1.1. The curves

We shall use $e$ and $q$ instead of $n$ and $s$ , respectively, of the $(n,s)$ -curves, which was usual in many previous papers on generalised sigma functions. Especially, the name $(n,s)$ -curve (which comes from singularity theory) is used by Buchstaber and Leykin in their papers, but we wish to avoid confusion with the many $n$ used as subscripts in sections from Section 6 onward, and the use of $s$ for Schur polynomials in Subsection 4.1.

So, we let $e$ and $q$ be two fixed positive integers such that $e\lt q$ and $\gcd (e,q)=1$ . We define, for these integers, a polynomial of indeterminates $X$ and $Y$

(1.1) \begin{equation} f(X,Y)=Y^e-p_1(X)Y^{e-1}-\cdots -p_{e-1}(X)Y-p_e(X), \end{equation}

where $p_j(X)$ is a polynomial of $X$ of degree $\lceil \tfrac {jq}{e}\rceil$ or smaller and its coefficients, which are also indeterminates, are denoted by

(1.2) \begin{equation} \begin{aligned} p_j(X)&=\sum _{k:jq-ek\gt 0}\mu _{jq-ek}\,X^k \ \ \ (1\leq j\leq e-1),\\[3pt] p_e(X)&=X^q+\mu _{e(q-1)}X^{q-1}+\cdots +\mu _{eq}. \end{aligned} \end{equation}

Please note that the sign at the front of each $p_j(X)$ with $j\neq e$ in $f(X,Y)$ is different from previous papers written by some of the authors. The base ring over which we work is quite general. For simplicity, the reader may start by taking the field $\mathbb {C}$ of complex numbers and assume the $\mu _i$ s to be constants belonging to this field. Let $\mathscr {C}=\mathscr {C}_{{\mu }}^{e,q}$ be the projective curve defined by

(1.3) \begin{equation} f(x,y)=0 \end{equation}

having a unique point $\infty$ at infinity. This means $(x,y)$ is a generic point of $\mathscr {C}$ in the classical terminology.

As the general elliptic curve is defined by an equation of the form

\begin{align*} y^2-(\mu _1x+\mu _3)y=x^3+\mu _2x^2+\mu _4x+\mu _6, \end{align*}

the curves $\mathscr {C}$ discussed here are a natural generalisation of elliptic curves.

The reason why we omit the terms of $\mu _j$ with $j\lt 0$ from $f(X,Y)$ is seen in the proof of 1.17.

Basically our situation is that all the coefficients $\mu _j$ of $f(X,Y)$ should be indeterminates. We denote by $\mathbb {Q}[{\mu }]$ the ring generated over the rationals $\mathbb {Q}$ by all the $\mu _j$ s. Then we shall treat $\mathscr {C}$ as a scheme over the $\mathrm {Spec}\,\mathbb {Q}[\mu ]$ . Since we need to use analytic methods from time to time, we freely switch the standing position where $\mu _j$ s are assumed to be complex numbers or indeterminates.

This $\mathscr {C}$ should be called an $(e,q)$ -curve following Buchstaber, Enolskii, and Leykin [Reference Buchstaber, Enolskii and Leykin10], or a plane telescopic curve after the paper [Reference Miura21]. Assuming all the $\mu _j$ s are complex numbers, the genus of $\mathscr {C}$ is $(e-1)(q-1)/2$ provided that it is non-singular. We will use $g$ to denote this quantity throughout this paper whether the curve $\mathscr {C}$ is non-singular or singular as well as in the case of the $\mu _j$ s being indeterminates: $g=(e-1)(q-1)/2$ . In this paper, we denote by

\begin{align*} f_X(x,y)\ \ \text {or}\ \ f_1(x,y) \ \ \ \ \text {[\,resp.}\ \ f_{\,Y}(x,y)\ \ \text {or}\ \ f_2(x,y)] \end{align*}

the polynomials obtained by substitution $X=x$ , $Y=y$ for the partial derivative of the polynomial $f(X,Y)$ with respect to $X$ [ resp. $Y$ ]

Now we introduce a weight function as follows. For a point $(x,y)$ in the curve $\mathscr {C}$ given by (1.1), we define the weight $\mathrm {wt}()$ on $\mathbb {Q}[\mu ][x,y]$ and $\mathbb {Q}[\mu ][X,Y]$ by

(1.4) \begin{equation} \mathrm {wt}(\mu _j)=-j, \ \ \ \mathrm {wt}(x)=\mathrm {wt}(X)=-e, \ \ \ \mathrm {wt}(y)=\mathrm {wt}(Y)=-q. \end{equation}

Then all the equations for functions, power series, differential forms, and so on in this paper are of homogeneous weight. We see that $\mathrm {wt}\big (f(X,Y)\big )=-eq$ . We will extend the notion of weight (1.4) in Subsection 4.2.

1.2. Definition of the discriminant

We shall define the discriminant of the curve $\mathscr {C}$ .

For a curve $\mathscr {C}_0$ given by some fixed constants $\mu _j\in \mathbb {C}$ , we always define its discriminant as the one obtained by substituting these constants to the discriminant $\varDelta$ defined in 1.5. For instance, if $(e,q)=(2,3)$ , then

\begin{align*} f(X,Y)=Y^2-(\mu _1X+\mu _3)Y-(X^3+\mu _2X^2+\mu _4X+\mu _6) \end{align*}

and its discriminant is given by

(1.6) \begin{equation} \begin{aligned} \varDelta &=-\mu _6{\mu _1}^6+\mu _3\mu _4{\mu _1}^5+(({-}{\mu _3}^2-12\mu _6)\mu _2+{\mu _4}^2){\mu _1}^4\\[3pt] &\ \ +(8\mu _3\mu _4\mu _2+{\mu _3}^3+36\mu _6\mu _3){\mu _1}^3+(({-}8{\mu _3}^2-48\mu _6){\mu _2}^2+8{\mu _4}^2\mu _2\\[3pt] &\ \ +({-}30{\mu _3}^2+72\mu _6)\mu _4){\mu _1}^2+(16\mu _3\mu _4{\mu _2}^2+(36{\mu _3}^3+144\mu _6\mu _3)\mu _2-96\mu _3{\mu _4}^2)\mu _1\\[3pt] &\ \ +({-}16{\mu _3}^2-64\mu _6){\mu _2}^3+16{\mu _4}^2{\mu _2}^2+(72{\mu _3}^2+288\mu _6)\mu _4\mu _2\\[3pt] &\ \ -64{\mu _4}^3-27{\mu _3}^4-216\mu _6{\mu _3}^2-432{\mu _6}^2. \end{aligned} \end{equation}

Specialising $\mu _1=\mu _3=\mu _2=0$ , we have the discriminant $\varDelta =-16(4{\mu _4}^3+27{\mu _6}^2)$ of the curve defined by the Weierstrass form $y^2=x^3+\mu _4x+\mu _6$ which appeared in (0.1).

For $(e,q)=(2,2g+1)$ , as in Section 1.3, we rewrite the equation as $y^2=x^{2g+1}+\cdots$ , where the right-hand side is a polynomial of $x$ only. Then the discriminant of this curve is a non-zero integer multiple of the discriminant of the right-hand side as a polynomial of $x$ only. For the $(3,4)$ -curve, we have Sylvester’s method as described in [Reference Gelfand, Kapranov and Zelevinsky15] pp.118-120, as explained to the authors by C. Ritzenthaler. However, Lemma 1.21 below gives a quite general method, which seems to cover the $(3,5)$ -curve and more. We do have explicit forms of the discriminants of the curves with $(e,q)=(2,3)$ , $(2,5)$ , $(2,7)$ , $(3,4)$ which we treat in this paper. Using the resultant of two forms, we mention here an alternative (but conjectural) construction for the interest of the reader, though it is essentially not used in this paper.

Definition 1.7. Let the coefficients $\mu _j$ of ( 1.1 ) be indeterminates, and define

\begin{align*} \begin{aligned} R_1&={\mathrm {rslt}}_X \left(\mathrm {rslt}_Y\big (f(X,Y), f_1(X,Y)\big ), \mathrm {rslt}_Y\big (f(X,Y), f_2(X,Y)\big ) \right), \\[3pt] R_2&=\mathrm {rslt}_Y \left(\mathrm {rslt}_X\big (f(X,Y), f_1(X,Y)\big ), \mathrm {rslt}_X\big (f(X,Y), f_2(X,Y)\big ) \right),\\[3pt] R&=\gcd (R_1,R_2)\ \ \ \ \textrm {in} \,\mathbb {Z}[{\mu }]. \end{aligned} \end{align*}

Here $\mathrm {rslt}_Z$ is the Sylvester resultant with respect to $Z$ .

Now we recall the conjecture from the paper [Reference Eilbeck, Enol’skii, Matsutani, Ônishi and Previato13].

1.3. The Weierstrass form of the curve and its modality

Starting from the equation $f(x,y)=0$ in (1.1) and removing the terms of $y^{e-1}$ and $x^{q-1}$ by replacing $y$ by $ y+\frac 1e\,p_1(x)$ , and $x$ by $ x+\frac 1q\mu _{(q-1)e}$ , respectively, we get a new equation $f(x,y)=0$ which is called the Weierstrass form of the original one. After making such transformations, we re-label the coefficients by $\mu _j$ .

