On Jacobi Inversion Formulae for Telescopic Curves

On Jacobi Inversion Formulae for Telescopic Curves

Takanori AYANO

T. Ayano

Osaka City University, Advanced Mathematical Institute,
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan<EMAIL_ADDRESS>
Received May 06, 2016, in final form August 23, 2016; Published online August
27, 2016

For a hyperelliptic curve of genus $g$, it is well known that the symmetric
products of $g$ points on the curve are expressed in terms of their
Abel–Jacobi image by the hyperelliptic sigma function (Jacobi inversion
formulae). Matsutani and Previato gave a natural generalization of the
formulae to the more general algebraic curves defined by $y^{r}=f(x)$, which
are special cases of $(n,s)$ curves, and derived new vanishing properties of
the sigma function of the curves $y^{r}=f(x)$. In this paper we extend the
formulae to the telescopic curves proposed by Miura and derive new vanishing
properties of the sigma function of telescopic curves. The telescopic curves
contain the $(n,s)$ curves as special cases.

sigma function; inversion of algebraic integrals; vanishing of sigma function;
Riemann surface; telescopic curve

14H42; 14H50; 14H55

##  1 Introduction

The theory of the elliptic function was the one of the main subjects of the
research of mathematics in the nineteenth century. Now the beautiful theory of
the elliptic function is constructed and is applied to many fields such as
mathematical physics, integrable system, number theory, engineering, and
cryptography. In integrable system, it is well known that the elliptic
function gives an exact solution of some nonlinear differential equations. In
cryptography, the cryptosystem using the elliptic curves is used widely.
Recently, with the scientific development, we have to analyze many complicated
nonlinear phenomena and it is necessary to give exact solutions of many
nonlinear differential equations in order to analyze the phenomena precisely.
In cryptography, it is necessary to make a wider class of algebraic curves
available to the cryptosystem for assuring the safety of cryptosystem.
Therefore it is very important to construct the basic theory of the Abelian
function, which is a generalization of the elliptic function to several
variables. The sigma function plays an important role in the theory of the
Abelian function.

The multivariate sigma function is introduced by F. Klein [13, 14] for
hyperelliptic curves as a generalization of the Weierstrass’s elliptic sigma
function. Recently, the hyperelliptic sigma function is generalized to the
more general plane algebraic curves called $(n,s)$ curves [5, 6, 7, 8, 21].
The sigma function is obtained by modifying Riemann’s theta function so as to
be modular invariant, i.e., it does not depend on the choice of a canonical
homology basis. Further the sigma function has some remarkable algebraic
properties that it is directly related with the defining equations of an
algebraic curve. From these algebraic properties, the sigma function is
expected to have many applications in mathematical physics etc. [7]. Further
the sigma function is useful to describe a solution of the inversion problem
of algebraic integrals. The Jacobi inversion problem for hyperelliptic curves
is described as follows.

Let $X$ be a hyperelliptic curve of genus $g$ defined by $y^{2}=f(x)$,

$\displaystyle
f(x)=x^{2g+1}+\lambda_{2g}x^{2g}+\cdots+\lambda_{1}x+\lambda_{0},\qquad\lambda_{i}\in\mathbb{C}.$

Let $du_{i}=-\frac{x^{g-i}}{2y}dx$, $1\leq i\leq g$, be the holomorphic one
forms on $X$ and $du={}^{t}(du_{1},\dots,du_{g})$. For $1\leq k\leq g$,
$P_{1},\dots,P_{k}\in X\backslash\infty$, and
$u^{[k]}=\sum\limits_{i=1}^{k}\int_{\infty}^{P_{i}}du$, one wants to express
the coordinates of $P_{i}$ in terms of $u^{[k]}$.

For $k=g$ and $P_{i}=(x_{i},y_{i})\in X$, we define the symmetric polynomial
$e_{i}$ by

$\displaystyle e_{i}=\sum_{1\leq\ell_{1}<\cdots<\ell_{i}\leq
g}x_{\ell_{1}}\cdots x_{\ell_{i}}.$

Let $\sigma(u)$ be the sigma function of $X$ and $S^{g}(X)$ the $g$-th
symmetric products of $X$. Then the following theorem is well-known [4].

[Jacobi inversion formulae] If $\sum\limits_{i=1}^{g}P_{i}\in
S^{g}(X\backslash\infty)$ is a general divisor, then we have

$\displaystyle\wp_{1,i}\big{(}u^{[g]}\big{)}=(-1)^{i-1}e_{i},\qquad 1\leq
i\leq g,$

where $\wp_{i,j}(u)=-\frac{\partial^{2}}{\partial u_{i}\partial
u_{j}}\log\sigma(u)$.

The inversion of algebraic integrals is deeply related to the problems of
mathematical physics (cf. [10, 11]).

Matsutani and Previato [17] gave a natural generalization of the above
formulae for any $1\leq k\leq g$ and the more general plane algebraic curves
defined by

$\displaystyle y^{r}=x^{s}+\lambda_{s-1}x^{s-1}+\cdots+\lambda_{0},$ (1.1)

where $r$ and $s$ are relatively prime positive integers and
$\lambda_{i}\in\mathbb{C}$. These curves are special cases of the $(n,s)$
curves. Furthermore, in [17], new vanishing properties of the sigma function
of the curves defined by (1.1) are derived by using the extended Jacobi
inversion formulae.

On the other hand, in [19], Miura introduced a certain canonical form, Miura
canonical form, for defining equations of any non-singular algebraic curve. A
telescopic curve [19] is a special curve for which Miura canonical form is
easy to determine. Let $m\geq 2$ and $(a_{1},\dots,a_{m})$ a sequence of
relatively prime positive integers satisfying certain condition. Then the
telescopic curve associated with $(a_{1},\dots,a_{m})$ or the
$(a_{1},\dots,a_{m})$ curve is the algebraic curve defined by certain $m-1$
equations in ${\mathbb{C}}^{m}$. For $m=2$, the telescopic curves are equal to
the $(n,s)$ curves.

In this paper we extend the formulae obtained in [17] to the telescopic curves
(Theorems 7.1 and 7.3). More specifically, for the telescopic curves, we give
formulae which express the $\wp$-function and the ratio of the derivative of
the sigma function by the ratio of the determinants of certain matrices
consisting of the algebraic functions. Under a certain condition, a coordinate
of one point on the telescopic curves can be expressed in terms of its
Abel–Jacobi image by the derivatives of the sigma function (Corollary 7.4).
Furthermore we derive new vanishing properties of the sigma function of the
telescopic curves as a corollary of the formulae (Corollaries 9.1 and 9.2).
Finally we comment that the Jacobi inversion formulae are derived for
$(3,4,5)$ curves in [16] and $(3,7,8)$, $(6,13,14,15,16)$ curves in [15],
which are not telescopic.

The present paper is organized as follows. In Section 2, the definition of the
telescopic curves is given. In Section 3, the fundamental differential of
second kind for the telescopic curves is reviewed and a coefficient of the
second kind differentials is determined explicitly. In Section 4, the
definition of the sigma function of telescopic curves and the expression of
the fundamental differential of second kind by the sigma function are given.
In Section 5, Frobenius–Stickelberger matrix is defined. In Section 6,
Riemann’s singularity theorem is reviewed. In Section 7, a generalization of
Jacobi inversion formulae to telescopic curves is given. In Section 8, as an
example, the formulae for the $(4,6,5)$ curves are given. In Section 9, some
new vanishing properties of the sigma function of telescopic curves are given.
In Section 10, as an example, the vanishing properties of the sigma function
of the $(4,6,5)$ curves are given.

##  2 Telescopic curves

In this section we briefly review the definition of telescopic curves
following [2, 19].

For $m\geq 2$, let $(a_{1},\dots,a_{m})$ be a sequence of positive integers
such that $\operatorname{gcd}(a_{1},\dots,a_{m})=1$, $a_{i}\geq 2$ for any
$i$, and

$\displaystyle\frac{a_{i}}{d_{i}}\in\frac{a_{1}}{d_{i-1}}{\mathbb{Z}}_{\geq
0}+\cdots+\frac{a_{i-1}}{d_{i-1}}{\mathbb{Z}}_{\geq 0},\qquad 2\leq i\leq m,$

where $d_{i}=\operatorname{gcd}(a_{1},\dots,a_{i})$.

Let

$\displaystyle B(A_{m})=\left\\{(\ell_{1},\dots,\ell_{m})\in{\mathbb{Z}}_{\geq
0}^{m}\,\big{|}\,0\leq\ell_{i}\leq\frac{d_{i-1}}{d_{i}}-1\ \text{for}\ 2\leq
i\leq m\right\\}.$

######  Lemma 2.1 ([2, 19]).

For any $a\in a_{1}\mathbb{Z}_{\geq 0}+\cdots+a_{m}\mathbb{Z}_{\geq 0}$, there
exists a unique element $(k_{1},\dots,k_{m})$ of $B(A_{m})$ such that

$\displaystyle\sum_{i=1}^{m}a_{i}k_{i}=a.$

By this lemma, for any $2\leq i\leq m$, there exists a unique sequence
$(\ell_{i,1},\dots,\ell_{i,m})\in B(A_{m})$ satisfying

$\displaystyle\sum_{j=1}^{m}a_{j}\ell_{i,j}=a_{i}\frac{d_{i-1}}{d_{i}}.$

######  Lemma 2.2 ([3]).

For any $2\leq i\leq m$, we have $\ell_{i,j}=0$ for $j\geq i$.

Consider $m-1$ polynomials in $m$ variables $x_{1},\dots,x_{m}$ given by

$\displaystyle
F_{i}(x)=x_{i}^{d_{i-1}/d_{i}}-\prod_{j=1}^{i-1}x_{j}^{\ell_{i,j}}-\sum\lambda^{(i)}_{j_{1},\dots,j_{m}}x_{1}^{j_{1}}\cdots
x_{m}^{j_{m}},\qquad 2\leq i\leq m,$ (2.1)

where $\lambda^{(i)}_{j_{1},\dots,j_{m}}\in\mathbb{C}$ and the sum of the
right-hand side is over all $(j_{1},\dots,j_{m})\in B(A_{m})$ such that

$\displaystyle\sum_{k=1}^{m}a_{k}j_{k}<a_{i}\frac{d_{i-1}}{d_{i}}.$

Let $X^{\rm{af\/f}}$ be the common zeros of $F_{2}$,…,$F_{m}$:

$\displaystyle
X^{\rm{af\/f}}=\big{\\{}(x_{1},\dots,x_{m})\in\mathbb{C}^{m}\,|\,F_{i}(x_{1},\dots,x_{m})=0,\,2\leq
i\leq m\big{\\}}.$

In [2, 19], $X^{\rm{af\/f}}$ is proved to be an affine algebraic curve. We
assume that $X^{\rm{af\/f}}$ is nonsingular. Let $X$ be the compact Riemann
surface corresponding to $X^{\rm{af\/f}}$. Then $X$ is obtained from
$X^{\rm{af\/f}}$ by adding one point, say $\infty$ [2, 19]. The genus of $X$
is given by [2, 19]

$\displaystyle
g=\frac{1}{2}\left\\{1-a_{1}+\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1\right)a_{i}\right\\}.$
(2.2)

We call $X$ the telescopic curve associated with $(a_{1},\dots,a_{m})$. The
numbers $a_{1},\dots,a_{m}$ are a generator of the semigroup of non-gaps at
$\infty$.

