Hilbert's twelfth problem

Let $K$ be a number field and fix an algebraic closure $\bar K$. Let $K^{\mathrm{ab}}\subset\bar K$ be the maximal abelian extension of $K$. For a modulus $\mathfrak m$ of $K$, let $K(\mathfrak m)$ be the ray class field of conductor dividing $\mathfrak m$. Class field theory gives

$$\operatorname{Gal}(K(\mathfrak m)/K)\cong\mathrm{Cl}_{\mathfrak m}(K),\qquad K^{\mathrm{ab}}=\bigcup_{\mathfrak m}K(\mathfrak m),$$

where $\mathrm{Cl}_{\mathfrak m}(K)$ is the ray class group. Class field theory proves these fields exist, but it does not produce generators for them.

Two cases have explicit answers.

• $K=\mathbb Q$ (Kronecker–Weber). $\mathbb Q(\mathfrak m)$ is generated by values of $e^{2\pi i z}$ at rational points:

$$\mathbb Q^{\mathrm{ab}}=\mathbb Q\bigl(e^{2\pi i/m}: m\ge1\bigr).$$

• $K$ imaginary quadratic (complex multiplication). Let $E$ be an elliptic curve over $\mathbb C$ with $\operatorname{End}(E)\cong\mathcal O_K$, and let $h$ be a Weber function on $E$ (for example, the $x$-coordinate in a Weierstrass model, suitably normalized). Then

$$K^{\mathrm{ab}}=K\bigl(j(E),\,h(P): P\in E_{\mathrm{tors}}\bigr),$$

and $j(E)$ and $h(P)$ are special values of the modular function $j$ and of elliptic functions.

Problem (Hilbert's twelfth problem). For every number field $K$, give an explicit construction of $K^{\mathrm{ab}}$ of the same kind. That is, specify analytic functions attached to $K$, defined without reference to class field theory, and explicit arguments depending on $K$ and $\mathfrak m$. Their special values must be computable to any precision by an algorithm from $K$ and $\mathfrak m$, and must generate $K(\mathfrak m)$ over $K$ for every modulus $\mathfrak m$. A solution must prove the generation statement.

Totally real fields have an unconditional answer by $p$-adic formulas (Dasgupta–Kakde, below). The problem is open for number fields that are neither totally real nor imaginary quadratic. These include CM fields of degree at least $4$ and fields that are neither totally real nor CM. A construction for totally real fields by archimedean special values, as predicted by Stark's conjectures, is also open [3, 6].

Background

Hilbert posed the problem as the twelfth of his 1900 list [1]. He asked how Kronecker's theorem on abelian extensions of $\mathbb Q$ and the theory of complex multiplication for imaginary quadratic fields extend to an arbitrary algebraic base field. Kronecker called the imaginary quadratic statement his "liebster Jugendtraum".

For a CM field $K$ of degree at least $4$, the theory of complex multiplication of abelian varieties, due to Shimura and Taniyama, produces explicit abelian extensions of the reflex field from values of Siegel modular functions at CM points [2]. These extensions do not in general exhaust the maximal abelian extension.

Stark's conjectures [3] predict that leading terms at $s=0$ of $L$-functions of abelian extensions are logarithms of units, the Stark units, in those extensions. For a totally real base field, the rank-one Stark units should generate abelian extensions, which would give an archimedean solution. Gross formulated $p$-adic analogues, the Gross–Stark conjectures. Darmon and Vonk proposed a theory of "real multiplication" using rigid meromorphic cocycles for real quadratic fields [4].