a quaternion algebra over $K$ (a $4$-dimensional central simple algebra with center $K$). One example is the matrix ing ${\rm M}_2(K)$ and every quaternion algebra over $K$ not isomorphic to that is a division ring. When $K$ does not have characteristic $2$ we can construct examples of quaternion algebras $(a,b)_K$ over $K$ for each $a, b \in K^\times$ in "the usual way". Unlike most fields (such as $\mathbf Q$ or other global fields), over each local field $\mathbf Q_v$ there are exactly two quaternion algebras up to isomorphism, such as ${\rm M}_2(\mathbf R) = (1,1)_\mathbf R$ and $\mathbf H = (-1,-1)_\mathbf R$ over $\mathbf R = \mathbf Q_{\infty}$. For $a, b \in \mathbf Q^\times$ and a place $v$, the Hilbert symbol $(a,b)_v$ being $1$ or $-1$ is equivalent to the quaternion algebra $(a,b)_{\mathbf Q_v}$ over $\mathbf Q_v$ being trivial (that is, being isomorphic to ${\rm M}_2(\mathbf Q_v)$) or nontrivial. A quaternion algebra over $K$ has order $1$ or $2$ in the Brauer group ${\rm Br}(K)$, although the converse is not true for general fields: a tensor product of two quaternion algebras over $K$ might not be equivalent to a quaternion algebra over $K$ in ${\rm Br}(K)$. When $K$ is a global field or a local field, however, ${\rm Br}(K)[2]$ is represented by quaternion algebras over $K$.

From class field theory there is a fundamental exact sequence of Brauer groups. $$ 0 \to {\rm Br}(\mathbf Q) \to \bigoplus_v {\rm Br}(\mathbf Q_v) \to \mathbf Q/\mathbf Z \to 0, $$ where the map out of ${\rm Br}(\mathbf Q)$ is $[A] \mapsto ([\mathbf Q_v \otimes_{\mathbf Q} A])_v$ and the map out of the direct sum is addition of components where we use the standard isomorphism ${\rm Br}(\mathbf Q_v) \to \mathbf Q/\mathbf Z$ for finite $v$ and ${\rm Br}(\mathbf Q_v) \to (1/2)\mathbf Z/\mathbf Z$ for $v = \infty$. For quaternion algebras we only need to think about the subgroup of elements killed by $2$: $$ 0 \to {\rm Br}(\mathbf Q)[2] \to \bigoplus_v {\rm Br}(\mathbf Q_v)[2] \to (1/2)\mathbf Z/\mathbf Z \to 0. $$

With this background in mind, let's describe the data of all the signs $\varepsilon_{i,v}$ in terms of quaternion algebras over $\mathbf Q$.

Properties (1) and (2) in the theorem from Serre's book are about the signs $\varepsilon_{i,v}$, not about the numbers $a_i$ -- the $a_i$ only show up in property (3).

Claim: for each $i$, properties (1) and (2) are equivalent to the data of a quaternion algebra over $\mathbf Q$.

To prove that claim, interpret the sign $\varepsilon_{i,v} \in \{\pm 1\}$ additively as an element of the group $(1/2)\mathbf Z/\mathbf Z \cong {\rm Br}(\mathbf Q_v)[2]$ (groups of order $2$ are isomorphic in only one way). By the second exact sequence with Brauer groups above, properties (1) and (2) are equivalent to saying that for each $i \in I$, the signs $\varepsilon_{i,\infty}, \varepsilon_{i,2}, \varepsilon_{i,3},\ldots,\varepsilon_{i,v}, \ldots$ correspond to a quaternion algebra $Q_i$ over $\mathbf Q$: property (1) says the sequence $(\varepsilon_{i,v})_v$ running over all $v$ belongs to $\bigoplus_v {\rm Br}(\mathbf Q_v)[2]$ and property (2) says this sequence is in the kernel of the map to $(1/2)\mathbf Z/\mathbf Z$. That kernel is the image of ${\rm Br}(\mathbf Q)[2]$ by exactness at the direct sum, so there is some quaternion algebra $Q_i$ over $\mathbf Q$ such that the quaternion algebra $\mathbf Q_v \otimes_\mathbf Q Q_i$ over $\mathbf Q_v$ is trivial or nontrivial according to $\varepsilon_{i,v}$ being $1$ or $-1$.