######  Example 2.3.

  1. (i)

The telescopic curve associated with a pair of relatively prime integers
$(n,s)$ is the $(n,s)$ curve introduced in [6].

  2. (ii)

For $A_{3}=(4,6,5)$, polynomials $F_{i}$ are given by

$\displaystyle
F_{2}(x)=x_{2}^{2}-x_{1}^{3}-\lambda^{(2)}_{0,1,1}x_{2}x_{3}-\lambda^{(2)}_{1,1,0}x_{1}x_{2}-\lambda^{(2)}_{1,0,1}x_{1}x_{3}-\lambda^{(2)}_{2,0,0}x_{1}^{2}-\lambda^{(2)}_{0,1,0}x_{2}$
$\displaystyle\hphantom{F_{2}(x)=}{}-\lambda^{(2)}_{0,0,1}x_{3}-\lambda^{(2)}_{1,0,0}x_{1}-\lambda^{(2)}_{0,0,0},$
$\displaystyle
F_{3}(x)=x_{3}^{2}-x_{1}x_{2}-\lambda^{(3)}_{1,0,1}x_{1}x_{3}-\lambda^{(3)}_{2,0,0}x_{1}^{2}-\lambda^{(3)}_{0,1,0}x_{2}-\lambda^{(3)}_{0,0,1}x_{3}-\lambda^{(3)}_{1,0,0}x_{1}-\lambda^{(3)}_{0,0,0}.$

For a meromorphic function $f$ on $X$, we denote by
$\operatorname{ord}_{\infty}(f)$ the order of a pole at $\infty$. Then we have
$\operatorname{ord}_{\infty}(x_{i})=a_{i}$. We enumerate the monomials
$x_{1}^{\alpha_{1}}\cdots x_{m}^{\alpha_{m}}$,
$(\alpha_{1},\dots,\alpha_{m})\in B(A_{m})$, according as the order of a pole
at $\infty$ and denote them by $\varphi_{i}$, $i\geq 1$. In particular we have
$\varphi_{1}=1$. The set $\\{\varphi_{i}\\}_{i=1}^{\infty}$ is a basis of
meromorphic functions on $X$ with a pole only at $\infty$.

Let $G$ be the $(m-1)\times m$ matrix defined by

$\displaystyle G=\left(\frac{\partial F_{i}}{\partial x_{j}}\right)_{2\leq
i\leq m,\,1\leq j\leq m}$

and $G_{k}$ the $(m-1)\times(m-1)$ matrix obtained by deleting the $k$-th
column from $G$. Then a basis of holomorphic one forms is given by

$\displaystyle du_{i}=-\frac{\varphi_{g+1-i}}{\det G_{1}}dx_{1},\qquad 1\leq
i\leq g.$

Let $(w_{1},\dots,w_{g})$ be the gap sequence at $\infty$:

$\displaystyle\\{w_{i}\,|\,1\leq i\leq g\\}={\mathbb{Z}}_{\geq
0}\backslash\left\\{\sum_{i=1}^{m}a_{i}{\mathbb{Z}}_{\geq 0}\right\\},\qquad
w_{1}<\cdots<w_{g}.$

In particular $w_{1}=1$, since $g\geq 1$. The following lemma is proved in
[2].

######  Lemma 2.4.

We have $w_{g}=2g-1$. In particular, $du_{g}$ has a zero of order $2g-2$ at
$\infty$.

From Lemma 2.4, we find that the vector of Riemann constants for a telescopic
curve with a base point $\infty$ is a half-period.

######  Lemma 2.5 ([3]).

It is possible to take a local parameter $z$ around $\infty$ such that

$\displaystyle x_{1}=\frac{1}{z^{a_{1}}},\qquad
x_{i}=\frac{1}{z^{a_{i}}}(1+O(z)),\qquad 2\leq i\leq m.$ (2.3)

######  Proposition 2.6 ([3]).

For $1\leq i\leq g$, the expansion of $du_{i}$ at $\infty$ is of the form

$\displaystyle du_{i}=z^{w_{i}-1}(1+O(z))dz.$

For the telescopic curve $X$ associated with $A_{m}=(a_{1},\dots,a_{m})$, we
define the partition by

$\displaystyle\mu(A_{m})=(w_{g},\dots,w_{1})-(g-1,\dots,0).$

######  Proposition 2.7 ([6, 21]).

The Young diagram of $\mu(A_{m})$ is symmetric.

######  Example 2.8.

For $(4,6,5)$ curves, we have $g=4$, $w_{1}=1$, $w_{2}=2$, $w_{3}=3$,
$w_{4}=7$ and $\mu((4,6,5))=(4,1,1,1)$. For $(4,6,7)$ curves, we have $g=5$,
$w_{1}=1$, $w_{2}=2$, $w_{3}=3$, $w_{4}=5$, $w_{5}=9$ and
$\mu((4,6,7))=(5,2,1,1,1)$. Therefore the Young diagrams of $(4,6,5)$ curves
and $(4,6,7)$ curves are as follows.

$(4,6,5)$ curves $(4,6,7)$ curves

##  3 Fundamental differential of second kind

A fundamental differential of second kind plays an important role in the
theory of the sigma function. We recall its definition.

######  Definition 3.1.

A two form $\omega(P,Q)$ on $X\times X$ is called a fundamental differential
of second kind if the following conditions are satisfied:

  1. (i)

$\omega(P,Q)=\omega(Q,P)$,

  2. (ii)

$\omega(P,Q)$ is holomorphic except $\\{(R,R)\,|\,R\in X\\}$ where it has a
double pole,

  3. (iii)

for $R\in X$, take a local coordinate $t$ around $R$, then the expansion
around $(R,R)$ is of the form

$\displaystyle\omega(P,Q)=\left(\frac{1}{(t_{P}-t_{Q})^{2}}+\text{regular}\right)dt_{P}dt_{Q}.$

A fundamental differential of second kind exists but is not unique. Let
$\omega_{1}(P,Q)$ be a fundamental differential of second kind. Then a two
form $\omega_{2}(P,Q)$ is a fundamental differential of second kind if and
only if there exists $\\{c_{ij}\\}_{i,j=1,\dots,g}\in\mathbb{C}$ such that
$c_{ij}=c_{ji}$ and

$\displaystyle\omega_{2}(P,Q)=\omega_{1}(P,Q)+\sum_{i,j=1}^{g}c_{ij}dv_{i}(P)dv_{j}(Q),$

where $\\{dv_{i}\\}_{i=1}^{g}$ is a basis of holomorphic one forms on $X$.

For a telescopic curve $X$, a fundamental differential of second kind is
algebraically constructed in [2]. We recall its construction. Note that the
construction inherits all steps of classical construction in [4] that was
recently recapitulated and generalized in [8, 21] for the $(n,s)$ curves.

Let $X$ be a telescopic curve of genus $g$. We define the 2-form
$\widehat{\omega}(P,Q)$ on $X\times X$ by

$\displaystyle\widehat{\omega}(P,Q)=d_{Q}\Omega(P,Q)+\sum_{i=1}^{g}du_{i}(P)dr_{i}(Q),$

where $P=(x_{1},\dots,x_{m})$, $Q=(y_{1},\dots,y_{m})$ are points on $X$,

$\displaystyle\Omega(P,Q)=\frac{\det H(P,Q)}{(x_{1}-y_{1})\det
G_{1}(P)}dx_{1},$

$H=(h_{ij})_{2\leq i,j\leq m}$ with

$\displaystyle
h_{ij}=\frac{F_{i}(y_{1},\dots,y_{j-1},x_{j},x_{j+1},\dots,x_{m})-F_{i}(y_{1},\dots,y_{j-1},y_{j},x_{j+1},\dots,x_{m})}{x_{j}-y_{j}},$

and $dr_{i}$ is a second kind differential with a pole only at $\infty$. The
set

$\displaystyle\left\\{\frac{\varphi_{i}(P)}{\det
G_{1}(P)}dx_{1}\right\\}_{i=1}^{\infty}$

is a basis of meromorphic one forms on $X$ with a pole only at $\infty$ [2,
21]. It is possible to take $\\{dr_{i}\\}_{i=1}^{g}$ such that
$\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$ [2, 21]. If we take
$\\{dr_{i}\\}_{i=1}^{g}$ such that
$\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$, then $\widehat{\omega}(P,Q)$
becomes a fundamental differential of second kind [2, 21].

We assign degrees as

$\displaystyle\deg x_{k}=\deg
y_{k}=a_{k},\qquad\deg\lambda_{j_{1},\dots,j_{m}}^{(i)}=a_{i}d_{i-1}/d_{i}-\sum_{k=1}^{m}a_{k}j_{k}.$

######  Lemma 3.2.

  1. $(i)$

For $1\leq k\leq m$, $\det G_{k}(Q)$ is homogeneous of degree
$\sum\limits_{i=2}^{m}a_{i}d_{i-1}/d_{i}-\sum\limits_{i=1}^{m}a_{i}+a_{k}$
with respect to the coefficients
$\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$ and the variables
$y_{1},\dots,y_{m}$.

  2. $(ii)$

$\det H(P,Q)$ is homogeneous of degree
$\sum\limits_{i=2}^{m}a_{i}(d_{i-1}/d_{i}-1)$ with respect to the coefficients
$\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$ and the variables
$x_{1},\dots,x_{m},y_{1},\dots,y_{m}$.

###### Proof.

For $2\leq i\leq m$ and $1\leq j\leq m$, $\frac{\partial F_{i}(y)}{\partial
y_{j}}$ is homogeneous of degree $a_{i}d_{i-1}/d_{i}-a_{j}$ with respect to
$\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$ and $y_{1},\dots,y_{m}$.
Therefore we obtain (i). For $2\leq i,j\leq m$, $h_{ij}$ is homogeneous of
degree $a_{i}d_{i-1}/d_{i}-a_{j}$ with respect to
$\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$ and
$x_{1},\dots,x_{m},y_{1},\dots,y_{m}$. Therefore we obtain (ii). ∎

We have

$\displaystyle
d_{Q}\Omega(P,Q)=\frac{\\!\Big{\\{}\sum\limits_{i=1}^{m}({-}1)^{i+1}(x_{1}\\!-\\!y_{1})\frac{\partial\det
H}{\partial y_{i}}(P,Q)\det G_{i}(Q)\Big{\\}}\\!+\\!\det G_{1}(Q)\det
H(P,Q)\\!}{(x_{1}-y_{1})^{2}\det G_{1}(P)\det G_{1}(Q)}dx_{1}dy_{1},$

where the numerator is homogeneous of degree
$2\sum\limits_{i=2}^{m}(d_{i-1}/d_{i}-1)a_{i}$ with respect to the
coefficients $\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$ and the
variables $x_{1},\dots,x_{m},y_{1},\dots,y_{m}$ [2]. We have
$\operatorname{ord}_{\infty}(\varphi_{g})=2g-2$ and
$\operatorname{ord}_{\infty}(\varphi_{g+1})=2g$ from Lemma 2.4. Therefore,
from (2.2), we have
$\operatorname{ord}_{\infty}(\varphi_{g}\varphi_{g+1})=4g-2=2\sum\limits_{i=2}^{m}(d_{i-1}/d_{i}-1)a_{i}-2a_{1}$.
Since

$\displaystyle du_{1}(P)=-\frac{\varphi_{g}(P)}{\det G_{1}(P)}dx_{1},$

if we take $\\{dr_{i}\\}_{i=1}^{g}$ such that
$\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$, then we find that $dr_{1}$ has
the following form

$\displaystyle dr_{1}(Q)=\sum_{i=1}^{g+1}c_{i}\frac{\varphi_{i}(Q)}{\det
G_{1}(Q)}dy_{1},\qquad c_{i}\in\mathbb{C}.$ (3.1)

Let

$\displaystyle\sum_{i=1}^{g}du_{i}(P)dr_{i}(Q)=\frac{\sum
c_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}x_{1}^{i_{1}}\cdots
x_{m}^{i_{m}}y_{1}^{j_{1}}\cdots y_{m}^{j_{m}}}{\det G_{1}(P)\det
G_{1}(Q)}dx_{1}dy_{1},$

where $(i_{1},\dots,i_{m}),(j_{1},\dots,j_{m})\in B(A_{m})$ and
$c_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}\in\mathbb{C}$.

