Remark on function field analogy

TL;DR

This paper explores the analogy between number fields and function fields, establishing an isomorphism between their Hilbert class fields and using K-theory and birational geometry to construct explicit generators.

Contribution

It introduces a novel isomorphism linking Hilbert class fields of class number one to function fields over algebraic curves, utilizing K-theory and birational geometry techniques.

Findings

Established an isomorphism between Hilbert class fields and function fields over curves.

Constructed explicit generators of Hilbert class fields from Drinfeld module torsion submodules.

Applied K-theory of Serre C*-algebras and birational geometry in the proof.

Abstract

We study the analogy between number fields and function fields in one variable over finite fields. The main result is an isomorphism between the Hilbert class fields of class number one and a family of the function fields $\mathbf{F}_q(C)$ over a desingularized algebraic curve $C$. Our proof is based on the K-theory of the Serre $C^*$-algebras and birational geometry of the curve $C$. We apply the isomorphism to construct explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module.

Equations

$$B(V,\mathcal{L},\sigma )=\underset{i\ge 0}{⨁}{H}^0\left(V,\mathcal{L}\otimes {\mathcal{L}}^{\sigma }\otimes \cdots \otimes {\mathcal{L}}^{{\sigma }^{i-1}}\right)$$

$$Mod(B(V,\mathcal{L},\sigma ))/Tors\cong Coh(V),$$

$$\rho :R[t,t^{-1};\sigma ]⟶\mathcal{B}(\mathcal{H}).$$

$$|{\mathbf{F}}_q[T]/g{\mathbf{F}}_q[T]|=q^{\mathrm{deg}g}.$$

$$f(x,y)=0,$$

$$\tau a=a^p\tau ,\forall a\in k.$$

$$\rho :A\to k⟨\tau ⟩,$$

$$JacC={\mathbf{C}}^g/\Lambda ,$$

$$f(JacC)={\mathbf{C}}^g/{\Lambda }_f,$$

$${\Lambda }_f=\omega \Lambda ,$$

$$(\alpha O_{\mathcal{H}(k)},\alpha O_{\mathcal{H}(k)}^{+},[u])$$

$${\Lambda }_f\subseteq \Lambda ,$$

$$R_f=\beta O_{{\mathbf{F}}_p(C)}.$$

$$O_{\mathcal{H}(k)}\to O_{{\mathbf{F}}_p(C)}$$

$$\mathcal{H}(k)\subset {\mathcal{H}}^2(k)\subset \cdots \subset {\mathcal{H}}^{m-1}(k)\cong {\mathcal{H}}^m(k),$$

$${\mathbf{F}}_p(C)\subset {\mathbf{F}}_{p^2}(C)\subset \cdots \subset {\mathbf{F}}_{p^{m-1}}(C)\cong {\mathbf{F}}_{p^{m-1}}(C).$$

$$j(\tau )=\frac{1}{q}+744+196884q+\dots ,whereq=e^{2\pi i\tau }.$$

$$\mathcal{H}(k)=k\left(j(\sqrt{-d})\right).$$

$$E_{\mu }(x)=\mu e^{2\pi x},$$

$$\mathcal{H}(k)=k\left(e^{2\pi i\sqrt{d}+\log \log \epsilon }\right)$$

$$\mathcal{H}(k)=k\left(e^{2\pi \theta +\log \log \epsilon }\right).$$

Full text

1. Introduction

It was first observed in [Dedekind & Weber 1882] [3] that the number fields and the fields of rational functions in one variable are related. For example, both fields are the Dedekind domains, i.e. every ideal factors into a product of the prime ideals. Unlike the number fields, the group of units of the function fields is no longer finitely generated. However, the behavior is back to normal for the rational functions over finite (Galois) fields. In fact, it was proved in [Artin & Whaples 1945] [1] that the two fields can be unified as global fields. This led to a fast progress driven by solution of the open problems on either side by recasting them in terms of the other [van der Geer, Moonen & Schoof 2005] [4]. The successful proof of an analog of the Riemann Hypothesis (Hasse & Weil) and the geometric Langlands Conjectures (Drinfeld & Lafforgue) are few examples. When and if such an analogy can be upgraded to an isomorphism is an interesting open problem.

