Degenerate abelian function fields

TL;DR

This paper investigates the properties of abelian function fields associated with singular curves, extending the classical theory from smooth Riemann surfaces to more general algebraic curves.

Contribution

It introduces a study of abelian function fields in the context of singular curves, expanding the understanding of their structure beyond smooth cases.

Findings

Extended the theory of abelian function fields to singular curves.

Analyzed the generation of these fields by fundamental abelian functions.

Provided insights into the relation between meromorphic functions on singular curves and their Jacobians.

Abstract

Originally, an abelian function field is the field of meromorphic functions on the Jacobi variety J(X) of a compact Riemann surface X. It is generated by the fundamental abelian functions belonging to the meromorphic function field on X. We study this relation for singular curves.

Equations

$$f_j(z+w)=R_j(f_0(z),f_1(z),\dots ,f_n(z),f_0(w),f_1(w),\dots ,f_n(w))$$

$$G={\mathbb{C}}^n/\Gamma ={\mathbb{C}}^p\times ({\mathbb{C}}^{\ast }{)}^q\times ({\mathbb{C}}^r/{\Gamma }_0),$$

$$\overline{G}=({\mathbb{P}}^1{)}^p\times ({\mathbb{P}}^1{)}^q\times \overline{{\mathbb{C}}^r/{\Gamma }_0}$$

$${\Gamma }_f:=\{\gamma \in {\mathbb{C}}^n;f(z+\gamma )=f(z)\text{for all z∈Cn}\}.$$

$$\overline{X}:=(X\setminus S)\cup \overline{S}.$$

$${\mathcal{O}}_{{\mathfrak{m}},Q}:=\begin{cases}(\rho_{*}{\mathcal{O}}_{X})_{Q}={\mathcal{O}}_{X,Q}&\mbox{if $Q\in X\setminus S$},\\ {\mathbb{C}}+{\mathcal{I}}_{Q}&\mbox{if $Q\in\overline{S}$}.\end{cases}$$

$$A:=H^0(X_{\mathfrak{m}},{\Omega }_{\mathfrak{m}}{)}^{\ast }/H_1(X_{\mathfrak{m}}\setminus \overline{S},\mathbb{Z}).$$

$$\left({\int }_{{\alpha }_i}{\rho }^{\ast }{\omega }_1,\dots ,{\int }_{{\alpha }_i}{\rho }^{\ast }{\omega }_{\pi }\right),i=1,\dots ,g,$$

$$\left({\int }_{{\beta }_i}{\rho }^{\ast }{\omega }_1,\dots ,{\int }_{{\beta }_i}{\rho }^{\ast }{\omega }_{\pi }\right),i=1,\dots ,g$$

$$\left({\int }_{{\gamma }_j}{\rho }^{\ast }{\omega }_1,\dots ,{\int }_{{\gamma }_j}{\rho }^{\ast }{\omega }_{\pi }\right),j=1,\dots ,s-1$$

$$A\cong {\mathbb{C}}^{\pi }/\Gamma \cong {\mathbb{C}}^p\times ({\mathbb{C}}^{\ast }{)}^q\times G,$$

$$A={\mathrm{Alb}}^{an}(X_{\mathfrak{m}})={\mathbb{C}}^p\times ({\mathbb{C}}^{\ast }{)}^q\times \mathcal{Q},$$

$$\phi :X\setminus S⟶A,P⟼\left[\left({\int }_{P_0}^P{\rho }^{\ast }{\omega }_1,\dots ,{\int }_{P_0}^P{\rho }^{\ast }{\omega }_{\pi }\right)\right].$$

$$\tau :({\mathbb{P}}^1{)}^{\pi }\ni (x_1,\dots ,x_{\pi })⟼(a_0:a_1:\cdots :a_{\pi })\in {\mathbb{P}}^{\pi },$$

$$\frac{a_1}{a_0}=-\sum _{i=1}^{\pi }{x}_i,\frac{a_2}{a_0}=\sum _{i<j}{x}_i{x}_j,\dots ,\frac{a_{\pi }}{a_0}=(-1{)}^n\prod _{i=1}^{\pi }{x}_i,$$

$${\mu }_0:X^{\pi }⟶C^{\pi }\subset ({\mathbb{P}}^1\times {\mathbb{P}}^1{)}^{\pi }\cong ({\mathbb{P}}^1{)}^{\pi }\times ({\mathbb{P}}^1{)}^{\pi }.$$