We want to determine the coefficients
$c_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}$ such that
$\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$ explicitly. For $(n,s)$ curves,
i.e., $m=2$, such coefficients are determined explicitly in [25]. Let
$\varphi_{g}(P)=x_{1}^{k_{1}}\cdots x_{m}^{k_{m}}$ and
$\varphi_{g+1}(Q)=y_{1}^{\ell_{1}}\cdots y_{m}^{\ell_{m}}$, where
$(k_{1},\dots,k_{m}),(\ell_{1},\dots,\ell_{m})\in B(A_{m})$. In this paper, in
order to derive the Jacobi inversion formulae for telescopic curves, we
determine the coefficient $c_{k_{1},\dots,k_{m};\ell_{1},\dots,\ell_{m}}$ for
telescopic curves.

######  Proposition 3.3.

We have

$\displaystyle c_{k_{1},\dots,k_{m},\ell_{1},\dots,\ell_{m}}=1.$

In order to prove Proposition 3.3, we need some lemmas.

######  Lemma 3.4.

For $1\leq k\leq m$, we have

$\displaystyle\det G_{k}(Q)=(-1)^{k+1}a_{k}y_{1}^{\gamma_{1}}\cdots
y_{m}^{\gamma_{m}}+\sum\alpha_{i_{1},\dots,i_{m}}y_{1}^{i_{1}}\cdots
y_{m}^{i_{m}},$

where $(\gamma_{1},\dots,\gamma_{m})$ is the unique element of $B(A_{m})$ such
that
$\sum\limits_{j=1}^{m}a_{j}\gamma_{j}=\sum\limits_{j=2}^{m}a_{j}d_{j-1}/d_{j}-\sum\limits_{j=1}^{m}a_{j}+a_{k}$,
$\alpha_{i_{1},\dots,i_{m}}\in\mathbb{C}$ and the sum of the right-hand side
is over all $(i_{1},\dots,i_{m})\in B(A_{m})$ such that
$\sum\limits_{j=1}^{m}a_{j}i_{j}<\sum\limits_{j=2}^{m}a_{j}d_{j-1}/d_{j}-\sum\limits_{j=1}^{m}a_{j}+a_{k}$.
If $\alpha_{i_{1},\dots,i_{m}}\neq 0$, then $\alpha_{i_{1},\dots,i_{m}}$
contains the coefficients of the defining equations.

See Appendix for proof.

######  Lemma 3.5.

We have

$\displaystyle\det
H(P,Q)=\prod_{i=2}^{m}\big{(}x_{i}^{d_{i-1}/d_{i}-1}+x_{i}^{d_{i-1}/d_{i}-2}y_{i}+\cdots+x_{i}y_{i}^{d_{i-1}/d_{i}-2}+y_{i}^{d_{i-1}/d_{i}-1}\big{)}$
$\displaystyle\hphantom{\det
H(P,Q)=}{}+\sum\beta_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}x_{1}^{i_{1}}\cdots
x_{m}^{i_{m}}y_{1}^{j_{1}}\cdots y_{m}^{j_{m}},$

where $\beta_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}\in\mathbb{C}$ and the sum
of the right-hand side is over all $(i_{1},\dots,i_{m}),(j_{1},\dots,j_{m})\in
B(A_{m})$ such that
$\sum\limits_{k=1}^{m}a_{k}(i_{k}+j_{k})<\sum\limits_{k=2}^{m}a_{k}(d_{k-1}/d_{k}-1)$.
If $\beta_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}\neq 0$, then
$\beta_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}$ contains the coefficients of the
defining equations.

###### Proof.

By the definition of $\det H(P,Q)$, when we expand the determinant $\det
H(P,Q)$, the terms which do not contain the coefficients of the defining
equations are

$\displaystyle\prod_{i=2}^{m}\frac{x_{i}^{d_{i-1}/d_{i}}-y_{i}^{d_{i-1}/d_{i}}}{x_{i}-y_{i}}=\prod_{i=2}^{m}\big{(}x_{i}^{d_{i-1}/d_{i}-1}+x_{i}^{d_{i-1}/d_{i}-2}y_{i}+\cdots+x_{i}y_{i}^{d_{i-1}/d_{i}-2}+y_{i}^{d_{i-1}/d_{i}-1}\big{)}.$

Therefore we obtain Lemma 3.5. ∎

######  Lemma 3.6.

Let

$\displaystyle\det G_{1}(Q)\det
H(P,Q)=\sum\gamma_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}x_{1}^{i_{1}}\cdots
x_{m}^{i_{m}}y_{1}^{j_{1}}\cdots y_{m}^{j_{m}},$

where $(i_{1},\dots,i_{m}),(j_{1},\dots,j_{m})\in B(A_{m})$. For $i_{1}\geq
1$, we have
$\gamma_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+2-i_{1},\ell_{2},\dots,\ell_{m}}=\gamma_{i_{1},\ell_{2},\dots,\ell_{m};k_{1}+\ell_{1}+2-i_{1},k_{2},\dots,k_{m}}=0$.
Furthermore we have

$\displaystyle\gamma_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1}+2,\ell_{2},\dots,\ell_{m}}=a_{1}.$

###### Proof.

Note that $\det G_{1}(Q)\det H(P,Q)$ is homogeneous of degree
$2\sum\limits_{i=2}^{m}a_{i}(d_{i-1}/d_{i}-1)$ and $\deg
x_{1}^{i_{1}}x_{2}^{k_{2}}\cdots
x_{m}^{k_{m}}y_{1}^{k_{1}+\ell_{1}+2-i_{1}}y_{2}^{\ell_{2}}\cdots
y_{m}^{\ell_{m}}=2\sum\limits_{i=2}^{m}a_{i}(d_{i-1}/d_{i}-1)$.

Therefore, if
$\gamma_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+2-i_{1},\ell_{2},\dots,\ell_{m}}\neq
0$, then
$\gamma_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+2-i_{1},\ell_{2},\dots,\ell_{m}}$
does not contain the coefficients of the defining equations. From Lemma 3.5,
we have $i_{1}=0$.

Similarly, if
$\gamma_{i_{1},\ell_{2},\dots,\ell_{m};k_{1}+\ell_{1}+2-i_{1},k_{2},\dots,k_{m}}\neq
0$, then $i_{1}=0$. From Lemmas 3.4, 3.5 and (2.1), we obtain
$\gamma_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1}+2,\ell_{2},\dots,\ell_{m}}=a_{1}$.
∎

######  Lemma 3.7.

Let

$\displaystyle\frac{\partial\det H}{\partial y_{i}}(P,Q)\det
G_{i}(Q)=\sum\delta_{i_{1},\dots,i_{m};j_{1},\dots,j_{m}}x_{1}^{i_{1}}\cdots
x_{m}^{i_{m}}y_{1}^{j_{1}}\cdots y_{m}^{j_{m}},$

where $(i_{1},\dots,i_{m}),(j_{1},\dots,j_{m})\in B(A_{m})$. For $i_{1}\geq
1$, we have
$\delta_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+1-i_{1},\ell_{2},\dots,\ell_{m}}=\delta_{i_{1},\ell_{2},\dots,\ell_{m};k_{1}+\ell_{1}+1-i_{1},k_{2},\dots,k_{m}}=0$.
Furthermore we have

$\displaystyle\delta_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1}+1,\ell_{2},\dots,\ell_{m}}=0\qquad\mbox{if}\quad
i=1$ (3.2)

and

$\displaystyle\delta_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1}+1,\ell_{2},\dots,\ell_{m}}=\left(\frac{d_{i-1}}{d_{i}}-1-k_{i}\right)(-1)^{i+1}a_{i}\qquad\mbox{if}\quad
2\leq i\leq m.$ (3.3)

###### Proof.

Note that $\frac{\partial\det H}{\partial y_{i}}(P,Q)\det G_{i}(Q)$ is
homogeneous of degree $2\sum\limits_{j=2}^{m}(d_{j-1}/d_{j}-1)a_{j}-a_{1}$ and
$\deg x_{1}^{i_{1}}x_{2}^{k_{2}}\cdots
x_{m}^{k_{m}}y_{1}^{k_{1}+\ell_{1}+1-i_{1}}y_{2}^{\ell_{2}}\cdots
y_{m}^{\ell_{m}}=2\sum\limits_{j=2}^{m}(d_{j-1}/d_{j}-1)a_{j}-a_{1}$.

Therefore, if
$\delta_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+1-i_{1},\ell_{2},\dots,\ell_{m}}\neq
0$, then
$\delta_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}+1-i_{1},\ell_{2},\dots,\ell_{m}}$
does not contain the coefficients of the defining equations. From Lemma 3.5,
we have $i_{1}=0$.

Similarly, if
$\gamma_{i_{1},\ell_{2},\dots,\ell_{m};k_{1}+\ell_{1}+1-i_{1},k_{2},\dots,k_{m}}\neq
0$, then $i_{1}=0$. From Lemmas 3.4, 3.5 and (2.1), we obtain (3.2) and (3.3).
∎

Let

$\displaystyle\widehat{\omega}(P,Q)=\frac{F(P,Q)}{(x_{1}-y_{1})^{2}\det
G_{1}(P)\det G_{1}(Q)}dx_{1}dy_{1}.$

######  Lemma 3.8.

For $0\leq i_{1}\leq k_{1}$, we have

$\displaystyle
c_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1},\ell_{2},\dots,\ell_{m}}=i_{1}c_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}$
$\displaystyle\hphantom{c_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1},\ell_{2},\dots,\ell_{m}}=}{}+(1-i_{1})c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}.$
(3.4)

###### Proof.

If $k_{1}=0,1$, then Lemma 3.8 holds obviously. Assume $k_{1}\geq 2$. For
$2\leq i_{1}\leq k_{1}$, from Lemmas 3.6 and 3.7, the coefficient of
$x_{1}^{i_{1}}x_{2}^{k_{2}}\cdots
x_{m}^{k_{m}}y_{1}^{k_{1}+\ell_{1}+2-i_{1}}y_{2}^{\ell_{2}}\cdots
y_{m}^{\ell_{m}}$ in $F(P,Q)$ is

$\displaystyle
c_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1},\ell_{2},\dots,\ell_{m}}\\!-2c_{i_{1}-1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1}+1,\ell_{2},\dots,\ell_{m}}\\!+c_{i_{1}-2,k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1}+2,\ell_{2},\dots,\ell_{m}}.\\!$