The aim of our note is a solution of the above problem for the Hilbert class fields of class number one, i.e. the principal ideal domains. Namely, we construct an isomorphism between such fields and the function fields $\mathbf{F}_{p^{m-1}}(C)$, where $C$ is a smooth algebraic curve obtained from the singular curve by the $m-1$ blow-ups (Theorem 1.1). To formalize our results, the following notation will be used.

Let $(x_{0},y_{0})$ be an isolated singularity of algebraic curve $f(x,y)=0$ given by a polynomial $f\in\mathbf{Z}[x,y]$. Consider an arithmetic scheme $Spec~{}\mathbf{Z}[x,y]/(f)$ and its natural morphism to $Spec~{}\mathbf{Z}$. Let $\mathscr{P}\in Spec~{}\mathbf{Z}[x,y]/(f)$ be the maximal ideal corresponding to the point $(x_{0},y_{0})$ lying over $(p)\in Spec~{}\mathbf{Z}$ for a prime $p\in\mathbf{Z}$.

Recall that the singularity $(x_{0},y_{0})$ can be resolved by a sequence of the blow-ups $\{\epsilon_{i}~{}|~{}1\leq i\leq m-1\}$, such that $f(x,y)=0$ defines a smooth algebraic curve $C$, see e.g. [Grassi 2009] [5, Example-Definition 10] or [Kollár 2007] [6, Section 1.7]. We denote by $\mathbf{F}_{p^{m-1}}(C)$ the geometric extension of the field of rational functions in one variable over the Galois field $\mathbf{F}_{p^{m-1}}$ [Rosen 2002] [10, Chapter 7].

On the other hand, the K-theory of the Serre $C^{*}$-algebra of an algebraic curve gives rise to a number field $k$ [8, Section 5.3.1]. Moreover, if the curve was singular, then its blow-up corresponds to the Hilbert class field $\mathscr{H}(k)$, i.e. the maximal abelian unramified extension of $k$. This is an implication of a general result for the birational maps between the algebraic surfaces [9, Theorem 1.3]. Let $\mathscr{H}^{i+1}(k):=\mathscr{H}(\mathscr{H}^{i}(k))$. If $C$ is a curve desingularized after $m-1$ blow-ups, then the number field $\mathscr{H}^{m-1}(k)$ must have class number one, since each birational map between smooth curves is an isomorphism. Our main result can be formulated as follows.

Theorem 1.1**.**

$\mathscr{H}^{m-1}(k)\cong\mathbf{F}_{q}(C),$* where $q=p^{m-1}$.*

The paper is organized as follows. A brief review of the preliminary facts is given in Section 2. Theorem 1.1 is proved in Section 3. We consider an application of Theorem 1.1 in Section 4, where explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module are constructed.

2. Preliminaries

This section contains a brief review of the Serre $C^{*}$-algebras, birational geometry and function fields. We refer the reader to [Grassi 2009] [5], [Kollár 2007] [6], [Rosen 2002] [10] and [Stafford & van den Bergh 2001] [11] for a detailed account.

2.1. Serre $C^{*}$-algebras ([8], [11])

Let $V$ be a projective variety over the field $k$. Denote by $\mathcal{L}$ an invertible sheaf of the linear forms on $V$. If $\sigma$ is an automorphism of $V$, then the pullback of $\mathcal{L}$ along $\sigma$ will be denoted by $\mathcal{L}^{\sigma}$, i.e. $\mathcal{L}^{\sigma}(U):=\mathcal{L}(\sigma U)$ for every $U\subset V$. The graded $k$-algebra