We can in fact say something stronger because the sequence is exact at ${\rm Br}(\mathbf Q)[2]$: $Q_i$ is unique up to isomorphism. In other words, for each $i \in I$ we can replace the signs $\varepsilon_{i,v}$ for all $v$ and properties (1) and (2) for that $i$ by a single (unique up to isomorphism) quaternion algebra $Q_i$ over $\mathbf Q$. So the theorem in Serre's book can be reformulated as being about a finite set of quaternion algebras $Q_i$ over $\mathbf Q$ and finite set of nonzero rational numbers $a_i$.

In the theorem from Serre's book, the condition $(a_i,x)_v = \varepsilon_{i,v}$ about a Hilbert symbol is equivalent to saying $(a_i,x)_{\mathbf Q_v} \cong \mathbf Q_v \otimes_{\mathbf Q} Q_i$ as quaternion algebras over $\mathbf Q_v$, so the rational quaternion algebras $(a_i,x)_\mathbf Q$ and $Q$ are "locally isomorphic everywhere". Two quaternion algebras $Q$ and $Q'$ over $\mathbf Q$ are isomorphic if and only if $\mathbf Q_v \otimes_\mathbf Q Q$ and $\mathbf Q_v \otimes_\mathbf Q Q'$ are isomorphic quaternion algebras over $\mathbf Q_v$ for all $v$. Therefore saying $(a_i,x)_v = \varepsilon_{i,v}$ for all $v$ is equivalent to saying $(a_i,x)_\mathbf Q$ and $Q_i$ are isomorphic quaterion algebras over $\mathbf Q$.

Now I can present a reformulation of the theorem from Serre's book. I am going to write the abstract finite index set $I$ as $\{1,\ldots,n\}$ for concreteness.

$\mathbf{Theorem}$. For a finite set of quaternion algebras $Q_1, \ldots, Q_n$ over $\mathbf Q$ and a finite set of nonzero rational numbers $a_1, \ldots, a_n$, in order that there exists an $x \in \mathbf Q^\times$ such that $Q_i \cong (a_i,x)_\mathbf Q$ for $i = 1, \ldots, n$, it is necessary and sufficient that for each place $v$ of $\mathbf Q$ there is an $x_v \in \mathbf Q_v^\times$ such that $\mathbf Q_v \otimes_\mathbf Q Q_i \cong (a_i,x_v)_{\mathbf Q_v}$ for $i = 1, \ldots, n$.

It is very important that the numbers $a_1, \ldots, a_n$ are part of the initial data. If you are given finitely many quaternion algebras $Q_1, \ldots, Q_n$ over $\mathbf Q$ then there are nonzero rational numbers $a_1, \ldots, a_n, x$ such that $Q_i \cong (a_i,x)_\mathbf Q$ for all $i$. This property, where the $a_i$'s are part of the conclusion, is called "strong linkage" for quaternion algebras over $\mathbf Q$ or we say that every finite set of quaternion algebras over $\mathbf Q$ has a "common slot" (the number $x$ appearing in all of those isomorphisms is the "common slot" and we aren't insisting on control over the other slots $a_1, \ldots, a_n$). All global fields have strong linkage (for global fields of characteristic $2$ a different construction of concrete quaternion algebras is needed to give the term "slot" a meaning). In the theorem of Serre's book, the desired condition is not just a "common slot" $x$ but we want to have control over the other slot too by insisting $Q_i$ have first slot $a_i$. This distinction between whether the numbers $a_i$ appear in the hypothesis of the theorem (as in Serre) or in the conclusion (the field $\mathbf Q$ has "strong linkage" for its quaternion algebras) is reminiscent of the difference between continuity and uniform continuity of a function $[a,b] \to \mathbf R$ being the ordering of the logical quantifiers.

Hilbert symbols make sense over all global fields (they are a bit more subtle for characteristic $2$ global fields in terms of concrete formulas), so it is straightforward to extend the theorem from Serre's book in terms of Hilbert symbols or in terms of quaternion algebras to all all global fields, and surely that extension to all global fields is still true. Off the top of my head I do not have a reference for you.