On the other hand, from $k_{1}+\ell_{1}+2-i_{1}>1$,
$k_{1}+\ell_{1}+2-i_{1}>\ell_{1}+1$, and Lemmas 3.6, 3.7, the coefficient of
$x_{1}^{k_{1}+\ell_{1}+2-i_{1}}x_{2}^{\ell_{2}}\cdots
x_{m}^{\ell_{m}}y_{1}^{i_{1}}y_{2}^{k_{2}}\cdots y_{m}^{k_{m}}$ in $F(P,Q)$ is
zero. Therefore, from $\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$, we have

$\displaystyle
c_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1},\ell_{2},\dots,\ell_{m}}=2c_{i_{1}-1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1}+1,\ell_{2},\dots,\ell_{m}}$
$\displaystyle\hphantom{c_{i_{1},k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1},\ell_{2},\dots,\ell_{m}}=}{}-c_{i_{1}-2,k_{2},\dots,k_{m};k_{1}+\ell_{1}-i_{1}+2,\ell_{2},\dots,\ell_{m}}.$
(3.5)

We prove the equation (3.4) by induction of $i_{1}$. For $i_{1}=0,1$, the
equation (3.4) holds obviously. Assume that the equation (3.4) holds for
$i_{1}=n,n+1,\;(0\leq n\leq k_{1}-2)$. From (3.5) and the assumption of
induction, we have

$\displaystyle
c_{n+2,k_{2},\dots,k_{m};k_{1}+\ell_{1}-n-2,\ell_{2},\dots,\ell_{m}}=2c_{n+1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-n-1,\ell_{2},\dots,\ell_{m}}-c_{n,k_{2},\dots,k_{m};k_{1}+\ell_{1}-n,\ell_{2},\dots,\ell_{m}}$
$\displaystyle\qquad{}=(2n+2)c_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}-2nc_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}$
$\displaystyle\qquad\quad{}-nc_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}+(n-1)c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}$
$\displaystyle\qquad{}=(n+2)c_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}+(-n-1)c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}.$

Therefore the equation (3.4) holds for $i_{1}=n+2$. ∎

###### Proof of Proposition 3.3.

From Lemmas 3.6 and 3.7, the coefficient of $x_{2}^{k_{2}}\cdots
x_{m}^{k_{m}}y_{1}^{k_{1}+\ell_{1}+2}y_{2}^{\ell_{2}}$ $\cdots
y_{m}^{\ell_{m}}$ in $F(P,Q)$ is

$\displaystyle
c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}+a_{1}-\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1-k_{i}\right)a_{i}.$

From $k_{1}+\ell_{1}+2>\ell_{1}+1$, $k_{1}+\ell_{1}+2>1$, and Lemmas 3.6, 3.7,
the coefficient of $x_{1}^{k_{1}+\ell_{1}+2}x_{2}^{\ell_{2}}\cdots
x_{m}^{\ell_{m}}$ $y_{2}^{k_{2}}\cdots y_{m}^{k_{m}}$ in $F(P,Q)$ is zero.
Therefore, from $\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$, we have

$\displaystyle
c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}+a_{1}-\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1-k_{i}\right)a_{i}=0.$
(3.6)

If $k_{1}=0$, then from (2.2)

$\displaystyle
c_{0,k_{2},\dots,k_{m};\ell_{1},\ell_{2},\dots,\ell_{m}}=\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1\right)a_{i}-\sum_{i=1}^{m}a_{i}k_{i}-a_{1}$
$\displaystyle\hphantom{c_{0,k_{2},\dots,k_{m};\ell_{1},\ell_{2},\dots,\ell_{m}}}{}=\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1\right)a_{i}-(2g-2)-a_{1}=1.$

Therefore, if $k_{1}=0$, then Proposition 3.3 holds.

Assume $k_{1}\geq 1$. From Lemmas 3.6 and 3.7, the coefficient of
$x_{1}x_{2}^{k_{2}}\cdots
x_{m}^{k_{m}}y_{1}^{k_{1}+\ell_{1}+1}y_{2}^{\ell_{2}}\cdots y_{m}^{\ell_{m}}$
in $F(P,Q)$ is

$\displaystyle
c_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}-2c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}+\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1-k_{i}\right)a_{i}.$

Since $k_{1}\geq 1$, we have $k_{1}+\ell_{1}+1>\ell_{1}+1$ and
$k_{1}+\ell_{1}+1>1$. Therefore, from Lemmas 3.6 and 3.7, the coefficient of
$x_{1}^{k_{1}+\ell_{1}+1}x_{2}^{\ell_{2}}\cdots
x_{m}^{\ell_{m}}y_{1}y_{2}^{k_{2}}\cdots y_{m}^{k_{m}}$ in $F(P,Q)$ is zero.
Therefore, from $\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$, we have

$\displaystyle
c_{1,k_{2},\dots,k_{m};k_{1}+\ell_{1}-1,\ell_{2},\dots,\ell_{m}}-2c_{0,k_{2},\dots,k_{m};k_{1}+\ell_{1},\ell_{2},\dots,\ell_{m}}+\sum_{i=2}^{m}\left(\frac{d_{i-1}}{d_{i}}-1-k_{i}\right)a_{i}=0.$
(3.7)

From Lemma 3.8 and equations (3.6), (3.7), (2.2), we have
$c_{k_{1},\dots,k_{m};\ell_{1},\dots,\ell_{m}}=1$. ∎

######  Proposition 3.9.

We can take

$\displaystyle dr_{1}(Q)=-\frac{\varphi_{g+1}(Q)}{\det G_{1}(Q)}dy_{1}$

such that $\widehat{\omega}(P,Q)=\widehat{\omega}(Q,P)$.

###### Proof.

From (3.1) and Proposition 3.3, $dr_{1}(Q)$ has the following form

$\displaystyle dr_{1}(Q)=-\frac{\varphi_{g+1}(Q)}{\det
G_{1}(Q)}dy_{1}-\sum_{i=1}^{g}c_{i}^{\prime}du_{i}(Q),$

for certain constants $c_{i}^{\prime}\in\mathbb{C}$. Let

$\displaystyle\omega_{1}(P,Q)=c_{1}^{\prime}du_{1}(P)du_{1}(Q)+\sum_{i=2}^{g}c_{i}^{\prime}du_{1}(P)du_{i}(Q)+\sum_{i=2}^{g}c_{i}^{\prime}du_{i}(P)du_{1}(Q).$

Then $\omega_{1}(P,Q)$ is holomorphic and $\omega_{1}(P,Q)=\omega_{1}(Q,P)$.
By adding $\omega_{1}(P,Q)$ to $\widehat{\omega}(P,Q)$, we obtain Proposition
3.9. ∎

######  Remark 3.10.

There is a certain freedom of choice of the second kind differentials
$\\{dr_{i}\\}_{i=1}^{g}$, i.e., we can add a linear combination of the
holomorphic one forms $\\{du_{i}\\}_{i=1}^{g}$ to $\\{dr_{i}\\}_{i=1}^{g}$. In
[9], for hyperelliptic curves, it is discussed what choice of
$\\{dr_{i}\\}_{i=1}^{g}$ is better for the problem that one considers.

##  4 Sigma function of telescopic curves

Let $X$ be a telescopic curve of genus $g\geq 1$ associated with
$(a_{1},\dots,a_{m})$. We take a fundamental differential of second kind

$\displaystyle\widehat{\omega}(P,Q)=d_{Q}\Omega(P,Q)+\sum_{i=1}^{g}du_{i}(P)dr_{i}(Q),$
(4.1)

such that

$\displaystyle dr_{1}(Q)=-\frac{\varphi_{g+1}(Q)}{\det G_{1}(Q)}dy_{1}.$ (4.2)

This choice is possible from Proposition 3.9. The set
$\\{du_{i},dr_{i}\\}_{i=1}^{g}$ becomes a symplectic basis of the cohomology
group $H^{1}(X,{\mathbb{C}})$ (see [2, 21]).

Take a symplectic basis $\\{\alpha_{i},\beta_{i}\\}_{i=1}^{g}$ of the homology
group and define the period matrices by

$\displaystyle
2\omega_{1}=\left(\int_{\alpha_{j}}du_{i}\right),\qquad\\!2\omega_{2}=\left(\int_{\beta_{j}}du_{i}\right),\qquad\\!-2\eta_{1}=\left(\int_{\alpha_{j}}dr_{i}\right),\qquad\\!-2\eta_{2}=\left(\int_{\beta_{j}}dr_{i}\right).$

The normalized period matrix is given by $\tau=\omega_{1}^{-1}\omega_{2}$.

Let $\delta=\tau\delta^{\prime}+\delta^{\prime\prime}$,
$\delta^{\prime},\delta^{\prime\prime}\in{\mathbb{R}}^{g}$ be the Riemann’s
constant with respect to the choice $(\\{\alpha_{i},\beta_{i}\\},\infty)$. We
set $\delta={}^{t}({}^{t}\delta^{\prime},{}^{t}\delta^{\prime\prime})$.

The sigma function $\sigma(u)$, $u=(u_{1},\dots,u_{g})$ is defined by

$\displaystyle\sigma(u)=C\exp\left(\frac{1}{2}{}^{t}u\eta_{1}\omega_{1}^{-1}u\right)\theta[\delta]\left((2\omega_{1})^{-1}u,\tau\right),$

where $\theta[\delta](u)$ is the Riemann’s theta function with the
characteristic $\delta$ defined by

$\displaystyle\theta[\delta](u)=\sum_{n\in\mathbb{Z}^{g}}\exp\big{\\{}\pi\sqrt{-1}\;{}^{t}(n+\delta^{\prime})\tau(n+\delta^{\prime})+2\pi\sqrt{-1}\;{}^{t}(n+\delta^{\prime})(u+\delta^{\prime\prime})\big{\\}},$

and $C$ is a constant. Since $\delta$ is a half-period from Lemma 2.4,
$\sigma(u)$ vanishes on the Abel–Jacobi image of the $(g-1)$-th symmetric
products of the telescopic curves. This property is important in the proof of
Proposition 4.3.

We have the following propositions.

######  Proposition 4.1 ([2, 21]).

For $m_{1},m_{2}\in\mathbb{Z}^{g}$, we have

$\displaystyle\frac{\sigma(u+2\omega_{1}m_{1}+2\omega_{2}m_{2})}{\sigma(u)}=(-1)^{2({}^{t}\delta^{\prime}m_{1}-{}^{t}\delta^{\prime\prime}m_{2})+{}^{t}m_{1}m_{2}}$
$\displaystyle\hphantom{\frac{\sigma(u+2\omega_{1}m_{1}+2\omega_{2}m_{2})}{\sigma(u)}=}{}\times\exp\big{\\{}{}^{t}(2\eta_{1}m_{1}+2\eta_{2}m_{2})(u+\omega_{1}m_{1}+\omega_{2}m_{2})\big{\\}}.$

######  Proposition 4.2 ([2, 21]).

We have $\sigma(-u)=(-1)^{|\mu(A_{m})|}\sigma(u)$.

The fundamental differential of second kind $\widehat{\omega}(P,Q)$ is
expressed by the sigma function as follows.