For example, if $(e,q)=(2,2g+1)$ , the new equation is

\begin{align*} f(x,y)=y^2-(x^{2g+1}+\mu _{4g-2}x^{2g-1}+\mu _{4g-4}x^{2g-2}+\cdots +\mu _{4g+2})=0; \end{align*}

and if $(e,q)=(3,4)$ , the new one is

\begin{align*} f(x,y)=y^3-(\mu _2x^2+\mu _5x+\mu _8)y-(x^4+\mu _6x^2+\mu _9x+\mu _{12})=0. \end{align*}

In these cases, the number of remaining $\mu _j$ s is $2g$ . However, in general, we can have some cases such that this number is less than $2g$ . The difference

\begin{align*} 2g-\textrm {``the number of}\ \mu _j^{\prime\prime} \end{align*}

is called the modality (a term used in singularity theory) of the $(e,q)$ -curve. We give here a simple formula giving modalities and, especially, determine all the curves of modality $0$ .

Proof. The number of $\mu _j$ s appearing in the Weierstrass form is

\begin{align*} \begin{aligned} \sum _{j=1}^{e-2}&\bigg (\bigg \lfloor \frac {(e-j)q}e\bigg \rfloor +1\bigg ) +(q-1) =\frac 12\sum _{j=1}^{e-1}\bigg (\bigg \lfloor \frac {(e-j)q}e\bigg \rfloor +\bigg \lfloor \frac {jq}e\bigg \rfloor \bigg )-\bigg \lfloor \frac {q}{e} \bigg \rfloor +(e-2)+(q-1)\\[3pt] &=\frac 12\sum _{j=1}^{e-1}(q-1)-\bigg \lfloor \frac {q}{e}\bigg \rfloor +e+q-3 =\frac 12(e-1)(q-1)+e+q-3-\bigg \lfloor \frac {q}{e}\bigg \rfloor . \end{aligned} \end{align*}

Subtracting the above result from $2g=(e-1)(q-1)$ gives the required expression. The latter part follows directly from this. This completes the proof.

On the case for a curve with positive modality, there is some description in [Reference Buchstaber and Leykin9] (the end of Section 2). Since it is not clear for us how positive modality causes difficulty, we do not discuss this theme here, though we give an example of positive modality in 1.16.

From now to the end of the paper, we always assume that the equation $f(x,y)=0$ of the curve $\mathscr {C}$ is given by a Weiserstrass form and the modality of $\mathscr{C}$ is 0.

1.4. Weight

We recall that (1.1), that is

\begin{align*} f(X,Y) =Y^e-X^q-\sum _{\substack {0\leq i\leq q-2\\[3pt]0\leq j\leq e-2\\[3pt]ie+jq\lt eq}}\mu _{eq-ie-jq}\,X^iY^j. \end{align*}

We define, for any pair $(i,j)$ with $0\leq i\leq q-2$ , $0\leq j\leq e-2$

(1.12) \begin{equation} \begin{aligned} M_{ei+qj}&=M_{ei+qj}(X,Y)=X^iY^j\ \textrm{and} \\[3pt] \mathrm {wt}(M)&=\{-\mathrm {wt}(X^iY^j)\,(\!=-\mathrm {wt}\big (M_{ei+qj}(X,Y)\big )\,|\,0\leq i\leq q-2, 0 \leq j\leq e-2\,\}\\[3pt] &=\{-(ei+qj)\,|\,0\leq i\leq q-2, 0\leq j\leq e-2\,\}. \end{aligned} \end{equation}

The sequence constituted by the elements in $-\mathrm {wt}(M)$ in increasing order is denoted by

\begin{align*} v_1(\!=0),\ v_2(\!=e), v_3, \ \cdots, \ v_{2g-2}, \ v_{2g-1}(\!=4g-2-e), \ v_{2g}(\!=4g-2). \end{align*}

Because of the assumption $\gcd (e,q)=1$ , for any $v\in \mathrm {wt}(M)$ there exists a unique pair $(i,j)$ such that $0\leq i\leq q-2$ , $0\leq j\leq e-2$ and $M_{v}=X^iY^j$ . We introduce the notation

(1.13) \begin{equation} \begin{aligned} M(X,Y)&={}^{t}{[\,M_{v_j}(X,Y) \ \ (j=1,2,\cdots, 2g)\,]}, \\[3pt] {}^{\mathrm {rev}}{M}(X,Y)&={}^{t}{[\,M_{v_j}(X,Y) \ \ (j=2g,2g-1,\cdots, 1)\,]}. \end{aligned} \end{equation}

We denote the Weierstrass gap sequence of the semigroup generated by $e$ and $q$ in the positive integers in the increasing order by

(1.14) \begin{equation} w_1(\!=1), \ w_2, \ \cdots, \ w_g(\!=2g-1), \end{equation}

which is also the Weierstrass gap sequence of $\mathscr {C}$ at $\infty$ . Namely, this is the unique (finite) increasing sequence of positive integers which cannot be written in the form $ae+bq$ with non-negative integers $a$ , $b$ . We denote the set of the terms in (1.14) by

\begin{align*} \mathrm {wgs}(e,\,q). \end{align*}

It is well-known that each term of the sequence is written in the form

\begin{align*} w_j=2g-1-v_{g-j+1} \ \ \ (1\leq j\leq g), \end{align*}

and that the assumption $\gcd (e,q)=1$ implies the Young tableaux associated with the sequence

\begin{align*} w_{2g}-(g-1), \ w_{2g-1}-(g-2), \ \cdots, \ w_2-1, \ w_1-0 \end{align*}

given by the Weierstrass gap sequences is symmetric with respect to the diagonal line from the top-left to the bottom-right. The terms in the sequence $\{v_j\}_{j=1}^{2g}$ are written also as

\begin{align*} v_j= \bigg \{\ \begin{aligned} &2g-1-w_{g-j+1} \ \ \ \textrm {if} \ 1\leq j\leq g,\\[3pt] &2g-1+w_{j-g} \ \ \ \ \ \textrm {if} \ g+1\leq j\leq 2g. \end{aligned} \end{align*}

We see that all the terms $\{M_{v_j}(X,Y)\}$ appear in $f(X,Y)$ provided the modality of the curve is $0$ . We shall give below sample values of the data above for the convenience of the reader to follow the calculation in Section 6.

Finally, we introduce the following notation on the coefficients $\mu _j$ ;

\begin{align*} \mu &=\{\,\mu _{eq-ie-jq}\,;\,0\leq i\leq q-2,\ 0\leq j\leq e-2,\ ie+jq\leq eq\,\}, \end{align*}

and

\begin{align*} \begin{aligned} \mathrm {wt}(\mu )&=\{-\mathrm {wt}(\mu _{eq-ie-jq})\,;\,0\leq i\leq q-2,\ 0\leq j\leq e-2, \ ie+jq\leq eq\,\}\\[3pt] &=\{-(eq-ie-jq)\,;\,0\leq i\leq q-2,\ 0\leq j\leq e-2, \ ie+jq\leq eq\,\}. \end{aligned} \end{align*}

Example 1.15. If $(e,q)=(2,2g+1)$ , then the $M_j$ s are given as follows:

If $(e,q)=(3,4)$ or $(3,5)$ , then the $M_j$ s are given as follows:

All these examples, and only these, are of modality $0$ as explained in (1.10).

Example 1.16. In contrast to the cases above, we have for $(e,q)=(3,7)$ :

However, the Weierstrass form of the $(3,7)$ -curve is given by

\begin{align*} \begin{aligned} Y^3&-(\mu _2X^4+\mu _5X^3+\mu _8X^2+\mu _{11}X +\mu _{14})Y\\[3pt] & -(X^7+\mu _6X^5+\mu _9X^4+\mu _{12}X^3+\mu _{15}X^3+\mu _{18}X+\mu _{21}) \end{aligned} \end{align*}

and this equation does not include a term in $M_{22}(X,Y)=X^5Y$ . This curve is of modality $1$ . We will not discuss this curve further in this paper.

1.5. The representation matrix for $f(X,Y)$ -multiplication

In this subsection, we shall define a certain matrix $T\in \mathrm {Mat}(2g,\mathbb {Q}[\mu ])$ whose determinant might be essentially the discriminant of $\varDelta$ .

Definition 1.18. The transpose of the representation matrix of the $({-}eq)f(X,Y)$ -plication map

(1.19) \begin{equation} \cdot ({-}eq)\,f(X,Y)\,:\,\mathbb {Q}[\boldsymbol {\mu }][X,Y]/(f_1,f_2)\longmapsto \mathbb {Q}[\boldsymbol {\mu }][X,Y]/(f_1,f_2) \end{equation}

with respect to the basis $M(X,Y)$ is denoted by $T\in \mathrm {Mat}(2g,\,\mathbb {Q}[\mu ])$ , that is

(1.20) \begin{equation} -eq\,f(X,Y)M(X,Y)\equiv M(X,Y)\,{}^{t}{T}\,\bmod {(f_1,\,f_2)}. \end{equation}

We define the subscript of the entries in $T$ as follows. The row-index $a$ runs through $\mathrm {wt}(M)$ in increasing order, and $b$ runs through $\mathrm {wt}(\widetilde {\mu })$ in decreasing order, and we write

\begin{align*} T=[\,T_{a,b}\,]\in \mathrm {Mat}(2g,\,\mathbb {Q}[\mu ]) \end{align*}

Then we have $\mathrm {wt}(T_{a,b}){=}-(a+b)$ . This is written in the usual manner as $T\,{=}\,[T_{v_i,\,eq-v_{2g+1-j}}]$ .

One of the reasons why we have an extra factor $-eq$ in (1.19) appears in 5.33 later.