[TABLE]

is called a twisted homogeneous coordinate ring of $V$ [Stafford & van den Bergh 2001] [11]. Such a ring is always non-commutative, unless the automorphism $\sigma$ is trivial. A multiplication of sections of $B(V,\mathcal{L},\sigma)=\oplus_{i=1}^{\infty}B_{i}$ is defined by the rule $ab=a\otimes b$, where $a\in B_{m}$ and $b\in B_{n}$. An invertible sheaf $\mathcal{L}$ on $V$ is called $\sigma$-ample, if for every coherent sheaf $\mathcal{F}$ on $V$, the cohomology group $H^{k}(V,~{}\mathcal{L}\otimes\mathcal{L}^{\sigma}\otimes\dots\otimes\mathcal{L}^{\sigma^{n-1}}\otimes\mathcal{F})$ vanishes for $k>0$ and $n>>0$. If $\mathcal{L}$ is a $\sigma$-ample invertible sheaf on $V$, then

[TABLE]

where $Mod$ is the category of graded left modules over the ring $B(V,\mathcal{L},\sigma)$, $Tors$ is the full subcategory of $Mod$ of the torsion modules and $Coh$ is the category of quasi-coherent sheaves on a scheme $V$. In other words, the $B(V,\mathcal{L},\sigma)$ is a coordinate ring of the variety $V$.

Let $R$ be a commutative graded ring, such that $V=Proj~{}(R)$. Denote by $R[t,t^{-1};\sigma]$ the ring of skew Laurent polynomials defined by the commutation relation $b^{\sigma}t=tb$ for all $b\in R$, where $b^{\sigma}$ is the image of $b$ under automorphism $\sigma$. It is known, that $R[t,t^{-1};\sigma]\cong B(V,\mathcal{L},\sigma)$ [Stafford & van den Bergh 2001] [11, Section 5]. Let $\mathcal{H}$ be a Hilbert space and $\mathscr{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. For a ring of skew Laurent polynomials $R[t,t^{-1};\sigma]$, consider a homomorphism:

[TABLE]

Recall that $\mathscr{B}(\mathcal{H})$ is endowed with a $\ast$-involution; the involution comes from the scalar product on the Hilbert space $\mathcal{H}$. We shall call representation (2.3) $\ast$-coherent, if (i) $\rho(t)$ and $\rho(t^{-1})$ are unitary operators, such that $\rho^{*}(t)=\rho(t^{-1})$ and (ii) for all $b\in R$ it holds $(\rho^{*}(b))^{\sigma(\rho)}=\rho^{*}(b^{\sigma})$, where $\sigma(\rho)$ is an automorphism of $\rho(R)$ induced by $\sigma$. Whenever $B=R[t,t^{-1};\sigma]$ admits a $\ast$-coherent representation, $\rho(B)$ is a $\ast$-algebra. The norm closure of $\rho(B)$ is a $C^{*}$-algebra denoted by $\mathscr{A}_{V}$. We refer to $\mathscr{A}_{V}$ as the Serre $C^{*}$-algebra of $V$ [8, Section 5.3.1].

Let $(K_{0}(\mathscr{A}_{V}),K_{0}^{+}(\mathscr{A}_{V}),u)$ be the dimension group of the $C^{*}$-algebra $\mathscr{A}_{V}$ [Blackadar 1986] [2, Sections 6.3 and 7.3]. Such a group defines the Serre $C^{*}$-algebra $\mathscr{A}_{V}$ up to an isomorphism. Moreover, if $V$ is defined over a number field, then its dimension group is order-isomorphic to the dimension group $(O_{K},O_{K}^{+},[u])$, where $O_{K}$ is the ring of integers of a number field $K$, $O_{K}^{+}$ the additive semigroup of positive integers for a real embedding $O_{K}\hookrightarrow\mathbf{R}$ and $[u]$ is a unit of $O_{K}$ [8, pp. 194-195].

2.2. Birational geometry ([5], [9])

Let $V$ be a projective variety over the field $k$. The map $\phi:V\dashrightarrow V^{\prime}$ is rational, if it is represented by rational functions. The rational maps cannot be composed unless they are dominant, i.e. the image of $\phi$ is Zariski dense in $V^{\prime}$. The map $\phi$ is called birational, if the inverse $\phi^{-1}$ is a rational map.