######  Proposition 4.3.

If $\sum\limits_{i=1}^{g}P_{i}\in S^{g}(X\backslash\infty)$ is a general
divisor, then for any $1\leq i\leq g$ we have

$\displaystyle\widehat{\omega}(P,Q)=d_{P}d_{Q}\log\sigma\left(\int_{Q}^{P}du-\sum_{j\neq
i}\int_{\infty}^{P_{j}}du\right),$ (4.3)

where $du={}^{t}(du_{1},\dots,du_{g})$.

###### Proof.

For simplicity we prove for $i=1$. Let
$e=-\sum\limits_{j=2}^{g}\int_{\infty}^{P_{j}}du$. Then, from Proposition 4.2,
we have $\sigma(e)=0$. Let

$\displaystyle
E(P,Q)=\sigma\left(\int_{Q}^{P}du-\sum_{j=2}^{g}\int_{\infty}^{P_{j}}du\right).$

Suppose $E(P,Q)$ vanishes identically with respect to $P$, $Q$. Then we have
$E(\infty,P_{1})=0$. Therefore there exist $g-1$ points
$P_{1}^{\prime},\dots,P_{g-1}^{\prime}\in X$ such that

$\displaystyle\sum_{i=1}^{g}\int_{\infty}^{P_{i}}du=\sum_{i=1}^{g-1}\int_{\infty}^{P_{i}^{\prime}}du.$

This contradicts the fact that $\sum\limits_{j=1}^{g}P_{j}\in
S^{g}(X\backslash\infty)$ is a general divisor. Consequently, $E(P,Q)$ does
not vanish identically with respect to $P$, $Q$. Therefore there exist $2g-2$
points $Q_{1},\dots,Q_{g-1}$, $R_{1},\dots,R_{g-1}\in X$ such that the divisor
of zeros of $E(P,Q)$ is the sum of $\\{(R,R)\,|\,R\in X\\}$,
$\\{Q_{j}\\}\times X$, $X\times\\{R_{j}\\}$ ($j=1,\dots,g-1$), including
multiplicities (cf. [20, p. 156]). Let $\widetilde{\omega}(P,Q)$ be the right-
hand side of (4.3). First we consider the series expansion of
$\widetilde{\omega}(P,Q)$ around a point $(R,R)$. Let $t$ be a local
coordinate around $R\in X$ such that $t(R)=0$ and $t_{P}$, $t_{Q}$ two copies
of $t$. Then we have the expansion
$E(P,Q)=(t_{P}-t_{Q})t_{P}^{k}t_{Q}^{\ell}f(t_{P},t_{Q})$ around $(R,R)$,
where $k$, $\ell$ are nonnegative integers and $f(t_{P},t_{Q})$ is a
holomorphic function of $t_{P}$, $t_{Q}$ satisfying $f(t_{P},t_{Q})\neq 0$ for
any $t_{P}$, $t_{Q}$. Hence, around $(R,R)$, we have the expansion

$\displaystyle\widetilde{\omega}(P,Q)=\frac{1}{(t_{P}-t_{Q})^{2}}+(\text{holomorphic
function of}\ t_{P},t_{Q}).$

Next we prove $\widetilde{\omega}(P,Q)$ is holomorphic around a point
$(S_{1},S_{2})$ satisfying $S_{1}\neq S_{2}$. For a local coordinate $t_{i}$
around $S_{i}$ such that $t_{i}(S_{i})=0$, $i=1,2$, we have the expansion
$E(P,Q)=t_{1}^{a}t_{2}^{b}g(t_{1},t_{2})$, where $a$, $b$ are nonnegative
integers and $g(t_{1},t_{2})$ is a holomorphic function of $t_{1}$, $t_{2}$
satisfying $g(t_{1},t_{2})\neq 0$ for any $t_{1}$, $t_{2}$. Hence
$\widetilde{\omega}(P,Q)$ is holomorphic around $(S_{1},S_{2})$ satisfying
$S_{1}\neq S_{2}$. Therefore $\widehat{\omega}(P,Q)-\widetilde{\omega}(P,Q)$
is holomorphic on $X\times X$. Consequently there exist constants
$\\{c_{ij}\\}$ such that

$\displaystyle\widehat{\omega}(P,Q)-\widetilde{\omega}(P,Q)=\sum_{ij}c_{ij}du_{i}(P)du_{j}(Q).$
(4.4)

From (4.1) we have

$\displaystyle\int_{\alpha_{j}}\widehat{\omega}=-\,{}^{t}du(P)(2\eta_{1}e_{j}),$

where the integration is with respect to the second variable and $e_{j}$ is
the $j$-th unit vector. On the other hand we have

$\displaystyle\int_{\alpha_{j}}\widetilde{\omega}=d_{P}\log\sigma\left(\int_{P_{0}}^{P}du-\sum_{j=2}^{g}\int_{\infty}^{P_{j}}du-2\omega_{1}e_{j}\right)-d_{P}\log\sigma\left(\int_{P_{0}}^{P}du-\sum_{j=2}^{g}\int_{\infty}^{P_{j}}du\right),$

where $P_{0}$ is a base point of $\alpha_{j}$. From Proposition 4.1 we have

$\displaystyle\int_{\alpha_{j}}\widetilde{\omega}=d_{P}\left\\{-\,{}^{t}(2\eta_{1}e_{j})\int_{P_{0}}^{P}du\right\\}=-\,{}^{t}(2\eta_{1}e_{j})du(P).$

Therefore we have $\int_{\alpha_{j}}(\widehat{\omega}-\widetilde{\omega})=0$.
If we set $C=(c_{ij})$, then from (4.4) we have ${}^{t}du(P)\cdot
C\cdot(2\omega_{1}e_{j})=0$. Hence we have $C\cdot(2\omega_{1}e_{j})=0$ for
any $j$, i.e., $C\omega_{1}=0$. Since $\omega_{1}$ is a regular matrix, we
have $C=0$. Therefore we have $\widehat{\omega}=\widetilde{\omega}$. ∎

We define the function

$\displaystyle\wp_{i,j}(u)=-\frac{\partial^{2}}{\partial u_{i}\partial
u_{j}}\log\sigma(u).$

Then we have

$\displaystyle\wp_{i,j}(u)=\frac{\sigma_{i}(u)\sigma_{j}(u)-\sigma_{i,j}(u)\sigma(u)}{\sigma(u)^{2}},$
(4.5)

where $\sigma_{i}(u)=\frac{\partial}{\partial u_{i}}\sigma(u)$ and
$\sigma_{i,j}(u)=\frac{\partial^{2}}{\partial u_{i}\partial u_{j}}\sigma(u)$.

We have

$\displaystyle d_{P}d_{Q}\log\sigma\left(\int_{Q}^{P}du-\sum_{j\neq
i}\int_{\infty}^{P_{j}}du\right)$
$\displaystyle\qquad{}=\sum_{k,\ell=1}^{g}\wp_{k,\ell}\left(\int_{Q}^{P}du-\sum_{j\neq
i}\int_{\infty}^{P_{j}}du\right)\frac{\varphi_{g+1-k}(P)\varphi_{g+1-\ell}(Q)}{\det
G_{1}(P)\det G_{1}(Q)}dx_{1}dy_{1}.$

From Proposition 4.3, we have

$\displaystyle\frac{F(P,Q)}{(x_{1}-y_{1})^{2}}=\sum_{k,\ell=1}^{g}\wp_{k,\ell}\left(\int_{Q}^{P}du-\sum_{j\neq
i}\int_{\infty}^{P_{j}}du\right)\varphi_{g+1-k}(P)\varphi_{g+1-\ell}(Q),$
(4.6)

as a meromorphic function of $(P,Q)\in X^{2}$. This formula is an analogue of
the formula of Klein (cf. [8, Theorem 3.4]).

##  5 Frobenius–Stickelberger matrix

For $P_{1},\dots,P_{k},P\in X$, we define the matrix (Frobenius–Stickelberger
matrix) as in the case of [17]

$\displaystyle\left(\begin{matrix}\varphi_{1}(P_{1})&\varphi_{2}(P_{1})&\cdots&\varphi_{k+1}(P_{1})\\\
\varphi_{1}(P_{2})&\varphi_{2}(P_{2})&\cdots&\varphi_{k+1}(P_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
\varphi_{1}(P_{k})&\varphi_{2}(P_{k})&\cdots&\varphi_{k+1}(P_{k})\\\
\varphi_{1}(P)&\varphi_{2}(P)&\cdots&\varphi_{k+1}(P)\end{matrix}\right).$
(5.1)

For $1\leq i\leq k+1$, let $\psi_{k+1}(P_{1},\dots,P_{k};P)$ and
$\psi_{k}^{(i)}(P_{1},\dots,P_{k})$ be the determinants of the matrix (5.1)
and the matrix obtained by deleting the last row and the $i$-th column from
(5.1), respectively. Note that $\psi_{k}^{(i)}(P_{1},\dots,P_{k})$ does not
vanish identically as a meromorphic function of $P_{1},\dots,P_{k}$. We define
$\mu_{k+1}(P_{1},\dots,P_{k};P)$ by

$\displaystyle\mu_{k+1}(P_{1},\dots,P_{k};P)=\frac{\psi_{k+1}(P_{1},\dots,P_{k};P)}{\psi_{k}^{(k+1)}(P_{1},\dots,P_{k})}$

and $\mu_{k,i}(P_{1},\dots,P_{k})$ by

$\displaystyle\mu_{k+1}(P_{1},\dots,P_{k};P)=\sum_{i=1}^{k+1}(-1)^{k+1-i}\mu_{k,i}(P_{1},\dots,P_{k})\varphi_{i}(P)$

and $\mu_{k,i}(P_{1},\dots,P_{k})=0$ for $i\geq k+2$. Then, for $1\leq i\leq
k+1$, we have

$\displaystyle\mu_{k,i}(P_{1},\dots,P_{k})=\frac{\psi_{k}^{(i)}(P_{1},\dots,P_{k})}{\psi_{k}^{(k+1)}(P_{1},\dots,P_{k})}.$

Note that $\mu_{k,i}(P_{1},\dots,P_{k})$ can be regarded as a meromorphic
function on $X^{k}$.

######  Proposition 5.1.

Let $z_{k}$ be the local parameter of $P_{k}$ around $\infty$ satisfying
(2.3). Then, as a meromorphic function of $P_{1},\dots,P_{k}$, we have the
expansion

$\displaystyle\mu_{k,i}(P_{1},\dots,P_{k})=\mu_{k-1,i}(P_{1},\dots,P_{k-1})z_{k}^{N(k)-N(k+1)}+O\big{(}z_{k}^{N(k)-N(k+1)+1}\big{)},$

where $1\leq i\leq k$ and $N(n)=\operatorname{ord}_{\infty}(\varphi_{n})$ for
a positive integer $n$.

