What is the motivation behind the Hilbert Symbol?

Question

The Hilbert Symbol it superficially similar to the Legendre Symbol: it measures whether or not solutions to some polynomial exist. In the case of the Legendre Symbol it was clear for me that it is a tool to figure out what the ring $$ \frac{\mathbb{F}_p[x]}{(x^2 - a)} $$ looks like for some fixed $a$. In particular, it tells us if we have a finite field extension or $\mathbb{F}_p\times\mathbb{F}_p$. Is there a similar story for the Hilbert Symbol which I am not seeing? The only way I'm able to understand this is whether or not $$ |\text{Hom}_{\textbf{Cring}}(\mathbb{Z}[x,y,z]/(ax^2 + by^2 -z^2),\mathbb{Z})| > 1 $$

For reference, I am using Serre's definition as $$ (a,b) = \begin{cases} 1 &\text{ if } ax^2 + by^2 = z^2 \text{ for some } (x,y,z)\neq (0,0,0) \\ -1 &\text{ otherwise } \end{cases} $$

Answer 1

The following relates more to Number Theory than Geometry, but for what it's worth the Hilbert symbol can be associated with an element of order 2 in the Brauer group of a number field $K$ thus representing whether a quaternion algebra in this case generated by say $\alpha, \beta, \text{ and } \alpha\beta$ where $\alpha^2=a$, $\beta^2 = b$, and $\alpha\beta = -\beta\alpha$ splits or not. Also there is a brief but nice discussion in Washington Cyclotomic Fields associating the Hilbert Symbol with an element of a second cohomology group (so again really just $Br(K)$). Finally there is some material in the "Exercises" by John Tate at the end of Cassels and Froehlich Algebraic Number Theory which develop many of the number theoretic properties of the Hilbert symbol in a fairly illustrative fashion.

Answer 2

The Hilbert symbol is a local object, attached to a local field $K_v$, i.e. the completion of a number field $K$ w.r.t. a $p$-adic valuation $v$. Its main motivation: the so called explicit reciprocity laws in class field theory.

Let us first recall how the local-global principle comes into play in CFT. The classical ("pre-world war II") global CFT built by Takagi et al. rests on the fundamental notions of the modulus $\mathcal M$ and the group $C_{\mathcal M}$ of generalized congruence classes attached to a finite abelian extension $L/K$ of number fields, but one big technical drawback of the modulii, which vary with $L$, is that they do not behave well w.r.t. the compositum of extensions over $K$. In the cohomological ("post world war II") CFT, the local-global principle takes root in algebraic number theory (Hasse dixit) with Chevalley's notion of idèles, which allows to start with CFT over local fields, and then "glue" the local results together to get CFT over number fields. The point is that the local reciprocity law for an abelian extension $L_w /K_v$ simply states that $K_{v}^*/NL_{w}^* \cong Gal(L_w /K_v)$, where $N$ denotes the norm of $L_w /K_v$, and the global reciprocity just as simply reads $C_K /NC_L \cong Gal(L/K)$, where $N$ is the norm and $C_F$ is the idèle class group of $F$. The above canonical isomorphisms are called (local or global) reciprocity isomorphisms. As written, they are compatible with inverse limits and allow to define reciprocity maps $ K_{v}^*\to G_{v}^{ab}$ and $C_K \to G_{K}^{ab}$, where $(.)^{ab}$ denotes the Galois group of the maximal abelian extension.

From now on, for simplicity, $K$ will denote a fixed local field. It remains to make as explicit as possible the local reciprocity map $\theta : K^* \to G_{K}^{ab}$. Because of the cohomological formulation of local CFT (= essentially the determination of the Brauer group of $K$), it is more convenient to describe $\theta$ by duality, more precisely, by giving all the values $\chi (\theta (b))$ for all $b \in K^*$ and all $1$-dimensional characters of $ G$ . This dual approach requires to appeal to Kummer theory, i.e. to assume that $K$ contains a group $\mu_n$ of $n$-th roots of unity. Then, for $a, b \in K^*$, the $n$-th Hilbert symbol is defined by $(a,b) := \theta (b)(\sqrt [n] {a} )/ \sqrt [n] {a} \in \mu_n$. The main functorial properties of the Hilbert symbol are given in exercise 2 of Cassels-Fröhlich. An explicit form of the Hilbert symbol is called an explicit reciprocity law. Examples : - if the residual characteristic $p$ of $K$ does not divide $n$, the $tame$ Hilbert symbol is explicitly known (ex. 2.8 of C-F) - this is not so for the wild Hilbert symbol (i.e. when $p$ divides $n$), even though many particular cases are known ; this means that explicit CFT is not known in general - for $n =2$ and $k = \mathbf Q_p$, everything is explicitly known ; this amounts to answer to the question of the existence of zeroes of the quadratic form $ax^2 + by^2 – z^2$ (your hint).

NB. The main relations verified by the Hilbert symbols (especially the so called Steinberg relation {a, 1-a} = 0) can be used in Milnor K-theory to characterize the group $K_{2}^M (F)$ of a field $F$ (Matsumoto’s theorem). Tate’s calculation of $K_{2}^M (\mathbf Q)$ can be interpreted as being essentially equivalent to the quadratic reciprocity law. A surprising isomorphism between $K_{n}^M (F) /r$ (where $r$ is any integer invertible in $F$) and the Galois cohomology of $F$ with coefficients $\mathbf Z/r \mathbf Z (n)$, conjectured by Milnor-Bloch-Kato, has been recently proved by Voevodsky et al. In the case of a number field $F$, this allowed to show deep results about the arithmetic meaning of special values of the Dedekind zeta function $\zeta _F (s)$.