A birational map $\epsilon:V\dashrightarrow V^{\prime}$ is called a blow-up, if it is defined everywhere except for a point $p\in V$ and an exceptional divisor $E\subset V^{\prime}$, such that $\epsilon^{-1}(V)=p$. Every birational map $\phi:V\dashrightarrow V^{\prime}$ is composition of a finite number of the blow-ups, i.e. $\phi=\epsilon_{1}\circ\dots\circ\epsilon_{k}$.

The variety $V$ is called a minimal model, if any birational map $V\dashrightarrow V^{\prime}$ is an isomorphism. In case $V$ is a curve or a surface, it is known that every birational class has a smooth minimal model. Namely, if $S$ is a surface, the minimal models exist and are unique unless $S$ is a ruled surface. By the Castelnuovo Theorem, the smooth projective surface $S$ is a minimal model if and only if $S$ does not contain a rational curves $C$ with the self-intersection index $-1$. If $C$ is a curve, then each birational map is an isomorphism, i.e. $C$ is the minimal model.

It is still not known if any variety has a minimal model; it has been shown that in dimension 3 and higher, the minimal model is not necessarily smooth. The minimal model always exists for certain types of varieties.

The birational map $V\dashrightarrow V^{\prime}$ induces a morphism $\mathscr{A}_{V}\dashrightarrow\mathscr{A}_{V^{\prime}}$ between the corresponding Serre $C^{*}$-algebras, see Section 2.1. The following result describes such a morphism for the varieties $V$ over number fields.

Theorem 2.1**.**

([9, Theorem 1.3])* The map $\mathscr{A}_{V}\dashrightarrow\mathscr{A}_{V^{\prime}}$ gives rise to an extension of the corresponding dimension groups $(O_{k},O_{k}^{+},[u])\dashrightarrow(O_{\mathscr{H}(k)},O_{\mathscr{H}(k)}^{+},[u])$, where $\mathscr{H}(k)$ is the Hilbert class field of $k$, i.e. the maximal abelian unramified extension of the field $k$.*

Remark 2.2*.*

Theorem 2.1 was proved for $V$ being a surface $S$ or a curve $C$ [9]. The proof extends to the higher dimensions, but the topic lies outside the scope of our paper. In what follows, we use Theorem 2.1 only for the case $V$ being a curve $C$.

2.3. Function fields ([10])

We briefly review the fields of rational functions in one variable over the finite fields $\mathbf{F}_{q}$.

2.3.1. The field $\mathbf{F}_{q}(T)$

Let $p$ be a prime number and $q=p^{n}$ for a positive integer $n$. Denote by $\mathbf{F}_{q}$ the field with $q$ elements, i.e. the field $\mathbf{Z}/p^{n}\mathbf{Z}$. Let $\mathbf{F}_{q}[T]$ be the polynomial ring over $\mathbf{F}_{q}$ and let $\mathbf{F}_{q}(T)$ be the field of fractions of $\mathbf{F}_{q}[T]$. The ring $\mathbf{F}_{q}[T]$ has many properties in common with the ring of integers $\mathbf{Z}$. Both are principal ideal domains, both have a finite group of units, and every residue class ring modulo a non-zero ideal is finite. Namely, if $\deg g$ is the degree of the polynomial $g\in\mathbf{F}_{q}[T]$, then

[TABLE]

2.3.2. Geometric extensions

Consider an extension of the field $\mathbf{F}_{q}(T)$ of the transcendence degree one. Any such has the form $\mathbf{F}_{q}(T)(y)$, where $y$ satisfies a polynomial equation:

[TABLE]

where $x=T$. In other words, the $\mathbf{F}_{q}(T)(y)$ is a field of the rational function on the algebraic curve (2.5). The extension $\mathbf{F}_{q}(T)(y)$ is called geometric, since it is independent of the extensions of the field of constants $\mathbf{F}_{q}$. The smooth algebraic curve (2.5) is denoted by $C$ and the corresponding geometric extension can be written as $\mathbf{F}_{q}(C)$. There exists a bijection between all geometric extension of the field $\mathbf{F}_{q}(T)$ and algebraic curves $C$ given by equation (2.5) for a polynomial $f\in\mathbf{Z}[x,y]$.