$$\begin{array}{ccc}{X}^{\pi }& \stackrel{\mu }{\to }& C^{\pi }\subset ({\mathbb{P}}^1\times {\mathbb{P}}^1{)}^{\pi }\cong ({\mathbb{P}}^1{)}^{\pi }\times ({\mathbb{P}}^1{)}^{\pi }\\ {\sigma }_0\downarrow & & \downarrow \tau \times \tau \\ X^{(\pi )}& \stackrel{\hat{\mu }}{\to }& {\mathbb{P}}^{\pi }\times {\mathbb{P}}^{\pi }\\ {\sigma }_1\downarrow {\sigma }_2& & \\ (X\setminus S{)}^{(\pi )}{\mathbb{P}}^{\pi }{\mathbb{P}}^{\pi }& & \\ \phi \downarrow & & \\ A={\mathrm{Alb}}^{an}(X_{\mathfrak{m}})& & \\ \sigma \uparrow & & \\ {\mathbb{C}}^{\pi }& & \end{array}$$

$${\wp }^x:={\sigma }_1\circ \hat{\mu }\circ {\phi }^{-1}\circ \sigma \text{and}{\wp }^y:={\sigma }_2\circ \hat{\mu }\circ {\phi }^{-1}\circ \sigma .$$

$${\wp }^x(z)=(1:{\xi }_1(z):\cdots :{\xi }_{\pi }(z))\text{and}{\wp }^y(z)=(1:{\eta }_1(z):\cdots :{\eta }_{\pi }(z)),$$

$${\xi }_1(z)=-\sum _{i=1}^{\pi }x(P_i),{\xi }_2(z)=\sum _{i<j}x(P_i)x(P_j),\dots ,{\xi }_{\pi }(z)=(-1{)}^{\pi }\prod _{i=1}^{\pi }x(P_i),$$

$${\eta }_1(z)=-\sum _{i=1}^{\pi }y(P_i),{\eta }_2(z)=\sum _{i<j}y(P_i)y(P_j),\dots ,{\eta }_{\pi }(z)=(-1{)}^{\pi }\prod _{i=1}^{\pi }y(P_i).$$

$$\tilde{K}:=\mathbb{C}({\xi }_1(z),\dots ,{\xi }_{\pi }(z),{\eta }_1(z),\dots ,{\eta }_{\pi }(z),{\xi }_1(w),\dots ,{\xi }_{\pi }(w),{\eta }_1(w),\dots ,{\eta }_{\pi }(w)).$$

$$B:={\mathbb{C}}^{\pi }/{\Gamma }_{K_0}={\mathbb{C}}^{p^{\prime }}\times ({\mathbb{C}}^{\ast }{)}^{q^{\prime }}\times {\mathcal{Q}}^{\prime },$$

$$\begin{array}{ccccc}A& \stackrel{{\tau }_1}{\to }& {\mathbb{C}}^{\pi }/\tilde{K}& \stackrel{{\tau }_2}{\to }& {\mathbb{C}}^{\pi }/{\Gamma }_K.\end{array}$$

$${\mathrm{Trans}}_{\mathbb{C}}{\sigma }^{\ast }(\mathrm{Mer}(\overline{A}){|}_A)={\mathrm{Trans}}_{\mathbb{C}}\tilde{K}=\pi .$$

Full text

1. Introduction

Abelian functions were discovered by solving Jacobi’s inversion problem. They are meromorphic functions on the Jacobi variety $J(X)$ of a compact Riemann surface $X$. We denote by ${\rm Mer}(J(X))$ the field of meromorphic functions on $J(X)$. Let ${\rm Mer}(X)$ be the field of meromorphic functions on $X$. The field ${\rm Mer}(J(X))$ is generated by the fundamental abelian functions belonging to ${\rm Mer}(X)$. We study this relation for singular curves.

In general, abelian function fields are fields of meromorphic functions on abelian varieties. We also know that an abelian function field admits an algebraic addition theorem. Weierstrass stated the following in his lectures in Berlin (see [8]):

Every system of $n$ (independent) functions in $n$ variables which admits an algebraic addition theorem is an algebraic combination of $n$ abelian (or degenerate abelian) functions with the same periods.

However, he did not publish his proof of the above statement. We did not know the precise meaning of degenerate abelian functions at that time.

We determined meromorphic function fields which admit an algebraic addition theorem in [2] and [3]. However, our statement was in a little weak form. We give its final form (Theorem 2.5) in this paper.