Proof. (1) We assume all the coefficients $\mu _j$ are constants in $\mathbb {C}$ . We have $\mathrm {det}(T)=0$ if and only if the rank (called the Tjurina number at $\mu$ ) of the co-kernel $\mathbb {Q}[\mu ][X,Y]/(f,f_1,f_2)=\mathbb {Q}[x,y]/(f_1,f_2)$ (which is the dimension of $\mathbb {C}[X,Y]/(f,f_1,f_2)$ over $\mathbb {C}$ ) of the map is positive. This is exactly the case that the ideal $(f,f_1,f_2)$ does not contain $1\in \mathbb {Q}[\mu ][X,Y]$ . By Hilbert’s Nullstellensatz (Theorem 5.4(i) in [Reference Matsumura20], for instance), we see this is equivalent to existence of a $(x,y)\in \mathbb {C}^2$ such that

(1.22) \begin{equation} f(x,y)=f_1(x,y)=f_2(x,y)=0. \end{equation}

Conversely $\varDelta =0$ means (1.22), and it implies the co-kernel is non-trivial. Therefore the zeroes of $\varDelta$ and those of $\det (T)$ coincide. So $\det (T)$ must be a non-zero rational constant multiple of a power of $\varDelta$ . (2) Since $\det (T)\neq 0$ by virtue of (1), this is obvious.

Now we present the following:

For each $(e,q)$ , if this conjecture is true then we have immediately from 1.21 that

\begin{align*} \det (T)=c\cdot \varDelta \ \ \ \text {for some $c\in \mathbb {Q}^{\times }$}. \end{align*}

We give the explicit value of $c$ above for the cases in Sections 6.3, 6.4, 6.5 and 6.7.

2.1. Preliminary on derivations

In this subsection, we prepare a general notation for the succeeding subsections. Let $A$ be a commutative ring. A map $D\,:\,A\longrightarrow A$ , $a\mapsto Da$ satisfying the following properties is called a derivation on $A$ ; for any $a$ , $b\in A$ ,

\begin{align*} D(a+b)=Da+Db, \ \ \ D(ab)=(Da)b+a(Db). \end{align*}

We denote by $\mathrm {Der}(A)$ the set of all derivations on $A$ , which is an $A$ -module. Since $\mathrm {Der}(A)$ is equipped with the natural Lie bracket

\begin{align*} [D_1,\ D_2]=D_1D_2-D_2D_1 \ \ \ (D_1, \ D_2\in \mathrm {Der}(A)), \end{align*}

$\mathrm {Der}(A)$ is a Lie algebra over $A$ . Let $M$ and $N$ be two $A$ -modules and $D\in \mathrm {Der}(A)$ be a fixed derivation on $A$ . A map $\widetilde {D}\,:\,M\longrightarrow N$ , $m\mapsto \widetilde {D}m$ satisfying

\begin{align*} \widetilde {D}(m_1+m_2)=\widetilde {D}m_1+\widetilde {D}m_2, \ \ \ \widetilde {D}(am)=(Da)m+a\widetilde {D}m \end{align*}

is called a derivation from $M$ to $N$ associated with $D$ . We denote by $\mathrm {Der}_A(M,N;D)$ the set of the derivations from $M$ to $N$ associated with $D$ . If $M=N$ , then we denote this simply by $\mathrm {Der}_A(M;D)=\mathrm {Der}_A(M,M;D)$ . Let $D_1$ , $D_2\in \mathrm {Der}(A)$ , and $\widetilde {D_1}\in \mathrm {Der}_A(M,N;D_1)$ , $\widetilde {D_2}\in \mathrm {Der}_A(M,N;D_2)$ . Then we see that

\begin{align*} a\widetilde {D_1}\in \mathrm {Der}_A(M,N;aD_1) \ \ (a\in A), \ \ \ \widetilde {D_1}+\widetilde {D_2}\in \mathrm {Der}_A(M,N;\,D_1+D_2). \end{align*}

Now let $B$ be an $A$ -algebra and take a derivative $D\in \mathrm {Der}(A)$ . Let $M$ and $N$ be two $B$ -modules, hence also $A$ -modules. Regarding $B$ as an $A$ -module, we take a derivation $\widetilde {D}\in \mathrm {Der}_A(B;D)$ associated with $D\in \mathrm {Der}(A)$ . Since we can regard naturally $\widetilde {D}\in \mathrm {Der}(B)$ , we have the inclusion

\begin{align*} \mathrm {Der}_B(M,N;\widetilde {D})\subset \mathrm {Der}_A(M,N;D). \end{align*}

For $\widetilde {D_1}\in \mathrm {Der}_A(M;D_1)$ , $\widetilde {D_2}\in \mathrm {Der}_A(M;D_2)$ , and for any $a\in A$ , $m\in M$ , we have

\begin{align*} [\widetilde {D_1},\ \widetilde {D_2}]\ (\!=\widetilde {D_1}\widetilde {D_2} -\widetilde {D_2}\widetilde {D_1}) \in \mathrm {Der}_A(M;[D_1,D_2]), \end{align*}

since $\widetilde {D_1}\widetilde {D_2}(am) =(D_1D_2a)m+(D_2a)(\widetilde {D_1}m)+(D_1a)(\widetilde {D_2}m) +a\widetilde {D_1}\widetilde {D}_2m$ . This implies that if $\widetilde {D_1}$ and $\widetilde {D_2}\in \mathrm {Der}_A(M;D)$ for a derivation $D\in \mathrm {Der}(A)$ , then

\begin{align*} [\widetilde {D_1},\ \widetilde {D_2}]\in \mathrm {Der}_A(M;0), \end{align*}

where $0$ stands for the $0$ -map from $A$ to $A$ itself, which is a derivation on $A$ . Namely, $[\widetilde {D_1},\ \widetilde {D_2}]$ is a $A$ -homomorphism. In the set of all maps from $M$ to $M$ , we take union of these sets over $\mathrm {Der}(A)$ and denote it as

\begin{align*} \mathrm {Der}_A(M)=\bigcup _{D\in \mathrm {Der}(A)}\mathrm {Der}_A(M;D). \end{align*}

This is also a Lie algebra over $A$ . Instead of writing $D\in \mathrm {Der}_A(M)$ , we can also say that the derivation $D$ acts on $M$ , for instance.

2.2. Chevalley’s lemma

We quote the following without proof from [Reference Chevalley11], p.112, Lemma 2.

Lemma 2.1. Let $K$ be a field and $R$ be a function field of one variable with $K$ the field of constants. Take a transcendental element $\xi$ in $R$ over $K$ and fix it. For $D\in \mathrm {Der}(K)$ , there exists a unique derivation $D_{\xi }\in \mathrm {Der}_K(R;D)$ satisfying

\begin{align*} D_{\xi }(\xi )=0. \end{align*}

Corollary 2.2. Suppose a derivation $D\in \mathrm {Der}(\mathbb {Q}[\mu ])$ and an element $\xi$ in $\mathbb {Q}[\mu, x,y]$ transcendental over $\mathbb {Q}(\mu )$ are given. There is a unique extension $D_{\xi }\in \mathrm {Der}_{\mathbb {Q}[\mu ]}(\mathbb {Q}(\mu, x,y),D)$ of $D$ satisfying

\begin{align*} D_{\xi }(\xi )=0. \end{align*}

Under the situation of 2.1, we extends $D_{\xi }$ to a derivation $\widetilde {D_{\xi }}$ on the space of differentials $R\,d\xi$ via

(2.3) \begin{equation} \widetilde {D_{\xi }}(\omega )=D_{\xi }\left(\frac {\omega }{d\xi }\right)d\xi \ \ \ \textrm {for} any \omega \in {R}\,d\xi . \end{equation}

Namely, we have $\widetilde {D_{\xi }}\in \mathrm {Der}_R(R\,d\xi ;\,D_{\xi })\subset \mathrm {Der}_K(R\,d\xi ;\,D)$ . However, in the following lemma, $\widetilde {D_{\xi }}$ is denoted by $D_{\xi }$ for simplicity.

Lemma 2.4. (Chevalley [Reference Chevalley11], p.125, Lemma 3) Here we use the notation above. Let $\xi$ and $\zeta$ be two transcendental elements in $R$ over $K$ . Then we have the following relation between $D_{\xi }$ and $D_{\zeta }$ . For any $w\in {R}$ , we have

\begin{align*} D_{\xi }(wd\xi ) - D_{\zeta }(wd\xi ) = d({-}wD_{\zeta }\xi ). \end{align*}

Proof. (From Manin [Reference Manin19]) The operator $D_{\xi }-D_{\zeta }+(D_{\zeta }\xi )\frac {d}{d\xi }$ is a derivative on $R$ which vanishes on $K$ and also kills $\xi$ . By the uniqueness of extension of a derivation on $K$ to $R$ , this vanishes on $R$ . So that

\begin{align*} (D_{\xi }-D_{\zeta })w+(D_{\zeta }\xi )\frac {dw}{d\xi }=0. \end{align*}

Moreover $(D_{\xi }-D_{\zeta })(wd\xi )=(D_{\xi }-D_{\zeta })w\cdot {d\xi } +w\cdot (D_{\xi }-D_{\zeta })d\xi$ . Since $\zeta$ is transcendental, we see $\frac {d}{d\zeta }D_{\zeta }=D_{\zeta }\frac {d}{d\zeta }$ by 2.16 below or [Reference Chevalley11], p.125, Lemma 1, and have

\begin{align*} D_{\xi }d\xi =D_{\xi }\left(\frac {d\xi }{d\xi }\right)d\xi =0,\ \ \ D_{\zeta }d\xi =D_{\zeta }\left(\frac {d\xi }{d\zeta }\right)d\zeta = \frac {d}{d\zeta }(D_{\zeta }\xi )d\zeta =d\big (D_{\zeta }(\xi )\big ). \end{align*}

Therefore

\begin{align*} (D_{\xi }-D_{\zeta })(wd\xi )=-(D_{\zeta }\,\xi )\frac {dw}{d\xi }d\xi -w{\cdot }d(D_{\zeta }\,\xi )=-d(wD_{\zeta }\,\xi ) \end{align*}

as desired.