2.3.3. Drinfeld module

The explicit class field theory for the function fields is strikingly simpler than for the number fields. The generators of the maximal abelian unramified extensions (i.e. the Hilbert class fields) are constructed using the concept of the Drinfeld module. Roughly speaking, such a module is an analog of the exponential function and a generalization of the Carlitz module. Nothing similar exists at the number fields side, where the explicit generators of abelian extensions are known only for the field of rationals (Kronecker-Weber theorem) and imaginary quadratic number fields (complex multiplication). Below we give some details on the Drinfeld modules.

Let $k$ be a field. A polynomial $f\in k[x]$ is said to be additive in the ring $k[x,y]$ if $f(x+y)=f(x)+f(y)$. If $char~{}k=p$, then it is verified directly that the polynomial $\tau(x)=x^{p}$ is additive. Moreover, each additive polynomial has the form $a_{0}x+a_{1}x^{p}+\dots+a_{r}x^{p^{r}}$. The set of all additive polynomials is closed under addition and composition operations. The corresponding ring is isomorphic to a ring $k\langle\tau\rangle$ of the non-commutative polynomials given by the commutation relation:

[TABLE]

Let $A=\mathbf{F}_{q}[T]$ and $k=\mathbf{F}_{q}(T)$. By the Drinfeld module one understands a homomorphism

[TABLE]

such that for all $a\in A$ the constant term of $\rho_{a}$ is $a$ and $\rho_{a}\not\in k$ for at least one $a\in A$. Consider a torsion module $\Lambda_{\rho}[a]=\{\lambda\in\bar{k}~{}|~{}\rho_{a}(\lambda)=0\}$. The following result describes the simplest case of the explicit class field theory for the function fields.

Theorem 2.3**.**

For each non-zero $a\in A$ the function field $k\left(\Lambda_{\rho}[a]\right)$ is an abelian extension of $k$ whose Galois group is isomorphic to a subgroup of $\left(A/(a)\right)^{*}$.

3. Proof

For the sake of clarity, let us outline the main ideas. Consider the case $m=2$ of Theorem 1.1, while the rest is proved by induction. One starts with a non-singular algebraic curve $C$ given by the equation $f(x,y)=0$, where $f\in\mathbf{Z}[x,y]$. The curve is obtained by the blow-up $C_{sing}\buildrel p\over{\dashrightarrow}C$ of a singular curve $C_{sing}$ at the point corresponding to the ideal $(p)\in Spec~{}\mathbf{Z}$ under the map $Spec~{}\mathbf{Z}[x,y]/(f)\to Spec~{}\mathbf{Z}$. Since $C$ is a smooth curve given by a polynomial $f\in\mathbf{Z}[x,y]$, it defines a geometric extension $\mathbf{F}_{p}(C)$ of the function field $\mathbf{F}_{p}(T)$. In fact, such an extension is functorial, i.e. morphisms of $C$ correspond to the isomorphisms of the field $\mathbf{F}_{p}(C)$. On the other hand, the K-theory of the Serre $C^{*}$-algebra of the curve $C$ gives rise to a number field $\mathscr{H}(k)$ of the class number one, see Sections 2.1 and 2.2. Such an assignment is also functorial. Thus one gets a pair of the global fields $\mathbf{F}_{p}(C)$ and $\mathscr{H}(k)$ attached to curve $C$ (and its Jacobian variety $Jac~{}C$) as it is shown in the commutative diagram in Figure 1. Since $Jac~{}C$ is an abelian variety, the horizontal arrow in Figure 1 is realized by an isomorphism between the two fields. Let us pass to a detailed argument by splitting the proof in a series of lemmas.