Another purpose of this paper is to complete the last section in the previous paper [6]. Let $S$ be a finite subset of a compact Riemann surface $X$. We construct a singular curve $X_{{\mathfrak{m}}}$ from $X$ by a modulus ${\mathfrak{m}}$ with support $S$. Let $A={\rm Alb}^{an}(X_{{\mathfrak{m}}})$ be the analytic Albanese variety of $X_{{\mathfrak{m}}}$. We proved in [6] that the period map $\varphi:X\setminus S\longrightarrow A$ gives a bimeromorphic map $\varphi:(X\setminus S)^{(\pi)}\longrightarrow A$, where $\pi$ is the genus of $X_{{\mathfrak{m}}}$ and $(X\setminus S)^{(\pi)}$ is the symmetric product of $X\setminus S$ of degree $\pi$. The analytic Albanese variety $A$ has the standard compactification $\overline{A}$ which is projective algebraic. Let ${\rm Mer}(\overline{A})$ be the field of meromorphic functions on $\overline{A}$. Its restriction ${\rm Mer}(\overline{A})|_{A}$ onto $A$ is a function field which admits an algebraic addition theorem. We show that ${\rm Mer}\left((X_{{\mathfrak{m}}})^{(\pi)}\right)\cong{\rm Mer}(\overline{A})$ through $\varphi$, where $(X_{{\mathfrak{m}}})^{(\pi)}$ is the symmetric product of $X_{{\mathfrak{m}}}$ of degree $\pi$ (Theorem 5.2).

We use notations in the previous paper [6].

2. Algebraic addition theorem

Let $K$ be a subfield of the meromorphic function field ${\rm Mer}({\mathbb{C}}^{n})$ on ${\mathbb{C}}^{n}$. We consider the following condition (T) concerning the transcendence degree ${\rm Trans}_{{\mathbb{C}}}K$ of $K$ over ${\mathbb{C}}$.

(T) $K$ is finitely generated over ${\mathbb{C}}$ and ${\rm Trans}_{{\mathbb{C}}}K=n$.

If $K$ satisfies condition (T), then we can take functions $f_{0},f_{1},\dots,f_{n}\in K$ with $K={\mathbb{C}}(f_{0},f_{1},\dots,f_{n})$.

Definition 2.1**.**

Let $K={\mathbb{C}}(f_{0},f_{1},\dots,f_{n})$ be a subfield of ${\rm Mer}({\mathbb{C}}^{n})$ satisfying condition (T). We say that $K$ admits an algebraic addition theorem (this is abbreviated to (AAT)) if for any $j=0,1,\dots,n$ there exists a rational function $R_{j}$ such that

[TABLE]

for all $z,w\in{\mathbb{C}}^{n}$.

The above definition does not depend on the choice of generators $f_{0},f_{1},\dots,f_{n}$ of $K$.

Any connected commutative complex Lie group $G$ of dimension $n$ is represented as

[TABLE]

where ${\mathbb{C}}^{r}/\Gamma_{0}$ is a toroidal group and $p+q+r=n$. By the standard compactification $\overline{{\mathbb{C}}^{r}/\Gamma_{0}}$ of ${\mathbb{C}}^{r}/\Gamma_{0}$ we obtain a compactification

[TABLE]

of $G$, which is called the standard compactification of $G$ (for details, see [4]).

Let $f\in{\rm Mer}({\mathbb{C}}^{n})$. We define the period group $\Gamma_{f}$ of $f$ by

[TABLE]

Definition 2.2**.**

A meromorphic function $f$ on ${\mathbb{C}}^{n}$ is said to be non-degenerate if its period group $\Gamma_{f}$ is discrete.

For a subfield $K$ of ${\rm Mer}({\mathbb{C}}^{n})$ we denote by $\Gamma_{K}:=\bigcap_{f\in K}\Gamma_{f}$ the period group of $K$. A subfield $K$ is said to be non-degenerate if it has a non-degenerate meromorphic function. We stated the following lemma without proof in [5]. Here we give its proof for the convenience of readers.

Lemma 2.3**.**

A subfield $K$ is non-degenerate if and only if $\Gamma_{K}$ is discrete.

Proof.

It is obvious by the definition that if $K$ is non-degenerate, then $\Gamma_{K}$ is discrete.

Conversely, suppose that $\Gamma_{K}$ is discrete. By induction on $n$ we show that $K$ has a non-degenerate function.

When $n=1$, the statement is trivial by the uniqueness theorem.

Let $n>1$. We assume that it holds for subfields of ${\rm Mer}({\mathbb{C}}^{k})$ with $k<n$. Take a non-constant $f\in K$. If $\Gamma_{f}$ is discrete, then $f$ is a non-degenerate function. If $\Gamma_{f}$ is not discrete, then there exist complex linear subspaces $V_{1}$ and $W_{1}$ such that ${\mathbb{C}}^{n}=V_{1}\oplus W_{1}$ with $0<\dim V_{1}<n$ and $\Gamma_{f}=V_{1}\oplus\Lambda_{1}$, where $\Lambda_{1}$ is a discrete subgroup of $W_{1}$. Since $\Gamma_{K|_{V_{1}}}$ is discrete, there exists $g\in K$ such that $\Gamma_{g|_{V_{1}}}$ is discrete by the assumption of induction.