For an explicit sample calculation, see Section 6.3 on the derivation $\frac {\partial }{\partial \mu _j}$ .

2.3. The horizontal derivation formula

We present a useful formula 2.1 called the horizontal derivation formula, which is important for our definition of the first de Rham cohomology equipped with a structure of a differential module and eventually giving a sophisticated algorithm to compute the Gauss-Manin connection $\varGamma _j$ (a sort of Christoffel symbol). The formula was obtained by Kouki Sato while working on [Reference Ônishi, Shibata and Sato28], and coauthors of that paper have permitted us to quote from it. This formula would be useful for various applications.

In addition to the matrix $T=[T_{ij}]$ in 1.18, we introduce $A_{i+e}$ and $B_{i+e}\in \mathbb {Q}[\mu, X,Y]$ for each $i\in \mathrm {wt}(M)$ defined by

(2.5) \begin{equation} -eq\cdot {}M_i\cdot {f}(X,Y)=\sum _{j\,\in \mathrm {wt}(M)}T_{ij}M_j+A_{i+e}\, f_X+B_{i+q}\,f_{\,Y}, \end{equation}

where $f(X,Y)$ is the defining polynomial of $\mathscr {C}$ and $M_i=M_i(X,Y)$ is the one defined in 1.12. In this notation, we have

\begin{align*} \mathrm {wt}(A_{i+e})=-(i+e), \ \ \ \mathrm {wt}(B_{i+q})=-(i+q). \end{align*}

We denote by

\begin{align*} \frac {\,\partial \,}{\partial \mu _j\,}\in \mathrm {Der}_{\mathbb {Q}[\mu ]}\left(\mathbb {Q}(\mu, x,y);\frac {\,\partial \,}{\partial \mu _j\,}\right) \end{align*}

the derivation uniquely determined by 2.2 as an extension of the derivation $\frac {\,\partial \,}{\partial \mu _j\,}\in \mathrm {Der}(\mathbb {Q}[\mu ])$ having the property

(2.6) \begin{equation} \frac {\,\partial \,}{\partial \mu _j\,}x=0. \end{equation}

Namely, we use the same notation for the extension. It is natural to define the weight to be

(2.7) \begin{equation} \mathrm {wt}\left (\frac {\,\partial \,}{\partial \mu _j\,}\right )=j. \end{equation}

Moreover, there is a unique extension $\widetilde {\ell _i}$ of the derivation

(2.8) \begin{equation} \ell _i=\sum _jT_{ij}\,\frac {\,\partial \,}{\partial \mu _j\,}\in \mathrm {Der}(\mathbb {Q}[\mu ]) \end{equation}

satisfying

\begin{align*} \widetilde {\ell _i}x=0, \end{align*}

by 2.2 again, for which we use the same notation

(2.9) \begin{equation} \widetilde {\ell _i}=\sum _jT_{ij}\,\frac {\,\partial \,}{\partial \mu _j\,} \in \mathrm {Der}_{\mathbb {Q}[\mu ]}(\mathbb {Q}(\mu, x,y);\ell _i). \end{equation}

Therefore, for a 1-form $\omega \in \mathbb {Q}(\mu, x,y)\,dx$ , defining $\overline {\ell _i}$ by

\begin{align*} \overline {\ell _j}(\omega )=\widetilde {\ell _j}\left(\frac {\omega }{\,dx\,}\right)dx \end{align*}

as (2.3) (be careful as this depends on the condition (2.6)), and we have an extension

(2.10) \begin{equation} \overline {\ell _i}\in \mathrm {Der}_{\mathbb {Q}[\mu ]}(\mathbb {Q}(\mu, x,y)dx;\ell _i). \end{equation}

However, we omit $\ \widetilde {\ }\; $ and ${\overline {\phantom{\ell _i} }}$ for simplicity, and use the notation

\begin{align*} \ell _i=\sum _jT_{ij}\,\frac {\,\partial \,}{\partial \mu _j\,} \end{align*}

for these extensions. This convention will be practical.

In this paper, for a polynomial $G=G(\mu, X,Y)\in \mathbb {Q}[\mu, X,Y]$ , we denote by $(G)_x=G_x$ , $(G)_y=G_y$ , and $(G)_{\mu _j}=G_{\mu _j}$ the polynomials given by the substitution $(X,Y)=(x,y)$ to the partial derivations $\frac {\partial }{\,\partial X}\,{G}(\mu, X,Y)$ , $\frac {\partial }{\,\partial X}\,{G}(\mu, X,Y)$ , and $\frac {\partial }{\,\partial \mu _j}\,{G}(\mu, X,Y)$ . This rule is applied also for $f(X,Y)$ . Moreover, the higher-order case is similarly defined. For example, $f_{yy}$ means the quantity given by the substitution $(X,Y)=(x,y)$ to $\frac {\partial ^2}{\,\partial Y^2\,}f(X,Y)$ .

Under this convention, we present the following formula, which is mainly used for 3.2.

Theorem 2.11. (the horizontal derivation formula) Let $N=N(X,Y)$ be a monomial of $X$ and $Y$ (with non-negative powers), and $P\in \mathbb {Q}[\mu ]$ be an arbitrary polynomial. Then we have the following formula :

(2.11) \begin{align} \ell _j \left(\frac {\,P\cdot N\,}{f_y}\,dx \right) & = P\,\frac {(N)_y\,B_{j+q}-N\big (M_j+(B_{j+q})_y\big )-(N\,A_{j+e})_x}{f_y}\,dx \nonumber \\[3pt] & \qquad \qquad + \ell _j(P)N\,\frac {\,dx\,}{f_y}+d\bigg (P\,\frac {\,NA_{j+e}\,}{f_y}\bigg ). \end{align}

Especially, if $e=2$ then we have

\begin{align*} \ell _j\bigg (\frac {\,M_k\,dx\,}{f_y}\bigg ) =\frac {\,-(M_k\,A_{j+2})_x-M_{j+k}\,}{2y}\,dx+d\bigg (\frac {\,M_k\,A_{j+2}\,}{f_y}\bigg ). \end{align*}

Remark 2.13. The formula ( 2.11 ) shows

(2.14) \begin{equation} \ell _j\in \mathrm {Der}_{\mathbb {Q}[\mu ]}\left(\mathbb {Q}[\mu, x,y]\frac {\,dx\,}{f_y},\ \mathbb {Q}[\mu, \,x,\,y]\frac {\,dx\,}{f_y} +d\,\left(\mathbb {Q}[\,\mu, \,x,\,y]\frac {1}{\,f_y\,}\right) \,;\ \ell _j\right). \end{equation}

Since $\det (T)$ equals $\varDelta$ up to a non-zero rational multiplicative constant (see 1.21(3) and 5.36 below) at least in our cases $(e,q)=(2,3)$ , $(2,5)$ , $(2,7)$ , and $(3,4)$ , we also have

\begin{align*} \frac {\,\partial \,}{\partial \mu _j\,} \in \mathrm {Der}_{\mathbb {Q}[\mu ]}\left( \mathbb {Q}\Big [\frac {\,1}{\,\varDelta \,},\,\mu, \,x,\,y,\,\frac {1}{\,f_2(x,y)\,}\Big ]\,;\ \frac {\,\partial \,}{\partial \mu _j\,}\right). \end{align*}

We refer the reader to [Reference Ônishi, Shibata and Sato28] on a proof of 2.1.

Here we note the following equality for the proof of the next lemma. Applying $\frac {\partial \,y}{\,\partial \mu _{eq-k}}$ to $f(x,y)=0$ , we have $f_y\,\frac {\partial \,y}{\,\partial \mu _{eq-k}}-M_k=0$ , so that

(2.15) \begin{equation} \frac {\partial \,y}{\,\partial \mu _{eq-k}}=\frac {\,M_k\,}{f_y}. \end{equation}

We give the following lemma, by which we understand (2.11) below more clearly is a special case of Lemma 1 in p.125 of [Reference Chevalley11] (proved by the uniqueness assertion in 2.1).

Lemma 2.17. Denote by $\frac {d}{\,dx\,}$ the derivation in $\mathrm {Der}_{\mathbb {Q}[\mu ]}(\mathbb {Q}(\mu, x,y);0)$ determined by the properties that $\frac {d\mu _k}{dx}=0$ for any $k\in \mathrm {wt}(\mu )$ and $\frac {d}{\,dx\,}x=1$ . Then for any $k\in \mathrm {wt}(\widetilde {\mu })$ , we have

\begin{align*} \frac {d}{\,dx\,}\frac {\partial y}{\,\partial \mu _k\,} =\frac {\partial }{\,\partial \mu _k\,}\frac {dy}{\,dx\,}. \end{align*}

Therefore, we have additionally

\begin{align*} \frac {d}{\,dx\,}\ell _j=\ell _j\frac {d}{\,dx\,} \end{align*}

as a map from $\mathbb {Q}(\mu, x,y)dx$ into itself.