Lemma 3.1**.**

Each isogeny of the Jacobian variety $Jac~{}C$ of curve $C$ gives rise to a multiplication $\alpha O_{\mathscr{H}(k)}$ ($\beta O_{\mathbf{F}_{p}(C)}$, resp.) of the ring of integers of the field $\mathscr{H}(k)$ ($\mathbf{F}_{p}(C)$, resp.) by an element $\alpha\in O_{\mathscr{H}(k)}$ ($\beta\in O_{\mathbf{F}_{p}(C)}$, resp.) Moreover, the degree of such isogeny is equal to the norm of the element $\alpha$ ($\beta$, resp.)

Proof.

(i) Let $C$ be an algebraic curve defining the function field $\mathbf{F}_{p}(C)$. Instead of $C$, we shall work with its Jacobian variety $Jac~{}C$, i.e. a complex torus

[TABLE]

where $g$ is the genus of $C$ and $\Lambda$ is a lattice of rank $2g$. Since $Jac~{}C$ defines (and is defined by) the curve $C$, one gets a commutative diagram in Figure 1.

(ii) Denote by $f$ an isogeny of the abelian variety $Jac~{}C$. Recall that the map $f$ defines a sub-lattice $\Lambda_{f}\subseteq\Lambda$ of a finite index, such that

[TABLE]

where the degree of $f$ is equal to the index of $\Lambda_{f}$ in the lattice $\Lambda$. We shall consider the action of isogeny $f$ on the fields $\mathscr{H}(k)$ and $\mathbf{F}_{p}(C)$, respectively.

(iii) Let $O_{\mathscr{H}(k)}$ be the ring of integers of the number field $\mathscr{H}(k)$. Denote by $f$ an isogeny of the Jacobian variety $Jac~{}C$. Let $(O_{\mathscr{H}(k)},O_{\mathscr{H}(k)}^{+},[u])$ be the dimension group of the corresponding Serre $C^{*}$-algebra, see Section 2.1 for the notation. Recall that the lattice $\Lambda$ in (3.1) has complex multiplication, i.e. the lattice in (3.2) can be written in the form:

[TABLE]

where $\omega$ is a complex number in a CM-field. On the other hand, it is known that the functor $Jac~{}C\to\mathscr{H}(k)$ transforms complex multiplication to a real multiplication, i.e. the dimension group of the variety (3.2) has the form:

[TABLE]

for an element $\alpha\in O_{\mathscr{H}(k)}$ [8, Section 1.4.1]. It follows from formulas (3.3) and (3.4) that each isogeny $f$ defines a correspondence $f\mapsto\alpha\in O_{\mathscr{H}(k)}$.

(iv) Let $O_{\mathbf{F}_{p}(C)}$ be the ring of integers of the function field $\mathbf{F}_{p}(C)$. Denote by $f$ an isogeny of the Jacobian variety $Jac~{}C$. Recall that the map $Jac~{}C\to\mathbf{F}_{p}(C)$ is functorial and therefore must preserve the inclusion relation of the corresponding lattices:

[TABLE]

see (3.1) and (3.2). In particular, the inclusion (3.5) gives rise to a subring $R_{f}\subseteq O_{\mathbf{F}_{p}(C)}$. It is easy to see, that the $R_{f}$ is an ideal of the ring $O_{\mathbf{F}_{p}(C)}$. Indeed, $\Lambda/\Lambda_{f}$ is an abelian group and therefore $\Lambda_{f}$ is the kernel of a homomorphism of $\Lambda\to\Lambda/\Lambda_{f}$. Thus the subring $R_{f}\subseteq O_{\mathbf{F}_{p}(C)}$ is the kernel of a ring homomorphism. In particular, the $R_{f}$ is an ideal of the ring $O_{\mathbf{F}_{p}(C)}$. But the ring $O_{\mathbf{F}_{p}(C)}$ was assumed to have the class number one, i.e. each ideal of $O_{\mathbf{F}_{p}(C)}$ is principal. Thus there exists an element $\beta\in O_{\mathbf{F}_{p}(C)}$, such that

[TABLE]

In other words, the isogeny $f$ in formula (3.6) gives rise to a correspondence $f\mapsto\beta\in O_{\mathbf{F}_{p}(C)}$.