Assume that $\Gamma_{g}$ is not discrete. Then there exist complex linear subspaces $V_{2}$ and $W_{2}$ such that ${\mathbb{C}}^{n}=V_{2}\oplus W_{2}$ and $\Gamma_{g}=V_{2}\oplus\Lambda_{2}$, where $\Lambda_{2}$ is a discrete subgroup of $W_{2}$. We have $W_{2}\supset V_{1}$ by the choice of $g$. We set $h_{1}:=fg$. Then $h_{1}$ is non-degenerate on $V_{1}\oplus V_{2}$, because $f$ is constant on $V_{1}$ and $g$ is constant on $V_{2}$. If ${\mathbb{C}}^{n}=V_{1}\oplus V_{2}$, then $h_{1}$ is the desired function. Otherwise, there exists a positive dimensional complex linear subspace $W$ such that ${\mathbb{C}}^{n}=V_{1}\oplus V_{2}\oplus W$.

If $\Gamma_{h_{1}}$ is not discrete, then there exist a complex linear subspace $U_{1}$ and a discrete subgroup $\Gamma_{1}$ such that $\Gamma_{h_{1}}=U_{1}\oplus\Gamma_{1}$. Since $h_{1}$ is non-degenerate on $V_{1}\oplus V_{2}$, we have $U_{1}\subset W$. Let $h_{2}:=h_{1}g=fg^{2}$. Then $h_{2}$ is non-degenerate on $V_{1}\oplus V_{2}$ by the same reason as above. On $U_{1}$ we have $h_{2}|_{U_{1}}=(h_{1}|_{U_{1}})(g|_{U_{1}})$. Since $h_{1}|_{U_{1}}$ is constant and $g|_{U_{1}}$ is non-degenerate, $\Gamma_{h_{2}|_{U_{1}}}$ is discrete. Hence, $h_{2}$ is non-degenerate on $V_{1}\oplus V_{2}\oplus U_{1}$. If $\Gamma_{h_{2}}$ is not discrete, we consider $h_{3}:=h_{2}g=fg^{3}$. Repeating this procedure, we finally obtain $m\in{\mathbb{N}}$ such that $fg^{m}$ is non-degenerate. ∎

Let $G={\mathbb{C}}^{n}/\Gamma$ be a connected commutative complex Lie group as above. We denote by ${\rm Mer}(G)$ the field of meromorphic functions on $G$. Let $\sigma:{\mathbb{C}}^{n}\longrightarrow G$ be the projection. Then, for any subfield $K$ of ${\rm Mer}({\mathbb{C}}^{n})$ with $\Gamma\subset\Gamma_{K}$ there exists a subfield $\kappa$ of ${\rm Mer}(G)$ such that $K=\sigma^{*}\kappa$. Let ${\rm Mer}(\overline{G})$ be the field of meromorphic functions on the standard compactification $\overline{G}$ of $G$. We denote by ${\rm Mer}(\overline{G})|_{G}$ the restriction of ${\rm Mer}(\overline{G})$ onto $G$.

Definition 2.4** ([5]).**

A subfield $K$ of ${\rm Mer}({\mathbb{C}}^{n})$ is said to be a W-type subfield if $K=\sigma^{*}({\rm Mer}(\overline{G})|_{G})$, where $G={\mathbb{C}}^{p}\times({\mathbb{C}}^{*})^{q}\times{\mathcal{Q}}$ with an $r$-dimensional quasi-abelian variety ${\mathcal{Q}}$ of kind 0, $n=p+q+r$ and $\sigma:{\mathbb{C}}^{n}\longrightarrow G$ is the projection (for the definition of quasi-abelian varieties of kind 0, see [4]).

The following theorem is the final form of the Weierstrass statement (cf. [2] and [3]). We have already obtained enough material to prove it in the previous papers ([1], [2] and [3]).

Theorem 2.5** (cf. Theorem 1.1 in [3]).**

Let $K$ be a non-degenerate subfield of ${\rm Mer}({\mathbb{C}}^{n})$ satisfying condition (T). If $K$ admits (AAT), then there exists a ${\mathbb{C}}$-linear isomorphism $\Phi:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}^{n}$ such that $\Phi^{*}K$ is a W-type subfield.

Proof.