Proof. This is exactly Lemma 2 in [Reference Chevalley11], p.125, which we can check by using (2.16) as follows. The left-hand side is

\begin{align*} \begin{aligned} \frac {\,\partial \,}{\,\partial \mu _k\,}\frac {\,dy\,}{\,dx\,} &=-\frac {\,\partial \,}{\,\partial \mu _k\,}\frac {\,f_x\,}{\,f_y\,} =-\frac {\,\frac {\,\partial f_x\,}{\,\partial \mu _k\,}\,}{\,f_y\,} +\frac {\,f_x\frac {\,\partial f_y\,}{\,\partial \mu _k\,}\,}{\,{f_y}^2\,}\\[3pt] &=-\frac {\,-(M_{eq-k})_x+f_{xy}\tfrac {\,\partial y\,}{\,\partial \mu _k\,}\,}{\,f_y\,} +\frac {\,f_x\cdot \big ({-}(M_{eq-k})_y+f_{yy}\frac {\,\partial y\,}{\,\partial \mu _k\,}\big )\,}{\,{f_y}^2\,}\\[3pt] &=-\frac {\,-(M_{eq-k})_x+f_{xy}\tfrac {\,M_{eq-k}\,}{\,f_y\,}\,}{\,f_y\,} +\frac {\,f_x\cdot \big ({-}(M_{eq-k})_y+f_{yy}\tfrac {\,M_{eq-k}\,}{\,f_y\,}\big )\,}{\,{f_y}^2\,}. \end{aligned} \end{align*}

On the other hand, we have

\begin{align*} \begin{aligned} \frac {\,d\,}{\,dx\,}\frac {\,\partial y\,}{\,\partial \mu _k\,} &=\frac {\,d\,}{\,dx\,}\frac {\,M_{eq-k}\,}{\,f_y\,} \ \ \ (\,\because \ 2.16)\\[3pt] &=\frac {\,(M_{eq-k})_x+(M_{eq-k})_y\frac {\,dy\,}{\,dx\,}\,}{\,f_y\,} -\frac {\,M_{eq-k}\big (f_{yx}+f_{yy}\frac {\,dy\,}{\,dx\,}\big )\,}{\,{f_y}^2\,}\\[3pt] &=\frac {\,(M_{eq-k})_x-(M_{eq-k})_y\frac {\,f_x\,}{\,f_y\,}\,}{\,f_y\,} -\frac {\,M_{eq-k}\big (f_{yx}-f_{yy}\frac {\,f_x\,}{\,f_y\,}\big )\,}{\,{f_y}^2\,}. \end{aligned} \end{align*}

These two coincide and the formula follows.

Remark 2.18. Thanks to 2.1, and

\begin{align*} \ell _j\bigg (d\bigg (\frac {h}{\,f_2\,}\bigg )\bigg ) =\ell _j\bigg (\frac {d}{\,dx\,}\bigg (\frac {h}{\,f_2\,}\bigg )\bigg )dx =\frac {d}{\,dx\,}\bigg (\ell _j\bigg (\frac {h}{\,f_2\,}\bigg )\bigg )dx =d\!\bigg (\ell _j\bigg (\frac {h}{\,f_2\,}\bigg )\bigg ) \end{align*}

for any $h\in \mathbb {Q}[\mu, x,y]$ by 2.16, we see that

(2.18) \begin{equation} \begin{aligned} \ell _j&\in \mathrm {Der}_{\mathbb {Q}[\mu ]}\left(\mathbb {Q}[\mu, \,x,\,y]\frac {dx}{\,f_y\,} +d\left(\mathbb {Q}[\,\mu, \,x,\,y]\frac {1}{\,f_y\,}\right);\,\ell _j\right) \ \ \text {and}\\[3pt] \ell _j&\in \mathrm {Der}_{\mathbb {Q}[\mu ]}\left( \left(\mathbb {Q}[\mu, \,x,\,y]\frac {dx}{\,f_y\,} +d\left(\mathbb {Q}[\,\mu, \,x,\,y]\frac {1}{\,f_y\,}\right);\,\ell _j\right)\Big / d\left(\mathbb {Q}[\,\mu, \,x,\,y]\frac {1}{\,f_y\,}\right) \right). \end{aligned} \end{equation}

While we use the same notation $\ell _j$ for elements in different sets above, it is supposed that there is no confusion.

Definition 2.20. For later use, we denote the Lie subalgebra generated by all the $\ell _j$ s over the ring $\mathbb {Q}[\mu ]$ in the Lie algebra obtained as the union of the above sets by $\boldsymbol {L}$ :

(2.21) \begin{equation} \boldsymbol {L}\subset \mathrm {Der}_{\mathbb {Q}[\mu ]}\left ( \mathbb {Q}[\mu, \,x,\,y]\frac {dx}{\,f_y\,} +d\,\bigg (\mathbb {Q}[\,\mu, \,x,\,y]\frac {1}{\,f_y\,}\bigg )\right ). \end{equation}

3.1. Differential forms of the first kind of the curve

Recall that any term of the sequence $w_1$ , $w_2$ , $\cdots$ , $w_g\in \mathrm {wgs}(e,q)$ in (1.14) is written as

\begin{align*} w_j=2g-1-a_je-b_jq, \end{align*}

with non-negative integers $a_j$ , $b_j$ . Using this notation, we define differential forms

(3.1) \begin{equation} \omega _{w_j}=\frac {x^{a_j}y^{b_j}}{f_2(x,y)}dx, \ \ \ (j=1, \ \cdots, \ g). \end{equation}

These are of the first kind (namely holomorphic everywhere on $\mathscr {C}$ ) and of weight $w_j$ .

3.2. Forms of the second kind and the first de Rham cohomology

In this paper, we should consider the curve $\mathscr {C}$ and other objects arising from $\mathscr {C}$ to be defined over the ring $\mathbb {Q}[{\mu }]$ ; in which the period matrix $\varOmega$ is exceptional as is explained later, and is defined over the field $\mathbb {C}$ of complex numbers.

Throughout this paper, we mean by the (differential) forms of the second kind all the forms without non-zero residue everywhere on $\mathscr {C}$ . Therefore we include in them the differential forms of the first kind. However, we mean by the differential forms of the third kind the forms having non-zero residue pole somewhere on $\mathscr {C}$ , on which we never assume that their order of such poles to be $1$ and that they does not contain any form of the second kind.

It is easy to see that the set of differential forms of the second kind on $\mathscr {C}$ with a pole only at $\infty$ is exactly

\begin{align*} \mathbb {Q}[\mu, x,y]\frac {dx}{\,f_2(x,y)\,}. \end{align*}

In concord with our principle that we shall treat our materials not over the fields $\mathbb {Q}(\mu )$ but over the ring $\mathbb {Q}[\mu ]$ , we have the following definition, thanks to the horizontal derivation formula 2.1 and 2.17.

Definition 3.2. We define the first de Rham cohomology of $\mathscr {C}$ over $\mathbb {Q}[\mu ]$ by

(3.3) \begin{equation} H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }]) =\frac {\mathbb {Q}[\mu, x,y]\frac {dx}{\,f_2(x,y)\,}+d\,\big (\mathbb {Q}[\mu, x,y] \frac {1}{\,f_2(x,y)\,}\big )} {d\,\big (\mathbb {Q}[\mu, x,y]\frac {1}{\,f_2(x,y)\,}\big )}. \end{equation}

We are never concerned with higher cohomologies in this paper. We have an important remark here.

Remark 3.4. In order to define the first de Rham cohomology over $\mathbb {Q}[\mu ]$ which endures for calculation for ‘horizontal’ differentiations, namely, for extensions of the elements in $\mathrm {Der}(\mathbb {Q}[\mu ])$ , the right-hand side of ( 3.3 ) is the ‘slimmest form’ as explained in the following. For an integer $k$ , we denote by $\omega _{\mathscr {C}}(k\cdot \infty )$ (resp. by $L_{\mathscr {C}}(k\cdot \infty )$ ) the module consists of the forms in $\mathbb {Q}[\mu, x,y]\frac {dx}{\,f_2(x,y)\,}$ (resp. the functions (the polynomials) in $\mathbb {Q}[\mu, x,y]$ ) whose order of the pole at $\infty$ is at most $k$ . Then we see by Theorem 8.2 in p.30 of [17] that

\begin{align*} H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }]) \simeq \frac {\,\omega _{\mathscr {C}}((2g-2)\cdot \infty )\,} {\,dL_{\mathscr {C}}((2g-1)\cdot \infty )\,}. \end{align*}

and this is a module of rank $2g$ over $\mathbb {Q}[\mu ]$ . We place emphasis on this that the derivatives $\ell _j$ s in ( 2.10 ) and the derivatives $L_j$ s which appear in ( 5.39 ) (and any elements in the Lie algebra generated by them) do not act on $\omega _{\mathscr {C}}((2g-2)\cdot \infty )$ but on the module $\mathbb {Q}[\mu, x,y]\frac {dx}{\,f_2(x,y)\,}+d\,\big (\mathbb {Q}[\mu, x,y] \frac {1}{\,f_2(x,y)\,}\big )$ in ( 3.3 ). So that the definition ( 3.3 ) is much suitable for explicit calculation on $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ as a differential $\boldsymbol {L}$ -module, where $\boldsymbol {L}$ is defined in ( 2.21 ).

3.3. Symplectic inner product

We want to choose good $g$ differential forms $\eta _{-w_j}$ ( $j=g$ , $\cdots$ , $1$ ) of the second kind and of weight $-w_j$ such that the $2g$ forms consists of them and the forms of the first kind in (3.1) give rise to a symplectic basis of the space $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ which is equipped with a natural symplectic inner product explained below. We denote $\boldsymbol {\omega }=(\omega _{w_1},\cdots, \omega _{w_g},\eta _{-w_g},\cdots, \eta _{-w_1})$ .