(v) Finally, let us show that the degree of isometry $f$ is equal to the norm of $\alpha\in O_{\mathscr{H}(k)}$ ($\beta\in O_{\mathbf{F}_{p}(C)}$, resp.) Indeed, $deg~{}(f)=|\Lambda/\Lambda_{f}|=N(\alpha)$, where $N(\alpha)$ is the norm of the algebraic integer $\alpha\in O_{\mathscr{H}(k)}$. The argument for the function fields is similar, where the norm $N(\beta)$ is calculated by the formula (2.4), i.e. $N(\beta)=p^{deg~{}\beta}$.

Lemma 3.1 is proved. ∎

Lemma 3.2**.**

$O_{\mathscr{H}(k)}\cong O_{\mathbf{F}_{p}(C)}.$**

Proof.

(i) Lemma 3.1 says that each isogeny $f$ of $Jac~{}C$ defines the elements $\alpha\in O_{\mathscr{H}(k)}$ and $\beta\in O_{\mathbf{F}_{p}(C)}$. Consider a map:

[TABLE]

given by the formula $\alpha\mapsto\beta$ as $f$ runs all possible isogenies of the variety $Jac~{}C$.

(ii) On the other hand, the abelian variety $Jac~{}C$ has complex multiplication and therefore each isogeny is represented by an element of the maximal commutative subring of the endomorphism ring of $Jac~{}C$. (The latter is isomorphic to an order in the ring of integers of the corresponding CM-field.) Thus the map (3.7) is a ring isomorphism.

Lemma 3.2 is proved. ∎

Corollary 3.3**.**

$\mathscr{H}(k)\cong\mathbf{F}_{p}(C).$**

Proof.

The proof follows from an embedding of the ring $O_{\mathscr{H}(k)}$ ($O_{\mathbf{F}_{p}(C)}$, resp.) in its field of fractions $\mathscr{H}(k)$ ($\mathbf{F}_{p}(C)$, resp.) and Lemma 3.2. ∎

Lemma 3.4**.**

$\mathscr{H}^{m-1}(k)\cong\mathbf{F}_{p^{m-1}}(C).$**

Proof.

(i) Recall that if the curve $C_{sing}$ has a singular point of multiplicity $m\geq 2$, then its desingularization $C_{sing}\buildrel p\over{\dashrightarrow}C$ consists of the composition of $m-1$ blow-ups. In view of Theorem 2.1 and using the notation $\mathscr{H}^{i+1}(k):=\mathscr{H}(\mathscr{H}^{i}(k))$, one gets a tower of the Hilbert class fields:

[TABLE]

where the field $\mathscr{H}^{m-1}(k)$ is the first in the tower to have the class number one.

(ii) On the other hand, (3.8) and Corollary 3.3 imply the following tower of the function fields:

[TABLE]

(iii) Notice that the geometric extensions of the function fields in the tower (3.9) are precluded, since the intermediate curves $\{C_{i}~{}|~{}1\leq i\leq m-2\}$ in the blow-up sequence C_{sing}:=C_{1}\buildrel p\over{\dashrightarrow}\dots C_{m-2}\buildrel p\over{\dashrightarrow}C are singular. Thus they cannot be used to define a geometric extension of the field $F_{p}(T)$.

Comparing (3.8) and (3.9), one gets the isomorphism $\mathscr{H}^{m-1}(k)\cong\mathbf{F}_{p^{m-1}}(C).$ Lemma 3.4 is proved. ∎

Theorem 1.1 follows from Lemma 3.4.

4. Complex multiplication revisited

Let $k=\mathbf{Q}(\sqrt{-d})$ be an imaginary quadratic number field. Recall that the theory of complex multiplication developed by Hilbert and Weber constructs generators of the field $\mathscr{H}(k)$ as follows.