We showed the following results in [1]. There exists a connected commutative complex Lie group $\Omega={\mathbb{C}}^{n}/\Gamma$ embedded in a complex projective space such that $K\cong{\mathbb{C}}(Y)$, where $Y$ is the Zariski closure of $\Omega$. Furthermore we can represent $\Omega$ as $\Omega={\mathbb{C}}^{p}\times({\mathbb{C}}^{*})^{q}\times{\mathcal{Q}}$, where ${\mathcal{Q}}={\mathbb{C}}^{r}/\Gamma_{0}$ is a quasi-abelian variety. Since ${\mathcal{Q}}$ is a connected closed complex Lie subgroup, the Zariski closure $Z$ of ${\mathcal{Q}}$ has the same dimension as ${\mathcal{Q}}$ by Theorem 4.5 in [2]. Therefore ${\mathcal{Q}}$ is a quasi-abelian variety of kind 0 (Theorem 8.1 in [3]). Let $\overline{{\mathcal{Q}}}$ be the standard compactification of ${\mathcal{Q}}$. Then we obtain $Y=({\mathbb{P}}^{1})^{p}\times({\mathbb{P}}^{1})^{q}\times\overline{{\mathcal{Q}}}$. Hence $K$ is a W-type subfield. ∎

3. Singular curves

Let $X$ be a compact Riemann surface with the structure sheaf ${\mathcal{O}}_{X}$. Take a finite subset $S$ of $X$. We consider an equivalence relation $R$ on $S$. We define the quotient set $\overline{S}:=S/R$ of $S$ by $R$. We set

[TABLE]

We induce to $\overline{X}$ the quotient topology by the canonical projection $\rho:X\longrightarrow\overline{X}$. Then $\overline{X}$ is a compact Hausdorff space.

Definition 3.1** ([9]).**

A modulus ${\mathfrak{m}}$ with support $S$ is the data of an integer ${\mathfrak{m}}(P)>0$ for each point $P\in S$.

Let $\rho_{*}{\mathcal{O}}_{X}$ be the direct image of ${\mathcal{O}}_{X}$ by the projection $\rho:X\longrightarrow\overline{X}$. For any $Q\in\overline{S}$ we denote by ${\mathcal{I}}_{Q}$ the ideal of $(\rho_{*}{\mathcal{O}}_{X})_{Q}$ formed by functions $f$ with ${\rm ord}_{P}(f)\geq{\mathfrak{m}}(P)$ for any $P\in\rho^{-1}(Q)$. We define a sheaf ${\mathcal{O}}_{\mathfrak{m}}$ on $\overline{X}$ by

[TABLE]

Then we obtain a 1-dimensional compact reduced complex space $(\overline{X},{\mathcal{O}}_{\mathfrak{m}})$, which we denote by $X_{{\mathfrak{m}}}$.

Conversely, any reduced and irreducible singular curve is obtained as above.

Let $g$ be the genus of $X$. For any $Q\in X_{{\mathfrak{m}}}$ we set $\delta_{Q}:=\dim\left((\rho_{*}{\mathcal{O}}_{X})_{Q}/{\mathcal{O}}_{{\mathfrak{m}},Q}\right)$. The genus of $X_{{\mathfrak{m}}}$ is defined by $\pi:=g+\delta$, where $\delta:=\sum_{Q\in X_{{\mathfrak{m}}}}\delta_{Q}$.

We denote by $\Omega_{{\mathfrak{m}}}$ the duality sheaf on $X_{{\mathfrak{m}}}$ (see Section 3.1 in [6]). We have $\dim H^{0}(X_{{\mathfrak{m}}},\Omega_{{\mathfrak{m}}})=\dim H^{1}(X_{{\mathfrak{m}}},{\mathcal{O}}_{{\mathfrak{m}}})=\pi$ (cf. [6]).

4. Analytic Albanese varieties

Let $X_{\mathfrak{m}}$ be a singular curve of genus $\pi=g+\delta$ constructed from $X$ by a modulus ${\mathfrak{m}}$ with support $S$, where $g$ is the genus of $X$. Take a basis $\{\omega_{1},\dots,\omega_{\pi}\}$ of $H^{0}(X_{\mathfrak{m}},\Omega_{\mathfrak{m}})$. We fix a canonical homology basis $\{\alpha_{1},\beta_{1},\dots,\alpha_{g},\beta_{g}\}$ of $X$. Let $S=\{P_{1},\dots,P_{s}\}$. We denote by $\gamma_{j}$ a small circle centered at $P_{j}$ with anticlockwise direction for $j=1,\dots,s$. Then the set $\{\alpha_{1},\beta_{1},\dots,\alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{s-1}\}$ forms a basis of $H_{1}(X\setminus S,{\mathbb{Z}})=H_{1}(X_{\mathfrak{m}}\setminus\overline{S},{\mathbb{Z}})$. Let $H^{0}(X_{\mathfrak{m}},\Omega_{\mathfrak{m}})^{*}$ be the dual space of $H^{0}(X_{\mathfrak{m}},\Omega_{\mathfrak{m}})$. We set