As before, $(x,y)$ is a generic point of $\mathscr {C}$ . It is known that the $\eta _{-w_j}$ s as well as the $\omega _{w_j}$ s are defined over $\mathbb {Q}[{\mu }]$ , namely, they are of the form $\frac {h(x,y)}{f_2(x,y)}dx$ with $h(x,y)\in \mathbb {Q}[{\mu },x,y]$ (see [Reference Ônishi27], [Reference Ônishi, Shibata and Sato28]). We already defined in (1.12) that

\begin{align*} M(x,y)=\{\,x^iy^j\,|\,0\leq i\leq q-2, \ 0\leq j\leq e-2\,\}. \end{align*}

The set $\big \{\frac {h(x,y)}{f_2(x,y)}\,dx\,|\,h(x,y)\in M(x,y)\big \}$ forms a basis of $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ as a $\mathbb {Q}[{\mu }]$ -module. Here we note that the order of pole of $x^{q-2}y^{e-2}$ at $\infty$ , which is the largest in $M(x,y)$ , is

\begin{align*} e(q-2)+q(e-2) = 2(e-1)(q-1)-2=2g-2, \end{align*}

and $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ is a free $\mathbb {Q}[{\mu }]$ -module of rank $2g$ .

The space $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ is equipped with the anti-symmetric inner product given by

\begin{align*} \omega \star \eta =\mathop {\mathrm {Res}}_{P=\infty } \left(\int _{\infty }^P\omega \right)\eta (P) \end{align*}

for any $\omega$ , $\eta \in H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ . This is formally defined using the formal expansion with respect to a local parameter at $\infty$ (see [Reference Ônishi27], [Reference Ônishi, Shibata and Sato28]), which is a formal interpretation of the product $\omega \star \eta ={\displaystyle \mathop {\mathrm {Res}}_{P\in \mathscr {C}^{\circ }}} \big (\int _{\infty }^P\omega \big )\eta (P)$ if we regard the $\mu _j$ s as complex numbers and $\mathscr {C}$ as a non-singular curve (i.e. a compact Riemann surface). Here $\mathscr {C}^{\circ }$ is the regular polygon obtained from the Riemann surface attached to the curve $\mathscr {C}$ with respect to the paths

(3.5) \begin{equation} \{\alpha _j,\,\beta _j\,|\,j=1,\cdots, g\} \end{equation}

which form a symplectic basis of the homology group $H_1(\mathscr {C},\mathbb {Z})$ as usual.

We choose $\eta _{-w_i}$ to satisfy the symplectic relations

(3.6) \begin{equation} \omega _{w_i}\star \omega _{w_j}=0, \ \ \eta _{-w_i}\star \eta _{-w_j}=0, \ \ \omega _{w_i}\star \eta _{-w_j}=-\eta _{-w_j}\star \omega _{w_i}=\delta _{ij}. \end{equation}

The choice of $\eta _{-w_j}$ is not unique but given by using so-called the fundamental $2$ -form of Klein. For a more concrete construction of these forms, we refer the reader to [Reference Ônishi27] (or to [Reference Ônishi, Shibata and Sato28] for more detailed description). Especially, as a $\mathbb {Q}[{\mu }]$ -module, the $\omega _{w_j}$ s and $\eta _{-w_j}$ s form a basis of $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }])$ .

4.1. Materials for the construction of the sigma functions

In this section, we recall the definition of the natural generalisation of (0.2) for the general curve $\mathscr {C}$ , which is a function of $g$ variables $u={}^{t}{[u_{w_g} \ \cdots \ u_{w_1}]}$ with $w_j\in \mathrm {wgs}(e,q)$ and written as $\sigma (u)$ $=\sigma (u_{w_g},\cdots, u_{w_1})$ . It is called, analogously, the sigma function for $\mathscr {C}$ . To define it precisely, we need to introduce the Schur polynomial, period matrices, and others.

We use the classical notation of matrices concerning theta series, so our notation is transposed from the notation of [Reference Buchstaber and Leykin9]. Specifically, we will denote the period matrix of a curve as (4.2) whereas Buchstaber and Leykin’s papers use the transpose of this matrix. The other differences between their notation and ours will follow from this.

Letting $\boldsymbol {T}=\sum _{j=1}^gu_{w_j}T^{w_j}$ with an indeterminate $T$ , we define $\{p_k\}$ by $p_k=0$ for negative $k$ and

\begin{align*} \sum _{k=0}^{\infty }p_kT^k=\sum _{n=0}\frac {\boldsymbol {T}^n}{n!}. \end{align*}

Then we define $s(u)=s(u_{w_g},\cdots, u_{w_1})$ (see Section 4 in [Reference Nakayashiki24]) by

(4.1) \begin{equation} s(u)=\det (\mathop {[\ p_{w_{g+1-i}+j-g}\ ]}_{1\leq i\leq g,\ 1\leq j\leq g}). \end{equation}

This is the Schur polynomial corresponding to the sequence $(w_g,\cdots, w_1)$ .

From now on in this Section, we assume all the $\mu _j$ s are complex numbers and $\mathscr {C}$ is non-singular. The period matrices are defined by $\varOmega =\bigg [\ \begin{matrix} \omega ^{\prime} &\ \omega ^{\prime\prime} \\[3pt] \eta ^{\prime} &\ \eta ^{\prime\prime} \end{matrix}\ \bigg ]$ with

(4.2) \begin{equation} \begin{aligned} \omega ^{\prime} &=\bigg [\int _{\alpha _j}\!\omega _{w_{g-i+1}}\bigg ], \ \ \omega ^{\prime\prime}=\bigg [\int _{\beta _j}\!\omega _{w_{g-i+1}}\bigg ], \\[3pt] \eta ^{\prime} &=\bigg [\int _{\alpha _j}\!\eta _{-w_{g-i+1}}\bigg ], \ \ \eta ^{\prime\prime}=\bigg [\int _{\beta _j}\!\eta _{-w_{g-i+1}}\bigg ], \end{aligned} \end{equation}

where the $\eta _{-w_j}$ ’s are ones of (3.6) and $\alpha _i$ s and $\beta _j$ ’s are ones of (3.5). We introduce the $g$ -dimensional space $\mathbb {C}^g$ with coordinates

(4.3) \begin{equation} u={}^{t}{[u_{w_g} \ u_{w_{g-1}} \ \cdots \ u_{w_1}]} \end{equation}

for the domain on which the sigma function is defined. We define a lattice in this space by

(4.4) \begin{equation} \varLambda =\omega ^{\prime}\mathbb {Z}^g+\omega ^{\prime\prime}\mathbb {Z}^g. \end{equation}

For any $u\in \mathbb {C}^g$ , we define $u^{\prime}$ , $u^{\prime\prime}\in \mathbb {R}^g$ by $u=\omega ^{\prime}u^{\prime}+\omega ^{\prime\prime}u^{\prime\prime}$ . Likewise, for $\ell \in \varLambda$ , we write $\ell =\omega ^{\prime}\ell ^{\prime}+\omega ^{\prime\prime}\ell ^{\prime\prime}$ with $\ell ^{\prime}$ , $\ell ^{\prime\prime}\in \mathbb {Z}^g$ . In addition, we write ${\omega ^{\prime}}^{-1}{}^{t}{(\omega _{w_g},\,\cdots, \,\omega _{w_1})}= {}^{t}{(\hat {\omega }_1,\,\cdots, \,\hat {\omega }_g)}$ , ${\omega ^{\prime}}^{-1}\omega ^{\prime\prime}=\big [\tau _{ij}\big ]$ , and define (see [Reference Mumford22], p.3.82 or [Reference Lewittes18], p43)

(4.5) \begin{equation} \delta _j=-\tfrac 12\tau _{jj} +\sum _{i=1}^g\int _{\alpha _i}\bigg (\int _{\infty }^{P}\hat {\omega }_j\bigg )\, \hat {\omega }_i(P),\ \ \ \delta ={\omega ^{\prime}}\,{}^{t}{[\delta _1 \ \cdots \ \delta _g]}\in \mathbb {C}^g. \end{equation}

Then we define the Riemann constant by $[\delta ^{\prime}\ \delta ^{\prime\prime}]$ . It is well-known that $\delta ^{\prime}$ , $\delta ^{\prime\prime}\in (\frac 12\mathbb {Z})^g$ for our curve. The $[\delta ^{\prime}\ \delta ^{\prime\prime}]$ can be taken independent of the values of $\mu _j$ s with a suitable choice of the paths of integrals in (4.5).

Using the above notation, we define a linear form $L(\, \ )$ on $\mathbb {C}^g\times \mathbb {C}^g$ by

(4.6) \begin{equation} L(u,v)={}^{t}{u}\,(\eta ^{\prime}v^{\prime}+\eta ^{\prime\prime}v^{\prime\prime}). \end{equation}

This is $\mathbb {C}$ -linear with respect to the first variable, and $\mathbb {R}$ -linear with respect to the second. Moreover, we define

(4.7) \begin{equation} \chi (\ell )=\exp \,\big (2\pi \boldsymbol {i}({}^{t}{\delta ^{\prime}}\ell ^{\prime\prime}-{}^{t}{\delta ^{\prime}}\ell ^{\prime} +\frac 12{}^{t}{\ell ^{\prime}}\ell ^{\prime\prime})\big )\in \{1,\,-1\} \end{equation}

for any $\ell \in \varLambda$ . Let

(4.8) \begin{equation} \kappa \,:\,\mathbb {C}^g\,\longmapsto \,\mathbb {C}^g/\varLambda \end{equation}

be the map given by modulo $\varLambda$ , and $\mathrm {Sym}^k\,\mathscr {C}$ be the $k$ -th symmetric product of $\mathscr {C}$ . Then we have the Abel-Jacobi mapping

(4.9) \begin{equation} \iota \,:\,\mathrm {Sym}^k\,\mathscr {C} \longrightarrow \ \mathbb {C}^g/\varLambda, \ \ \ (P_1,\,\cdots \!,\,P_k)\longmapsto \ \sum _{j=1}^k\bigg ( \int _{\infty }^{P_j}\omega _{w_g}, \,\cdots, \, \int _{\infty }^{P_j}\omega _{w_1} \bigg ), \end{equation}

whose image is denoted by $\Theta ^{[k]}$ . We denote $\Theta =\Theta ^{[g-1]}$ , which is called the standard theta divisor of the Jacobian variety $\mathbb {C}^g/\varLambda$ of $\mathscr {C}$ . For $k=1$ , the map $\iota$ is an isomorphism from $\mathscr {C}$ to $\Theta ^{[1]}$ , by which we frequently identify these two.