Denote by $\mathscr{E}_{\tau}=\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$ an elliptic curve of the complex modulus $\tau=\sqrt{-d}$. Notice that such curves correspond to the case $g=1$ in formula (3.1) and the lattice $\Lambda=\mathbf{Z}+\mathbf{Z}\tau$ has complex multiplication by the ring $O_{k}$ of algebraic integers of the field $k$. Consider the $j$-invariant of $\mathscr{E}_{\tau}$ given by the convergent power series:

[TABLE]

Roughly speaking, the main theorem of complex multiplication says that:

[TABLE]

In other words, the value of the transcendental function (4.1) at the point $\tau=\sqrt{-d}$ is an algebraic number generating the Hilbert class field of $k$.

On the other hand, the Drinfeld modules indicate that the generators of $\mathscr{H}(k)$ do not require dramatic formulas like (4.1) but instead good old exponents will do. In this section we construct such generators using the torsion submodules of the Drinfeld module and Theorem 1.1. The corresponding exponential function was first obtained in [7].

Recall that the Drinfeld module defines explicit generators of the abelian extensions of the function fields (Section 2.3.3). Such modules generalize the Carlitz modules used in construction of the cyclotomic function fields, and can be viewed as an analog of the exponential function $E(x)=e^{x}$. Namely, the torsion module $\Lambda_{\rho}[a]\in\overline{\mathbf{F}_{q}(C)}$ is a generator of the Hilbert class field of the function field $\mathbf{F}_{q}(C)$, where $a\in\mathbf{F}_{q}(C)$ and $\rho$ is a homomorphism given by formula (2.7), see Theorem 2.3.

We are looking for an image of the $\Lambda_{\rho}[a]$ under the isomorphism $\mathbf{F}_{q}(C)\cong\mathscr{H}(k)$ of Theorem 1.1 by considering a family of the exponential functions of the form:

[TABLE]

where $\mu$ is a scaling parameter depending on the arithmetic of the ground field $k$. The following result solves the problem for the imaginary quadratic fields $k=\mathbf{Q}(\sqrt{-d})$.

Theorem 4.1**.**

([7])* The image of the torsion module $\Lambda_{\rho}[a]$ is given by the exponential function (4.3) evaluated at the point $x=\sqrt{-d}$, where $\mu=\log\varepsilon$ and $\varepsilon$ is the fundamental unit of the real quadratic number field $\mathbf{Q}(\sqrt{d})$. In other words,*

[TABLE]

for all but a finite number of values of the square-free discriminant $d$ of the imaginary quadratic field $k=\mathbf{Q}(\sqrt{-d})$.

Remark 4.2*.*

Formula (4.4) was proved using the Sklyanin algebras ([Stafford & van den Bergh 2001] [11, Example 8.3]) and the corresponding noncommutative tori ([8, Section 1.1]). We refer the reader to [7] for the details.

The image of the torsion module $\Lambda_{\rho}[a]$ for other number fields can be described as follows.

Conjecture 4.3**.**

Let $\theta$ be an algebraic number given by its minimal polynomial $p(x)=x^{n}-a_{1}x^{n-1}+\dots-a_{1}x+a_{0}$, where the integers $a_{i}$ are non-negative. Denote by $\varepsilon$ a unit of the splitting field of the polynomial $q(x)=x^{n}-a_{1}x^{n-1}-\dots-a_{1}x-a_{0}$. If $k\cong\mathbf{Q}(\theta)$ is a number field, then:

[TABLE]

Remark 4.4*.*

We refer the reader to [8, pp. 187-188] for the motivation of Conjecture 4.3.

Example 4.5**.**

Let $p(x)=x^{2}+d$, where $d>0$ is a square-free integer. One gets $\theta=i\sqrt{d}$ and the polynomial $q(x)=x^{2}-d$ has the real quadratic splitting field $\mathbf{Q}(\sqrt{d})$. It is easy to see, that in this case (4.5) coincides with the formula (4.4).