[TABLE]

Consider $2g+s-1$ vectors

[TABLE]

[TABLE]

and

[TABLE]

in ${\mathbb{C}}^{\pi}$. Let $\Gamma$ be a subgroup of ${\mathbb{C}}^{\pi}$ generated by these vectors over ${\mathbb{Z}}$. Then $\Gamma$ is a discrete subgroup of ${\mathbb{C}}^{\pi}$. We have $A\cong{\mathbb{C}}^{\pi}/\Gamma$ as a complex Lie group. We call $A$ with the structure as a complex Lie group the analytic Albanese variety of $X_{{\mathfrak{m}}}$, and write it as ${\rm Alb}^{an}(X_{{\mathfrak{m}}})$. The theorem of Remmert-Morimoto says that

[TABLE]

where $G$ is a toroidal group of dimension $r$ and $p+q+r=\pi$. In [6] we showed that $G$ is a quasi-abelian variety of kind 0. Then we write

[TABLE]

where ${\mathcal{Q}}$ is an $r$-dimensional quasi-abelian variety of kind 0. We define a period map $\varphi$ with base point $P_{0}\in X\setminus S$ by

[TABLE]

The period map $\varphi$ is extended to a bimeromorphic map $\varphi:(X\setminus S)^{(\pi)}\longrightarrow A$, where $(X\setminus S)^{(\pi)}$ is the symmetric product of $X\setminus S$ of degree $\pi$ (Theorem 5.19 in [6]).

5. Degenerate abelian function fields

We set $K_{{\mathfrak{m}}}:=\rho^{*}{\rm Mer}(X_{{\mathfrak{m}}})\subset{\rm Mer}(X).$ Then there exist $x,y\in K_{{\mathfrak{m}}}$ such that $K_{{\mathfrak{m}}}={\mathbb{C}}(x,y).$ Since $x$ and $y$ are algebraically dependent, we have an irreducible polynomial $f$ such that $f(x,y)=0.$ Let $C$ be the closure of $\{f=0\}$ in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}.$ The $\pi$-dimensional complex projective space ${\mathbb{P}}^{\pi}$ is identified with the symmetric product $({\mathbb{P}}^{1})^{(\pi)}$ of ${\mathbb{P}}^{1}$ of degree $\pi$ by the map induced from the following rational map

[TABLE]

[TABLE]

where $(x_{1},\dots,x_{\pi})$ is the inhomogeneous coordinates of $({\mathbb{P}}^{1})^{\pi}$ and $(a_{0}:a_{1}:\cdots:a_{\pi})$ is the homogeneous coordinates of ${\mathbb{P}}^{\pi}$. By the definition of $C$ we have a holomorphic map $\mu:X\longrightarrow C$. This gives a holomorphic map

[TABLE]

Let $\sigma_{0}:X^{\pi}\longrightarrow X^{(\pi)}$ be the canonical projection. Then we obtain the following commutative diagram:

[TABLE]

where $\hat{\mu}:X^{(\pi)}\longrightarrow{\mathbb{P}}^{\pi}\times{\mathbb{P}}^{\pi}$ is the holomorphic map induced from $\mu$ and $\sigma_{i}:{\mathbb{P}}^{\pi}\times{\mathbb{P}}^{\pi}\longrightarrow{\mathbb{P}}^{\pi}$ is the projection onto the $i$-th component for $i=1,2$. Therefore we can define meromorphic maps $\wp^{x}:{\mathbb{C}}^{\pi}\longrightarrow{\mathbb{P}}^{\pi}$ and $\wp^{y}:{\mathbb{C}}^{\pi}\longrightarrow{\mathbb{P}}^{\pi}$ by

[TABLE]

If we represent $\wp^{x}$ and $\wp^{y}$ in homogeneous coordinates of ${\mathbb{P}}^{\pi}$ as

[TABLE]

then $\xi_{1}(z),\dots,\xi_{\pi}(z),\eta_{1}(z),\dots,\eta_{\pi}(z)\in\sigma^{*}{\rm Mer}(A).$ Let $[z]$ be a generic point of $A$, where $z\in{\mathbb{C}}^{\pi}$. Then there exists uniquely $(P_{1},\dots,P_{\pi})\in(X\setminus S)^{(\pi)}$ with $\varphi((P_{1},\dots,P_{\pi}))=[z]$. In this case we have