###### Proof.

As a meromorphic function of $P_{1},\dots,P_{k}$, we have

$\displaystyle\mu_{k,i}(P_{1},\dots,P_{k})=\frac{\psi_{k}^{(i)}(P_{1},\dots,P_{k})}{\psi_{k}^{(k+1)}(P_{1},\dots,P_{k})}=\frac{\varphi_{k+1}(P_{k})\psi_{k-1}^{(i)}(P_{1},\dots,P_{k-1})+O\big{(}z_{k}^{-N(k+1)+1}\big{)}}{\varphi_{k}(P_{k})\psi_{k-1}^{(k)}(P_{1},\dots,P_{k-1})+O\big{(}z_{k}^{-N(k)+1}\big{)}}$
$\displaystyle\hphantom{\mu_{k,i}(P_{1},\dots,P_{k})}{}=\frac{\big{\\{}z_{k}^{-N(k+1)}+O\big{(}z_{k}^{-N(k+1)+1}\big{)}\big{\\}}\psi_{k-1}^{(i)}(P_{1},\dots,P_{k-1})+O\big{(}z_{k}^{-N(k+1)+1}\big{)}}{\big{\\{}z_{k}^{-N(k)}+O\big{(}z_{k}^{-N(k)+1}\big{)}\big{\\}}\psi_{k-1}^{(k)}(P_{1},\dots,P_{k-1})+O\big{(}z_{k}^{-N(k)+1}\big{)}}$
$\displaystyle\hphantom{\mu_{k,i}(P_{1},\dots,P_{k})}{}=\frac{z_{k}^{-N(k+1)}}{z_{k}^{-N(k)}}\cdot\frac{\psi_{k-1}^{(i)}(P_{1},\dots,P_{k-1})+O(z_{k})}{\psi_{k-1}^{(k)}(P_{1},\dots,P_{k-1})+O(z_{k})}$
$\displaystyle\hphantom{\mu_{k,i}(P_{1},\dots,P_{k})}{}=\mu_{k-1,i}(P_{1},\dots,P_{k-1})z_{k}^{N(k)-N(k+1)}+O\big{(}z_{k}^{N(k)-N(k+1)+1}\big{)}.$
∎

##  6 Riemann’s singularity theorem

Let $X$ be a telescopic curve of genus $g\geq 1$. For a divisor $D$, let
$L(D)$ be the vector space consisting of meromorphic functions $f$ on $X$ such
that $\operatorname{div}(f)+D\geq 0$ and the zero function on $X$, and
$\ell(D)$ the dimension of $L(D)$.

For $1\leq k\leq g-1$ and $P_{1},\dots,P_{k}\in X\backslash\infty$, let

$\displaystyle u^{[k]}=\sum_{i=1}^{k}\int_{\infty}^{P_{k}}du$

and

$\displaystyle n_{k}=\ell(P_{1}+\cdots+P_{k}+(g-k-1)\infty).$

Then the following theorem holds.

######  Theorem 6.1 (Riemann’s singularity theorem, cf. [1, 18, 20]).

  1. $1.$

For every multi-index $(\alpha_{1},\dots,\alpha_{m})$ with
$\alpha_{i}\in\\{1,\dots,g\\}$ and $m<n_{k}$,

$\displaystyle\frac{\partial^{m}}{\partial u_{\alpha_{1}}\cdots\partial
u_{\alpha_{m}}}\sigma\big{(}u^{[k]}\big{)}=0.$

  2. $2.$

There exists a multi-index $(\beta_{1},\dots,\beta_{n_{k}})$, which in general
depends on $P_{1}+\cdots+P_{k}$, such that

$\displaystyle\frac{\partial^{n_{k}}}{\partial u_{\beta_{1}}\cdots\partial
u_{\beta_{n_{k}}}}\sigma\big{(}u^{[k]}\big{)}\neq 0.$

The following proposition is stated for the hyperelliptic curves in [24] and
for the curves $y^{r}=f(x)$ in [17, 18]. The same statement is also satisfied
for telescopic curves. The proof is similar to [24, Proposition 5.2].

######  Proposition 6.2.

If $\psi_{k}^{(k+1)}(P_{1},\dots,P_{k})\neq 0$, then we have
$n_{k}=\sharp\\{n\,|\,0\leq N(n)\leq g-k-1\\}$, where
$N(n)=\operatorname{ord}_{\infty}(\varphi_{n})$ for a positive integer $n$ and
$\sharp$ means the number of elements.

##  7 Jacobi inversion formulae for telescopic curves

For $1\leq k\leq g$ and $P_{1},\dots,P_{k}\in X\backslash\infty$, let

$\displaystyle u^{[k]}=\sum_{j=1}^{k}\int_{\infty}^{P_{j}}du.$

###  7.1 $\boldsymbol{k=g}$

######  Theorem 7.1.

As a meromorphic function of $P_{1},\dots,P_{g}$, we have

$\displaystyle\wp_{1,i}\big{(}u^{[g]}\big{)}=(-1)^{i-1}\mu_{g,g+1-i}(P_{1},\dots,P_{g}),\qquad
1\leq i\leq g.$ (7.1)

###### Proof.

Let $S$ be the set of $(P_{1},\dots,P_{g})\in(X\backslash\infty)^{g}$ such
that $\sum\limits_{i=1}^{g}P_{i}$ is a general divisor and $P_{i}\neq P_{j}$
for any $i$, $j$ $(i\neq j)$. First we prove the equation (7.1) for
$(P_{1},\dots,P_{g})\in S$. From Lemma 3.2(ii), we have
$\operatorname{ord}_{\infty}(\det
H(P,Q))\leq\sum\limits_{i=2}^{m}(d_{i-1}/d_{i}-1)a_{i}$ and
$\operatorname{ord}_{\infty}(x_{1}\frac{\partial\det H}{\partial
y_{k}}(P,Q))\leq\sum\limits_{i=2}^{m}(d_{i-1}/d_{i}-1)a_{i}-a_{k}+a_{1}$ with
respect to $P$. On the other hand, we have
$\operatorname{ord}_{\infty}(x_{1}^{2}\varphi_{g}(P))=-1+a_{1}+\sum\limits_{i=2}^{m}(d_{i-1}/d_{i}-1)a_{i}$.
Since $a_{i}\geq 2$ for any $i$, we have $\operatorname{ord}_{\infty}(\det
H(P,Q))<\operatorname{ord}_{\infty}(x_{1}^{2}\varphi_{g}(P))$ and
$\operatorname{ord}_{\infty}(x_{1}\frac{\partial\det H}{\partial
y_{k}}(P,Q))<\operatorname{ord}_{\infty}(x_{1}^{2}\varphi_{g}(P))$ with
respect to $P$. We let $P\to\infty$ after dividing the both sides of (4.6) by
$\varphi_{g}(P)$ and $Q=P_{i}$. Then, from (4.2), we obtain

$\displaystyle\varphi_{g+1}(P_{i})=\sum_{\ell=1}^{g}\wp_{1,\ell}\big{(}u^{[g]}\big{)}\varphi_{g+1-\ell}(P_{i})=\sum_{j=1}^{g}\wp_{1,g+1-j}\big{(}u^{[g]}\big{)}\varphi_{j}(P_{i}),$

where we use the fact that $\wp_{1,\ell}(u)$ is an even function from
Proposition 4.2. From $(P_{1},\dots,P_{g})\in S$, we have
$\psi_{g}^{(g+1)}(P_{1},\dots,P_{g})\neq 0$ (cf. [1, p. 154]). From
$\mu_{g+1}(P_{1},\dots,P_{g};P_{i})=0$ for any $i$, we have

$\displaystyle\mu_{g+1}(P_{1},\dots,P_{g};P_{i})=\varphi_{g+1}(P_{i})+\sum_{j=1}^{g}(-1)^{g+1-j}\mu_{g,j}(P_{1},\dots,P_{g})\varphi_{j}(P_{i})=0.$

Therefore we have

$\displaystyle\sum_{j=1}^{g}\wp_{1,g+1-j}\big{(}u^{[g]}\big{)}\varphi_{j}(P_{i})=\sum_{j=1}^{g}(-1)^{g-j}\mu_{g,j}(P_{1},\dots,P_{g})\varphi_{j}(P_{i}).$

Therefore we have

$\displaystyle A\cdot\left(\begin{matrix}\alpha_{1}\\\ \vdots\\\
\alpha_{g}\end{matrix}\right)=\left(\begin{matrix}0\\\ \vdots\\\
0\end{matrix}\right),$

where $A=(\varphi_{j}(P_{i}))_{1\leq i,j\leq g}$ and
$\alpha_{j}=\wp_{1,g+1-j}\big{(}u^{[g]}\big{)}-(-1)^{g-j}\mu_{g,j}(P_{1},\dots,P_{g})$.
From $(P_{1},\dots,P_{g})$ $\in S$, we have $\det A\neq 0$. Therefore we have
$\alpha_{j}=0$, i.e.,
$\wp_{1,g+1-j}\big{(}u^{[g]}\big{)}=(-1)^{g-j}\mu_{g,j}(P_{1},\dots,P_{g})$
for any $j$. We set $i=g+1-j$, then we have
$\wp_{1,i}(u^{[g]})=(-1)^{i-1}\mu_{g,g+1-i}(P_{1},\dots,P_{g})$.

Let $T$ be the set of $(P_{1},\dots,P_{g})\in(X\backslash\infty)^{g}$ such
that $\psi_{g}^{(g+1)}(P_{1},\dots,P_{g})\neq 0$. Then we have $S=T$ (cf. [1,
p. 154]). Since the equation (7.1) holds for any $(P_{1},\dots,P_{g})\in T$,
it holds as a meromorphic function of $P_{1},\dots,P_{g}$. ∎

######  Remark 7.2.

As discussed in [17], for a hyperelliptic curve, $\mu_{g,g+1-i}$ is equal to
the symmetric polynomial $e_{i}$. Therefore Theorem 7.1 is a natural
generalization of the Jacobi inversion formulae for hyperelliptic curves to
telescopic curves.

###  7.2 $\boldsymbol{k\leq g-1}$

Let $a=\min\\{a_{1},\dots,a_{m}\\}$. Hereafter we assume $g-a\leq k\leq g-1$.

######  Theorem 7.3.

$\sigma_{g-k}\big{(}u^{[k]}\big{)}$ does not vanish identically with respect
to $P_{1},\dots,P_{k}$ and we have, as a meromorphic function of
$P_{1},\dots,P_{k}$,

$\displaystyle\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k}\big{(}u^{[k]}\big{)}}=(-1)^{k+i-g}\mu_{k,g+1-i}(P_{1},\dots,P_{k}),\qquad
1\leq i\leq g.$

###### Proof.

First we prove for $k=g-1$. Let $z_{g}$ be the local parameter of $P_{g}$
around $\infty$ satisfying (2.3). From [3, Theorem 2],
$\sigma_{1}\big{(}u^{[g-1]}\big{)}$ does not vanish identically with respect
to $P_{1},\dots$, $P_{g-1}$ and we have the expansion

$\displaystyle\sigma\left(u^{[g-1]}+\int_{\infty}^{P_{g}}du\right)=\sigma_{1}\big{(}u^{[g-1]}\big{)}z_{g}+O\big{(}z_{g}^{2}\big{)}.$

From Theorem 7.1 and (4.5), we have

$\displaystyle\frac{\sigma_{i}\left(u^{[g-1]}+{\int_{\infty}^{P_{g}}du}\right)\sigma_{1}\left(u^{[g-1]}+{\int_{\infty}^{P_{g}}du}\right)-\sigma_{1,i}\left(u^{[g-1]}+{\int_{\infty}^{P_{g}}du}\right)\sigma\left(u^{[g-1]}+{\int_{\infty}^{P_{g}}du}\right)}{\sigma\left(u^{[g-1]}+{\int_{\infty}^{P_{g}}du}\right)^{2}}$
$\displaystyle\qquad{}=(-1)^{i-1}\mu_{g,g+1-i}(P_{1},\dots,P_{g}),$

as a meromorphic function of $P_{1},\dots,P_{g}$. Therefore, from Proposition
5.1, we have