4.2. Weight revisited

In this subsection, we assume all $\mu _j$ s in (1.1) are complex numbers. We denote by $\mathscr {C}_{\mu }$ the curve given by (1.1) and assume it is non-singular, namely, we assume it gives a Riemann surface. Here, we shall extend the notion of the weight $\mathrm {wt}$ of (1.4) as follows. Firstly, we define the weight of $u_j$ , a coordinate of $u$ in (4.3), by

(4.10) \begin{equation} \mathrm {wt}(u_j)=j. \end{equation}

We explain below that this definition is consistent with the settings of weight so far. For a later use in (5.40) for instance, we prepare another notation

\begin{align*} \mathrm {wt}(u)=\{\,\mathrm {wt}(u_{w_j})\,|\,j=1,\,\cdots, \,g\,\}=\mathrm {wgs}(e,q), \end{align*}

too. Let $\varepsilon$ be an arbitrary non-zero complex number and $\mathscr {C}_{\varepsilon \mu }$ be the curve defined by (1.1) with every $\mu _j$ being replaced by $\varepsilon ^{-j}\mu _j$ . Then the map

\begin{align*} \boldsymbol {\varepsilon }\,:\, \mathscr {C}_{\mu }\longrightarrow \mathscr {C}_{\varepsilon \mu }, \ \ \ (x,y)\longmapsto (\varepsilon ^{-e}x, \varepsilon ^{-q}y) \end{align*}

is an isomorphism (i.e. an isomorphism of compact Riemann surfaces). Now we extend the defining domain of $\mathrm {wt}$ to the space $\mathbb {C}^g$ of (4.3) as follows. We take a fixed reference point $(\{{\mu _j}^{(0)}\},\,u^{(0)})$ of $(\mu, \,u)=(\{\mu _j\},\,u)$ of (1.1) and (4.3), and assume $\mathscr {C}_{\mu ^{(0)}}$ is also non-singular. Let

(4.11) \begin{equation} \mathscr {M}=\mathscr {M}(\mu ^{(0)},\,u^{(0)}) \end{equation}

be the germ of the analytic functions of $(\mu, \,u)$ at $(\mu ^{(0)},\,u^{(0)})$ , where we regard them as complex variables. Let $\varPhi (\mu, u)\in \mathscr {M}$ . We denote by $\varPhi (\varepsilon \mu, \varepsilon {u})$ the function obtained from $\varPhi (\mu, u)$ by replacing all $\mu _j$ by $\varepsilon ^{-j}\mu$ and all $u_i$ by $\varepsilon ^iu_i$ for $i=w_g$ , $\cdots$ , $w_1$ . We say that the function obtained is induced by the mapping $\boldsymbol {\varepsilon }$ . If there is a constant integer $w$ such that

\begin{align*} \varPhi (\varepsilon \mu, \varepsilon {u})=\varepsilon ^{w}\,\varPhi (\mu, {u}) \end{align*}

for any $\varepsilon \in \mathbb {C}$ , then we say the function $\varPhi (\mu, u)$ is of weight $w$ , and denote this as

\begin{align*} \mathrm {wt}(\varPhi (\mu, u))=w. \end{align*}

Now we explain that the definition of $\mathrm {wt}$ is actually an extension of the weight defined at (1.4). The coordinates $x$ and $y$ of $\mathscr {C}_{\mu }$ are naturally seen as functions of $u_g$ of $u$ in (4.3) on certain restricted domain in $\mathbb {C}^g$ for any $\mu _j\in \mathbb {C}$ , which should be denoted by $\kappa ^{-1}\iota (\mathscr {C})$ by using the notation (4.8) and (4.9). Then, observing the weight of $x$ and $y$ as power series of $u_g$ , their weights are $-e$ and $-q$ , respectively. The function $u_g\mapsto x$ above is seen as the restriction to $\kappa ^{-1}\iota (\mathscr {C})$ of the Abelian function

(4.12) \begin{equation} u\longmapsto \frac {x_1\cdots x_g} {\displaystyle \sum _{1\leq i_1\lt \cdots \lt i_{g-1}\leq g}\!\!\!\!\!x_{i_1}\cdots {}x_{i_{g-1}}}, \end{equation}

where $u=\iota ((x_1,y_1),\,\cdots, \,(x_g,y_g))\,\bmod {\varLambda }$ . The function (4.12) is easily checked to be of weight $-e$ . Under similar observation, the function $u_g\mapsto y$ is seen of weight $-q$ . Summarising the above, the notion of weight of (1.4) is completely compatible with the former notion above as well as one on $\mathscr {M}$ .

Take any circuit integrals of an entry of one of the matrices in (4.2) and consider the map $\boldsymbol {\varepsilon }$ . For example, we choose $\int _{\alpha _j}\omega _{w_i}$ . Note that it is an element in $\mathscr {M}$ which is a function on $\mu _j$ s and independent of $u$ . While the map $\boldsymbol {\varepsilon }$ changes $\omega _{w_i}$ to $\varepsilon ^{w_i}\omega _{w_i}$ as $\mathrm {wt}(\omega _{w_i})=w_i$ , the deformation of the path $\alpha _j$ of the integral no longer affects this change of value integral because of Cauchy’s theorem. So that, the mapping $\boldsymbol {\varepsilon }$ induces a change of the integral

\begin{align*} \int _{\alpha _j}\omega _{w_i} \longmapsto \ \varepsilon ^{w_i}\!\int _{\alpha _j}\omega _{w_i}, \end{align*}

for any non-zero $\varepsilon$ . Therefore, we have

\begin{align*} \mathrm {wt}\left(\,\int _{\alpha _j}\omega _{w_i}\right)=w_i. \end{align*}

By the argument above, the weight of such a circuit integral does not depend on its path of integral as well as the choice of the reference points ${\mu _j}^{(0)}$ s and $u^{(0)}$ . Because of (4.9) for $k=g$ is surjective (Jacobi’s theorem) and any values $\int _{\alpha _j}\omega _{w_i}$ s are appeared as the coordinates $u_{w_j}$ the definition of weight (4.10) is justified.

4.3. Construction of the sigma function

Using the notions defined in the previous subsections, we define the sigma function by the following characterisation. Now we fix the curve $\mathscr {C}=\mathscr {C}_{\mu }^{e,q}$ .

The theorem 4.1 was proved in various ways in the literature, each version has a slightly different point of view. It is convenient to summarise here these versions from our point of view. We define

(4.15) \begin{equation} \begin{aligned} \tilde {\sigma }(u)&=\tilde {\sigma }(u,\varOmega ) =\left(\frac {(2\pi )^g}{\mathrm {det}\,{\omega ^{\prime}}}\right)^{\tfrac 12} \exp \big ({-}\tfrac 12\,{}^{t}{u}\,\eta ^{\prime}{\omega ^{\prime}}^{-1}u\big )\\[3pt] &\cdot \sum _{n\in \mathbb {Z}^g} \exp \big (\tfrac 12\,{}^{t}{(n+\delta ^{\prime\prime})}\,{\omega ^{\prime}}^{-1}\omega ^{\prime\prime}(n+\delta ^{\prime\prime}) +{}^{t}{(n+\delta ^{\prime\prime})}({\omega ^{\prime}}^{-1}u+\delta ^{\prime})\big ), \end{aligned} \end{equation}

where $\mathrm {det}$ denotes the determinant. Note that the arguments in any exponential are of weight $0$ , so that the infinite series part of (4.15) is of weight $0$ as well. It is easy to show that this function has property (1) (see 5.21 below) and that this function satisfies (2) using (5.3) below. Frobenius’ method shows that the solutions of the equation (2) form a one dimensional space (see p.93 of [Reference Lang17]). Although this function is constructed by using $\varOmega$ , it is independent of $\varOmega$ . Namely, the function $\tilde {\sigma }(u,\varOmega )$ is invariant under a modular transform, that is the transform of $\varOmega$ by $\mathrm {Sp}(2g,\mathbb {Z})$ which come from changing the choice of $\alpha _i$ s and $\beta _i$ s. Therefore, it is expressed as a power series of $u$ with coefficients being functions of only the $\mu _j$ s. On the latter part of (3) and (4), we refer the reader to [Reference Nakayashiki24]. In that paper, $\tilde {\sigma }(u)$ times some constant is expressed as a determinant of infinite size (see also [Reference Ônishi27]). See also the paper [Reference Nakayashiki23] by Nakayashiki, in which he proved the sigma function is no other than (4.15) times a non-zero constant depending on $\mu _j$ s with emphasising the later part of (3). That $\tilde {\sigma }(u)$ has the property (5) come from a well-known property of Riemann theta function. Using notation we have explained previously, we define

(4.16) \begin{equation} \hat {\sigma }(u)=\varDelta ^{-\frac 18}\tilde {\sigma }(u), \end{equation}

with an appropriate choice of the $\frac 18$ th root of the discriminant $\varDelta$ . It is known for $g=1$ and $2$ that this function exactly satisfies all the properties in 4.1, namely, we have $\sigma (u)=\hat {\sigma }(u)$ . For $g=1$ , it is shown as in [Reference Ônishi26] by a transformation formulae for $\eta (\tau )$ and the theta series described in pp.176–180 of [Reference Rademacher29], and for $g=2$ the paper [Reference Grant16] by D. Grant, in which the property (4) is shown by using Thomae’s formula.