[TABLE]

[TABLE]

We denote $K:={\mathbb{C}}(\xi_{1}(z),\dots,\xi_{\pi}(z),\eta_{1}(z),\dots,\eta_{\pi}(z))\subset\sigma^{*}{\rm Mer}(A)$. Let $\kappa$ be the corresponding subfield of ${\rm Mer}(A)$, i.e. $K=\sigma^{*}\kappa$. Then we have $\varphi^{*}\kappa={\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)})$ by the above relation, where $(X_{{\mathfrak{m}}})^{(\pi)}$ is the symmetric product of $X_{{\mathfrak{m}}}$ of degree $\pi$. We note that $(X_{{\mathfrak{m}}})^{(\pi)}$ is an irreducible projective algebraic variety (cf. III. 14 in [9]). We may consider ${\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)})$ as a subfield of ${\rm Mer}(X^{(\pi)})$. It is obvious that $\varphi^{*}\kappa$ is finitely generated over ${\mathbb{C}}$ and ${\rm Trans}_{{\mathbb{C}}}\varphi^{*}\kappa=\pi$.

Proposition 5.1**.**

$K$* admits (AAT).*

Proof.

Letting $(z,w)\in{\mathbb{C}}^{\pi}\times{\mathbb{C}}^{\pi}$, we define

[TABLE]

We have a subfield $\widetilde{\kappa}$ of ${\rm Mer}(A\times A)$ such that $\widetilde{K}=(\sigma\times\sigma)^{*}\widetilde{\kappa}$. Take any $\psi\in\kappa$. It suffices to show that $\psi(\varphi(P)+\varphi(Q))\in(\varphi\times\varphi)^{*}\widetilde{\kappa}$.

We take $P,Q\in(X\setminus S)^{(\pi)}$ such that $\varphi(P)$ and $\varphi(Q)$ are generic points of $A$. Fixing $\varphi(Q)$, we consider $\psi(\varphi(P)+\varphi(Q))$ as a function of $P$. Then it is meromorphically extended to $(X_{{\mathfrak{m}}})^{(\pi)}$. Similarly, it is a meromorphic function on $(X_{{\mathfrak{m}}})^{(\pi)}$ as a function of $Q$ if we fix $\varphi(P)$. The set of $P\in(X\setminus S)^{(\pi)}$ such that $\varphi(P)$ is not generic is an analytic set of positive codimension. Furthermore, $X^{(\pi)}\setminus(X\setminus S)^{(\pi)}$ is an analytic subset of $X^{(\pi)}$. Then $\psi(\varphi(P)+\varphi(Q))$ extends meromorphically to $X^{(\pi)}\times X^{(\pi)}$ (for example, see Theorem 2 in [10] or Corollary 3.4 in [7]). There exists $\widetilde{\xi}\in{\rm Mer}(X^{(\pi)}\times X^{(\pi)})$ such that ${\rm Mer}(X^{(\pi)}\times X^{(\pi)})={\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)}\times(X_{{\mathfrak{m}}})^{(\pi)})(\widetilde{\xi})$. Then $\psi(\varphi(P)+\varphi(Q))$ is a rational function of generators of ${\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)}\times(X_{{\mathfrak{m}}})^{(\pi)})$ and $\widetilde{\xi}$. However, it is independent of $\widetilde{\xi}$ on $\left((X_{{\mathfrak{m}}})^{(\pi)}\times(X\setminus S)^{(\pi)}\right)\cup\left((X\setminus S)^{(\pi)}\times(X_{{\mathfrak{m}}})^{(\pi)}\right)$. Therefore we obtain $\psi(\varphi(P)+\varphi(Q))\in(\varphi\times\varphi)^{*}\widetilde{\kappa}$. ∎

Let $\overline{A}$ be the standard compactification of $A$. Then a W-type subfield $\sigma^{*}({\rm Mer}(\overline{A})|_{A})$ is considered as a degenerate abelian function field, for we have the following theorem.

Theorem 5.2**.**

We have $K\cong\sigma^{*}({\rm Mer}(\overline{A})|_{A})$, hence ${\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)})\cong{\rm Mer}(\overline{A})$.

Proof.