$\displaystyle\frac{\big{\\{}\sigma_{i}\big{(}u^{[g-1]}\big{)}+O(z_{g})\big{\\}}\cdot\big{\\{}\sigma_{1}\big{(}u^{[g-1]}\big{)}+O(z_{g})\big{\\}}+O(z_{g})}{\sigma_{1}\big{(}u^{[g-1]}\big{)}^{2}z_{g}^{2}+O(z_{g}^{3})}$
$\displaystyle\qquad{}=(-1)^{i-1}z_{g}^{-2}\mu_{g-1,g+1-i}(P_{1},\dots,P_{g-1})+O\big{(}z_{g}^{-1}\big{)}.$

By comparing the coefficient of $z_{g}^{-2}$ of the above equation, we find
that $\sigma_{i}\big{(}u^{[g-1]}\big{)}$ does not vanish identically with
respect to $P_{1},\dots,P_{g-1}$ and we have, as a meromorphic function of
$P_{1},\dots,P_{g-1}$,

$\displaystyle\frac{\sigma_{i}\big{(}u^{[g-1]}\big{)}}{\sigma_{1}\big{(}u^{[g-1]}\big{)}}=(-1)^{i-1}\mu_{g-1,g+1-i}(P_{1},\dots,P_{g-1}).$

Next we prove Theorem 7.3 for $g-a\leq k\leq g-2$ by induction of $k$ as in
the case of [17]. Assume that Theorem 7.3 holds for $k$ satisfying $g-a+1\leq
k\leq g-1$. Then, for $i\geq g-k$, $\sigma_{i}\big{(}u^{[k]}\big{)}$ does not
vanish identically with respect to $P_{1},\dots,P_{k}$. From the assumption of
induction, we have, as a meromorphic function of $P_{1},\dots,P_{k}$,

$\displaystyle\frac{\sigma_{g-k+1}\big{(}u^{[k]}\big{)}}{\sigma_{g-k}\big{(}u^{[k]}\big{)}}=-\mu_{k,k}(P_{1},\dots,P_{k}).$

From Proposition 5.1, we have

$\displaystyle\mu_{k,k}(P_{1},\dots,P_{k})=z_{k}^{N(k)-N(k+1)}+O\big{(}z_{k}^{N(k)-N(k+1)+1}\big{)}.$

By the assumption of induction we have, as a meromorphic function of
$P_{1},\dots,P_{k}$,

$\displaystyle\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k}\big{(}u^{[k]}\big{)}}=(-1)^{k+i-g}\mu_{k,g+1-i}(P_{1},\dots,P_{k}).$
(7.2)

By multiplying the both sides of (7.2) by
$\sigma_{g-k}\big{(}u^{[k]}\big{)}/\sigma_{g-k+1}\big{(}u^{[k]}\big{)}$, we
have, as a meromorphic function of $P_{1},\dots,P_{k}$,

$\displaystyle\frac{\sigma_{g-k}\big{(}u^{[k]}\big{)}}{\sigma_{g-k+1}\big{(}u^{[k]}\big{)}}\cdot\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k}\big{(}u^{[k]}\big{)}}=(-1)^{k+i-g}\frac{\sigma_{g-k}\big{(}u^{[k]}\big{)}}{\sigma_{g-k+1}\big{(}u^{[k]}\big{)}}\cdot\mu_{k,g+1-i}(P_{1},\dots,P_{k}).$

Therefore, from Proposition 5.1, we have

$\displaystyle\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k+1}\big{(}u^{[k]}\big{)}}=(-1)^{k-1+i-g}\big{\\{}z_{k}^{N(k+1)-N(k)}+O\big{(}z_{k}^{N(k+1)-N(k)+1}\big{)}\big{\\}}\cdot\mu_{k,g+1-i}(P_{1},\dots,P_{k})$
$\displaystyle\hphantom{\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k+1}\big{(}u^{[k]}\big{)}}}{}=(-1)^{k-1+i-g}\mu_{k-1,g+1-i}(P_{1},\dots,P_{k-1})+O(z_{k}).$
(7.3)

Since $\psi_{k-1}^{(k)}(P_{1},\dots,P_{k-1})$ does not vanish identically with
respect to $P_{1},\dots,P_{k-1}$, there exist
$\tilde{P}_{1},\dots,\tilde{P}_{k-1}\in X\backslash\infty$ such that
$\psi_{k-1}^{(k)}(\tilde{P}_{1},\dots,\tilde{P}_{k-1})\neq 0$. Let
$\tilde{u}^{[k-1]}=\sum\limits_{i=1}^{k-1}\int_{\infty}^{\tilde{P}_{i}}du$.
From $g-a<k$, we have $g-k<a$. Therefore, from Theorem 6.1 and Proposition
6.2, there exists $i_{0}$ such that $\sigma_{i_{0}}(\tilde{u}^{[k-1]})\neq 0$.
Therefore $\sigma_{i_{0}}\big{(}u^{[k-1]}\big{)}$ does not vanish identically
with respect to $P_{1},\dots,P_{k-1}$. Since the equation (7.3) holds for
$i=i_{0}$, we find that $\sigma_{g-k+1}\big{(}u^{[k-1]}\big{)}$ does not
vanish identically with respect to $P_{1},\dots,P_{k-1}$. Take the limit
$P_{k}\to\infty$ in (7.3), then we have

$\displaystyle\frac{\sigma_{i}\big{(}u^{[k-1]}\big{)}}{\sigma_{g-k+1}\big{(}u^{[k-1]}\big{)}}=(-1)^{k-1+i-g}\mu_{k-1,g+1-i}(P_{1},\dots,P_{k-1}).$

Therefore Theorem 7.3 holds for $k-1$. ∎

######  Corollary 7.4.

If $g=a-1,a,a+1$, then we have

$\displaystyle\frac{\sigma_{g}\big{(}u^{[1]}\big{)}}{\sigma_{g-1}\big{(}u^{[1]}\big{)}}=-x_{i_{0}}(P_{1}),$
(7.4)

where $i_{0}$ is determined by $a_{i_{0}}=\arg\min\\{a_{1},\dots,a_{m}\\}$.

###### Proof.

If $g=a-1,a,a+1$, then Theorem 7.3 holds for $k=1$. ∎

######  Remark 7.5.

Corollary 7.4 asserts that the $x_{i_{0}}$ coordinate of $P_{1}$ is expressed
by the sigma function. For example, Corollary 7.4 holds for $(4,6,5)$ curves.

######  Remark 7.6.

For $(2,5)$ and $(2,7)$ curves, it is known that $x_{2}$ coordinate of $P_{1}$
can be expressed explicitly by the sigma function (see [23, Lemma 3.2.4] and
[7, p. 221]). For example, for $(2,5)$ curves, it is known that

$\displaystyle
x_{2}(P_{1})=\frac{1}{2}\cdot\frac{\sigma\big{(}2u^{[1]}\big{)}}{\sigma_{1}\big{(}u^{[1]}\big{)}^{4}}.$

On the other hand, for $(2,5)$ and $(2,7)$ curves, it is known that the
expression of $x_{2}$ coordinate of $P_{1}$ can also be derived by
differentiating the both sides of (7.4) (see [7, p. 221] and [17, Remark
5.4]). For example, for $(2,5)$ curves, it is known that

$\displaystyle
x_{2}(P_{1})=\frac{1}{2}\cdot\frac{\sigma_{11}\big{(}u^{[1]}\big{)}x_{1}(P_{1})^{2}+2\sigma_{12}\big{(}u^{[1]}\big{)}x_{1}(P_{1})+\sigma_{22}\big{(}u^{[1]}\big{)}}{\sigma_{1}\big{(}u^{[1]}\big{)}}.$

Although the similar expressions for the other coordinates of telescopic
curves are not obtained currently, we will consider a generalization of these
results to telescopic curves in a subsequent work.

######  Remark 7.7.

Theorem 7.1 holds for $\widehat{\omega}(P,Q)$ satisfying (4.2). On the other
hand, Theorem 7.3 holds for any choice of $\widehat{\omega}(P,Q)$.

######  Remark 7.8.

As mentioned in [18], Theorem 7.3 for $k=g-1$ can also be proved by [21,
Theorem 1] and [12, Theorem 1].

######  Remark 7.9.

In this paper, we consider the Jacobi inversion formulae for the telescopic
curves, which the Young diagrams are symmetric, i.e., the vector of Riemann
constants for a base point $\infty$ is a half-period. On the other hand, in
[15, 16], the Jacobi inversion formulae are derived for $(3,4,5)$ curves and
$(3,7,8)$ curves, which the Young diagrams are not symmetric, i.e., the vector
of Riemann constants for a base point $\infty$ is not a half-period.

##  8 Example: $\boldsymbol{(4,6,5)}$-curve

In this section we give an explicit example of the Jacobi inversion formulae
in the case of a $(4,6,5)$-curve $X$. The genus of $X$ is 4 and
$\varphi_{1}=1$, $\varphi_{2}=x_{1}$, $\varphi_{3}=x_{3}$,
$\varphi_{4}=x_{2}$, $\varphi_{5}=x_{1}^{2}$. Therefore the Jacobi inversion
formulae are as follows.

For $k=4$, $i=1$, we have

$\displaystyle\wp_{1,1}\big{(}u^{[4]}\big{)}=\frac{\left|\begin{matrix}1&x_{1}(P_{1})&x_{3}(P_{1})&x_{1}^{2}(P_{1})\\\
1&x_{1}(P_{2})&x_{3}(P_{2})&x_{1}^{2}(P_{2})\\\
1&x_{1}(P_{3})&x_{3}(P_{3})&x_{1}^{2}(P_{3})\\\
1&x_{1}(P_{4})&x_{3}(P_{4})&x_{1}^{2}(P_{4})\end{matrix}\right|}{\left|\begin{matrix}1&x_{1}(P_{1})&x_{3}(P_{1})&x_{2}(P_{1})\\\
1&x_{1}(P_{2})&x_{3}(P_{2})&x_{2}(P_{2})\\\
1&x_{1}(P_{3})&x_{3}(P_{3})&x_{2}(P_{3})\\\
1&x_{1}(P_{4})&x_{3}(P_{4})&x_{2}(P_{4})\end{matrix}\right|}.$

For $k=3$, $i=2$, we have

$\displaystyle\frac{\sigma_{2}\big{(}u^{[3]}\big{)}}{\sigma_{1}\big{(}u^{[3]}\big{)}}=-\frac{\left|\begin{matrix}1&x_{1}(P_{1})&x_{2}(P_{1})\\\
1&x_{1}(P_{2})&x_{2}(P_{2})\\\
1&x_{1}(P_{3})&x_{2}(P_{3})\end{matrix}\right|}{\left|\begin{matrix}1&x_{1}(P_{1})&x_{3}(P_{1})\\\
1&x_{1}(P_{2})&x_{3}(P_{2})\\\
1&x_{1}(P_{3})&x_{3}(P_{3})\end{matrix}\right|}.$

For $k=2$, we have

$\displaystyle\frac{\sigma_{3}\big{(}u^{[2]}\big{)}}{\sigma_{2}\big{(}u^{[2]}\big{)}}=\frac{x_{3}(P_{1})-x_{3}(P_{2})}{x_{1}(P_{2})-x_{1}(P_{1})},\qquad\frac{\sigma_{4}\big{(}u^{[2]}\big{)}}{\sigma_{2}\big{(}u^{[2]}\big{)}}=\frac{x_{1}(P_{1})x_{3}(P_{2})-x_{1}(P_{2})x_{3}(P_{1})}{x_{1}(P_{2})-x_{1}(P_{1})}.$

For $k=1$, we have

$\displaystyle\frac{\sigma_{4}\big{(}u^{[1]}\big{)}}{\sigma_{3}\big{(}u^{[1]}\big{)}}=-x_{1}(P_{1}).$

##  9 Vanishing of $\boldsymbol{\sigma_{i}}$

In [3, 22], the vanishing and the expansion of the sigma functions of $(n,s)$
curves and telescopic curves on the Abel–Jacobi image are studied. In this
section, we show that from Theorem 7.3 we can derive some new vanishing
properties of $\sigma_{i}$ for telescopic curves immediately.