Since $K$ admits (AAT) (Proposition 5.1), there exists a ${\mathbb{C}}$-linear isomorphism $\Phi:{\mathbb{C}}^{\pi}\longrightarrow{\mathbb{C}}^{\pi}$ such that $K_{0}:=\Phi^{*}K$ is a W-type subfield by Theorem 2.5. Let $\Gamma_{K_{0}}$ be the period group of $K_{0}$. Then we have

[TABLE]

where ${\mathcal{Q}}^{\prime}={\mathbb{C}}^{r^{\prime}}/(\Gamma_{K_{0}})_{0}$ is an $r^{\prime}$-dimensional quasi-abelian variety of kind 0 and $p^{\prime}+q^{\prime}+r^{\prime}=\pi$. Furthermore we have $K_{0}=\sigma_{K_{0}}^{*}({\rm Mer}(\overline{B})|_{B})$ by the definition of W-type subfields, where $\sigma_{K_{0}}:{\mathbb{C}}^{\pi}\longrightarrow B$ is the projection.

From $K\subset\sigma^{*}{\rm Mer}(A)$ and $A={\mathbb{C}}^{\pi}/\Gamma$ it follows that $\Gamma\subset\Gamma_{K}$. We note $\Phi(\Gamma_{K})=\Gamma_{K_{0}}$. We set $\widetilde{\kappa}:=(\varphi^{-1})^{*}\left({\rm Mer}(X^{(\pi)})|_{(X\setminus S)^{(\pi)}}\right)\subset{\rm Mer}(A)$. Since $\varphi^{*}\kappa={\rm Mer}((X_{{\mathfrak{m}}})^{(\pi)})|_{(X\setminus S)^{(\pi)}}\subset{\rm Mer}(X^{(\pi)})|_{(X\setminus S)^{(\pi)}}$, we have $\kappa\subset\widetilde{\kappa}$. If we set $\widetilde{K}:=\sigma^{*}\widetilde{\kappa}$, then we have $K\subset\widetilde{K}\subset\sigma^{*}{\rm Mer}(A)$. Therefore, $\Gamma\subset\Gamma_{\widetilde{K}}\subset\Gamma_{K}$, where $\Gamma_{\widetilde{K}}$ is the period group of $\widetilde{K}$. Hence we obtain the following sequence of homomorphisms:

[TABLE]

Since ${\rm Trans}_{{\mathbb{C}}}K={\rm Trans}_{{\mathbb{C}}}\widetilde{K}=\pi$, $\tau_{2}:{\mathbb{C}}^{\pi}/\widetilde{K}\longrightarrow{\mathbb{C}}^{\pi}/\Gamma_{K}$ is an isogeny (Proposition 3 in [5]). By Proposition 7.1 in [6] we have $\varphi^{*}({\rm Mer}(\overline{A})|_{A})\subset{\rm Mer}(X^{(\pi)})|_{(X\setminus S)^{(\pi)}}$. Then we obtain ${\rm Mer}(\overline{A})|_{A}=(\varphi^{-1})^{*}(\varphi^{*}({\rm Mer}(\overline{A})|_{A})\subset\widetilde{\kappa}$. Therefore, we have $\sigma^{*}({\rm Mer}(\overline{A})|_{A})\subset\widetilde{K}$. Since $\sigma^{*}({\rm Mer}(\overline{A})|_{A})$ is a W-type subfield, we have

[TABLE]

The period group of $\sigma^{*}({\rm Mer}(\overline{A})|_{A})$ is $\Gamma$. Then $\tau_{1}:A={\mathbb{C}}^{\pi}/\Gamma\longrightarrow{\mathbb{C}}^{\pi}/\Gamma_{\widetilde{K}}$ is also an isogeny by Proposition 3 in [5]. Therefore, $\tau_{2}\circ\tau_{1}:A\longrightarrow{\mathbb{C}}^{\pi}/\Gamma_{K}$ is an isogeny. We note that $B$ and ${\mathbb{C}}^{\pi}/\Gamma_{K}$ are isogenous for $K_{0}=\Phi^{*}K$. Thus we see that $A$ and $B$ are isogenous. Since an isogeny is fiber preserving (Proposition 2 in [5]), we have ${\rm Mer}(\overline{A})|_{A}\cong{\rm Mer}(\overline{B})|_{B}$ and {\rm Mer}(\overline{B})|_{B}\cong{\rm Mer}\left(\overline{({\mathbb{C}}^{\pi}/\Gamma_{K})}\right)\bigr{|}_{{\mathbb{C}}^{\pi}/\Gamma_{K}}. Hence we obtain $K\cong\sigma^{*}{\rm Mer}(\overline{A})|_{A}$. ∎

The generators $\{\xi_{i}(z),\eta_{i}(z);i=1,\dots,\pi\}$ is just the fundamental degenerate abelian functions belonging to ${\rm Mer}(X_{{\mathfrak{m}}})$ as in the non-singular case.