######  Corollary 9.1.

If $g-a\leq k\leq g-1$ and $i\geq g-k$, then $\sigma_{i}\big{(}u^{[k]}\big{)}$
does not vanish identically with respect to $P_{1},\dots,P_{k}$.

###### Proof.

For $i\geq g-k$, $\mu_{k,g+1-i}(P_{1},\dots,P_{k})$ does not vanish
identically with respect to $P_{1},\dots,P_{k}$. Therefore Corollary 9.1
follows from Theorem 7.3. ∎

For $g-a\leq k\leq g-1$ and $i>g-k$, we consider the expansion

$\displaystyle\sigma_{g-k}\left(u^{[k-1]}+\int_{\infty}^{P_{k}}du\right)=C_{k}\big{(}u^{[k-1]}\big{)}z_{k}^{\alpha_{k}}+O\big{(}z_{k}^{\alpha_{k}+1}\big{)}$

and

$\displaystyle\sigma_{i}\left(u^{[k-1]}+\int_{\infty}^{P_{k}}du\right)=C_{k,i}\big{(}u^{[k-1]}\big{)}z_{k}^{\beta_{k,i}}+O\big{(}z_{k}^{\beta_{k,i}+1}\big{)},$

where $C_{k}\big{(}u^{[k-1]}\big{)}$ and $C_{k,i}\big{(}u^{[k-1]}\big{)}$ do
not vanish identically with respect to $P_{1},\dots,P_{k-1}$.

######  Corollary 9.2.

  1. $(i)$

We have $\alpha_{k}=\beta_{k,i}+N(k+1)-N(k)$. In particular, if $g-a<k\leq
g-1$ and $i>g-k$, then we have $\beta_{k,i}=0$ and $\alpha_{k}=N(k+1)-N(k)$.

  2. $(ii)$

We have, as a meromorphic function of $P_{1},\dots,P_{k-1}$,

$\displaystyle\frac{C_{k,i}\big{(}u^{[k-1]}\big{)}}{C_{k}\big{(}u^{[k-1]}\big{)}}=(-1)^{k+i-g}\mu_{k-1,g+1-i}(P_{1},\dots,P_{k-1}).$
(9.1)

###### Proof.

From Theorem 7.3, we have

$\displaystyle\frac{\sigma_{i}\big{(}u^{[k]}\big{)}}{\sigma_{g-k}\big{(}u^{[k]}\big{)}}=(-1)^{k+i-g}\mu_{k,g+1-i}(P_{1},\dots,P_{k}).$

Therefore we have

$\displaystyle\frac{C_{k,i}\big{(}u^{[k-1]}\big{)}z_{k}^{\beta_{k,i}}+O\big{(}z_{k}^{\beta_{k,i}+1}\big{)}}{C_{k}\big{(}u^{[k-1]}\big{)}z_{k}^{\alpha_{k}}+O\big{(}z_{k}^{\alpha_{k}+1}\big{)}}$
$\displaystyle\qquad{}=(-1)^{k+i-g}\mu_{k-1,g+1-i}(P_{1},\dots,P_{k-1})z_{k}^{N(k)-N(k+1)}+O\big{(}z_{k}^{N(k)-N(k+1)+1}\big{)}.$

Therefore we obtain $\beta_{k,i}-\alpha_{k}=N(k)-N(k+1)$ and (9.1). On the
other hand, if $g-a<k\leq g-1$ and $i>g-k$, then from Corollary 9.1
$\sigma_{i}\big{(}u^{[k-1]}\big{)}$ does not vanish identically with respect
to $P_{1},\dots,P_{k-1}$. Therefore, if $g-a<k\leq g-1$ and $i>g-k$, then
$\beta_{k,i}=0$. ∎

##  10 Example: $\boldsymbol{(4,6,5)}$-curve

By applying Corollary 9.1 for the $(4,6,5)$ curves, we have
$\sigma_{3}\big{(}u^{[1]}\big{)}\not\equiv 0$,
$\sigma_{4}\big{(}u^{[1]}\big{)}\not\equiv 0$,
$\sigma_{2}\big{(}u^{[2]}\big{)}\not\equiv 0$,
$\sigma_{3}\big{(}u^{[2]}\big{)}\not\equiv 0$,
$\sigma_{4}\big{(}u^{[2]}\big{)}\not\equiv 0$,
$\sigma_{1}\big{(}u^{[3]}\big{)}\not\equiv 0$,
$\sigma_{2}\big{(}u^{[3]}\big{)}\not\equiv 0$,
$\sigma_{3}\big{(}u^{[3]}\big{)}\not\equiv 0$,
$\sigma_{4}\big{(}u^{[3]}\big{)}\not\equiv 0$.

By applying Corollary 9.2 for the $(4,6,5)$ curves, we have

$\displaystyle\sigma_{1}\big{(}u^{[3]}\big{)}=C_{3}\big{(}u^{[2]}\big{)}z_{3}+O\big{(}z_{3}^{2}\big{)},\qquad\sigma_{2}\big{(}u^{[2]}\big{)}=C_{2}\big{(}u^{[1]}\big{)}z_{2}+O\big{(}z_{2}^{2}\big{)},$
$\displaystyle\sigma_{3}\big{(}u^{[1]}\big{)}=C_{1}z_{1}^{4}+O\big{(}z_{1}^{5}\big{)},$

where $C_{3}\big{(}u^{[2]}\big{)}\not\equiv 0$,
$C_{2}\big{(}u^{[1]}\big{)}\not\equiv 0$, and $C_{1}\neq 0$.

##  Appendix A Proof of Lemma 3.4

From (2.1), for $2\leq i\leq m$, we have

$\displaystyle\frac{\partial F_{i}}{\partial
y_{n}}=\begin{cases}-\ell_{i,n}y_{1}^{\ell_{i,1}}\cdots
y_{n}^{\ell_{i,n}-1}\cdots y_{i-1}^{\ell_{i,i-1}}-\sum
j_{n}\lambda_{j_{1},\dots,j_{m}}^{(i)}y_{1}^{j_{1}}\cdots
y_{n}^{j_{n}-1}\cdots y_{m}^{j_{m}},&1\leq n\leq i-1,\\\
(d_{i-1}/d_{i})y_{i}^{d_{i-1}/d_{i}-1}-\sum
j_{i}\lambda_{j_{1},\dots,j_{m}}^{(i)}y_{1}^{j_{1}}\cdots
y_{i}^{j_{i}-1}\cdots y_{m}^{j_{m}},&n=i,\\\ -\sum
j_{n}\lambda_{j_{1},\dots,j_{m}}^{(i)}y_{1}^{j_{1}}\cdots
y_{n}^{j_{n}-1}\cdots y_{m}^{j_{m}},&i+1\leq n\leq m.\end{cases}$

Let $\epsilon_{k}$ be the coefficient of $y_{1}^{\gamma_{1}}\cdots
y_{m}^{\gamma_{m}}$ in $\det G_{k}(Q)$. Since $\det G_{k}(Q)$ is homogeneous
of degree
$\sum\limits_{i=2}^{m}a_{i}d_{i-1}/d_{i}-\sum\limits_{i=1}^{m}a_{i}+a_{k}$ and
$\sum\limits_{i=1}^{m}a_{i}\gamma_{i}=\sum\limits_{i=2}^{m}a_{i}d_{i-1}/d_{i}-\sum\limits_{i=1}^{m}a_{i}+a_{k}$,
$\epsilon_{k}$ does not contain
$\big{\\{}\lambda_{j_{1},\dots,j_{m}}^{(i)}\big{\\}}$. Therefore
$\epsilon_{k}$ is the determinant of the $(m-1)\times(m-1)$ matrix obtained by
deleting the $k$-th column from the $(m-1)\times m$ matrix $M$

$\displaystyle M:=\left(\begin{matrix}-\ell_{2,1}&d_{1}/d_{2}&0&\cdots&0\\\
-\ell_{3,1}&-\ell_{3,2}&d_{2}/d_{3}&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\ell_{m,1}&-\ell_{m,2}&\cdots&-\ell_{m,m-1}&d_{m-1}/d_{m}\end{matrix}\right).$

By multiplying some elementary matrices on the left, the matrix $M$ becomes

$\displaystyle\widetilde{M}=\left(\begin{matrix}z_{2}&d_{1}/d_{2}&0&\cdots&0\\\
z_{3}&0&d_{2}/d_{3}&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
z_{m}&0&0&\cdots&d_{m-1}/d_{m}\end{matrix}\right)$

for certain $z_{2},\dots,z_{m}\in\mathbb{C}$. For $k=1$, we have

$\displaystyle\epsilon_{1}=\frac{d_{1}}{d_{2}}\cdot\frac{d_{2}}{d_{3}}\cdots\frac{d_{m-1}}{d_{m}}=\frac{d_{1}}{d_{m}}=a_{1}.$

For $k\geq 2$, we have

$\displaystyle\epsilon_{k}=(-1)^{k}z_{k}\cdot\frac{d_{1}}{d_{2}}\cdots\check{\frac{d_{k-1}}{d_{k}}}\cdots\frac{d_{m-1}}{d_{m}}=(-1)^{k}z_{k}\cdot
a_{1}\frac{d_{k}}{d_{k-1}},$

where a check on top of a letter signifies deletion.

Since

$\displaystyle M\left(\begin{matrix}a_{1}\\\ \vdots\\\
a_{m}\end{matrix}\right)=\left(\begin{matrix}0\\\ \vdots\\\
0\end{matrix}\right),$

we have

$\displaystyle\widetilde{M}\left(\begin{matrix}a_{1}\\\ \vdots\\\
a_{m}\end{matrix}\right)=\left(\begin{matrix}0\\\ \vdots\\\
0\end{matrix}\right).$

Therefore we have $z_{k}a_{1}+(d_{k-1}/d_{k})a_{k}=0$ for $2\leq k\leq m$.
Therefore we have $\epsilon_{k}=(-1)^{k+1}a_{k}$.

### Acknowledgements

The author would like to thank Professor Shigeki Matsutani for answering a
question on the paper [18] kindly and sending his unpublished paper. The
author would like to thank Professor Atsushi Nakayashiki for inviting him the
conference “Curves, Moduli and Integrable Systems” at Tsuda College and giving
valuable discussions. The author would like to thank Professor Masato Okado
for the support of travel costs for a presentation at Tsukuba University. The
author would like to thank Professor Yoshihiro Onishi for inviting him Meijo
University and giving valuable discussions. The author would like to thank the
anonymous referees for reading our paper carefully and giving many valuable
comments. In particular, the author is deeply grateful for their warm
encouragement.

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