However, it is not known for $g\geq 3$ if the function (4.16) is really the sigma function. The paper [Reference Buchstaber and Leykin9] is the first one to seriously attack this problem.

In this paper, we show, following the idea of [Reference Buchstaber and Leykin9], that $\sigma (u)=\hat {\sigma }(u)$ for the genus $3$ curves that is for the $(2,7)$ -curve and the $(3,4)$ -curve. This means (4.16) satisfies especially (4) up to an absolutely numerical multiplicative constant. The strategy of the proof is as follows: We construct a system of linear partial differential equations (heat equations) satisfied by $\hat {\sigma }(u)$ and show that the solution space is of dimension one (over the base field) by explicit construction of a recursive system for the coefficients of the power series expansion of any unknown solution and showing the uniqueness of the solution of this system. Then we see $\sigma (u)=\hat {\sigma }(u)$ up to a non-zero multiplicative absolutely numerical constant. This result might be seen as a generalisation of Thomae’s formula ( [Reference Thomae30]). Section 6 is devoted to these last steps.

5.1. Generalisation of the frobenius-stickelberger theory

This and the following sections are devoted to explaining the theory of Buchstaber and Leykin [Reference Buchstaber and Leykin9], on the differentiation of Abelian functions with respect to their parameters, as clearly as we can. That generalises the work of Frobenius and Stickelberger [Reference Frobenius and Stickelberger14], discussed above, on the elliptic function case of this problem.

For higher genus cases, we do not have a naive generalisation of (0.8) and (0.9), which are mentioned in the Introduction. However, we can give a natural generalisation of the relations (0.10) to the curve $\mathscr {C}_{\mu }$ as explained in the next section.

As we discussed in Subsection 3.2, any element $L$ in $\boldsymbol {L}$ of (2.21) operates linearly on the space $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[\mu ])$ . For complex variables $\mu _j$ s, we let $L$ operate firstly on the forms with variables $\mu _j$ s of representative of basis of $H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[\mu ])$ , then we restore $\mu _j$ s to the original values in $\mathbb {C}$ .

Taking a derivative $L\in \boldsymbol {L}$ , we define $\Gamma ^L\in \mathrm {Mat}(2g,\mathbb {Q}[{\mu }])$ as the representation matrix of the action of $L$ by

(5.1) \begin{equation} L(\boldsymbol {\omega })=\boldsymbol {\omega }\,{}^{t}{\Gamma ^L} \ \ \ \textrm {for} \ \ \boldsymbol {\omega }=(\omega _{w_g},\,\cdots, \,\omega _{w_1},\,\eta _{-w_g},\,\cdots, \,\eta _{-w_1}) \in H_{\mathrm {dR}}^1(\mathscr {C}/\mathbb {Q}[{\mu }]). \end{equation}

In [Reference Buchstaber and Leykin9], the matrix $-\!{}^{t}(\Gamma ^L)$ defined by (5.1) is called the Gauss-Manin connection for the derivation (vector field) $L$ .

From here to the end of Section 5, we assume that all the $\mu _j$ s are complex variables which vary as $\mathscr {C}$ is non-singular. We can then use the periods $\omega _{ij}$ s and $\eta _{ij}$ s.

By integrating (5.1) along each element in the chosen symplectic basis of $H_1(\mathscr {C},\,\mathbb {Z})$ , we get the natural action

(5.2) \begin{equation} L(\varOmega )=\Gamma ^L\,\varOmega \end{equation}

of $L$ on the space of $\varOmega$ s for all the choices of symplectic basis of $H_1(\mathscr {C},\,\mathbb {Z})$ . So, we see also how $L$ operates on the field $\mathbb {Q}(\{{\omega ^{\prime}}_{ij}\},\,\{{\omega ^{\prime\prime}}_{ij}\})$ . Of course, since $\varOmega$ is the period matrix of a symplectic basis, these elements must satisfy the constraint

(5.3) \begin{equation} {}^{t}{\varOmega } J \varOmega = 2 \pi i J, \ \ \ \textrm {where} J=\bigg [\begin{matrix} & 1_g\\[3pt] -1_g & \end{matrix}\bigg ] \end{equation}

by (3.6). This is none other than the generalisation of Frobenius-Stickelberger’s relation (0.10) and is to say that $(2\pi {i})^{-\frac 12}\varOmega \in \mathrm {Sp}(2g,\mathbb {C})$ . It follows immediately that

(5.4) \begin{equation} \varOmega ^{-1} =\frac {1}{2\pi i}J^{-1}\, {}^{t}{\varOmega }J =\frac {1}{2\pi i} \bigg [ \begin{array}{r@{\quad}r} {}^{t}{\eta ^{\prime\prime}} &\ -{}^{t}{\omega ^{\prime\prime}}\\[3pt] -{}^{t}{\eta ^{\prime}} & {}^{t}{\omega ^{\prime}} \end{array} \bigg ]. \end{equation}

After operating $L$ on both sides of (5.3), using (5.2) and (5.3), we see that the matrix $\Gamma ^L$ satisfies

(5.5) \begin{equation} {}^{t}\Gamma ^LJ + J \Gamma ^L = 0, \ \ \textrm {i.e.}\ \ {}^{t}{(\Gamma ^LJ)}=\Gamma ^LJ \end{equation}

because $L({}^{t}{\varOmega })={}^{t}{\varOmega }\,{}^{t}{\Gamma ^L}$ , which is to say that $\Gamma ^L$ is in the Lie algebra $\mathfrak {sp}(2g,\mathbb {Q}[\mu ])$ of $\mathrm {Sp}(2g,\mathbb {Q}[\mu ])$ . Thus we may write

(5.6) \begin{equation} -\Gamma ^LJ=\bigg [\, \begin{matrix} \alpha & \ \beta \\[3pt] {}^{t}{\beta }& \ \gamma \end{matrix}\, \bigg ], \ \ \ \Gamma ^L = \bigg [ \begin{array}{c@{\quad}r} -\beta & \ \alpha \\[3pt] -\gamma & {}^{t}{\beta } \end{array} \bigg ] \end{equation}

with ${}^{t}{\alpha }=\alpha$ and ${}^{t}{\gamma }=\gamma$ .

Conversely, starting from a matrix

\begin{align*} \Gamma = \bigg [ \begin{array}{c@{\quad}r} -\beta & \ \alpha \\[3pt] -\gamma & {}^{t}{\beta } \end{array} \bigg ]\in \mathfrak {sp}(2g,\mathbb {Q}[{\mu }]), \end{align*}

with ${}^{t}{\alpha }=\alpha$ and ${}^{t}{\gamma }=\gamma$ , we get uniquely an operator $L\in \boldsymbol {L}$ such that $\Gamma ^L=\Gamma$ . So far, this is a natural generalisation of the situation investigated by Frobenius-Stickelberger [Reference Frobenius and Stickelberger14].

5.2. The primary heat equation

In this Section, we review the general heat equations satisfied by the sigma functions.

If we want to find second-order linear partial differential equations (heat equations) satisfied by the sigma function, we should proceed in as general a way as possible. Here, note that the equation (0.6) is satisfied not only by the Jacobi theta function (0.3) but also by each individual term of the sum in (0.3). This corresponds a statement in the proof of Theorem 13 in page 274 of [Reference Buchstaber and Leykin9]. Here we will review their theory of such equations, correcting a few minor errors, and apply it explicitly to more general curves than considered in [Reference Buchstaber and Leykin9]. We shall start from this point of view.

Take an derivative $L\in \boldsymbol {L}$ and assume it is of homogeneous weight, say $\mathrm {wt}(L)=k$ (see (2.7)), we have the symmetric matrix $-\Gamma ^L J= \bigg [\, \begin{matrix} \alpha & \beta \\[3pt] {}^{t}{\beta }& \gamma \end{matrix}\, \bigg ]$ with

\begin{align*} \alpha =\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\alpha _{-i,k-j}]}, \ \ \beta =\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\beta _{j,k-i}]}, \ \ \gamma =\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\gamma _{-i,k+j}]} \end{align*}

and we also associate with it a second-order differential operator $H^L$ , given by

(5.8) \begin{equation} \begin{aligned} H^L&=\frac 12[\,{}^{t}{\partial _u}\ {}^{t}{u}\,] \bigg [ \begin{array}{r@{\quad}c} \alpha & \ \beta \\[3pt] {}^{t}{\beta } & \ \gamma \end{array} \bigg ] \bigg [ \begin{array}{c} \partial _u \\[3pt] u \end{array} \bigg ]\\[3pt] &=\sum _{i\in \mathrm {wt}(u)}\sum _{j\in \mathrm {wt}(u)} \left(\tfrac 12\,{\alpha }_{-i,\,k-j}\ \frac {\partial ^2}{\partial u_i\partial u_j} +{\beta }_{-j,\,k-i}\ u_i\frac {\partial }{\partial u_j} +\tfrac 12\,{\gamma }_{-i,\,k+j}\ u_iu_j\right) +\tfrac 12\,{\mathrm {Tr}\,\beta }. \end{aligned} \end{equation}

Here $\partial _u$ denotes the column vector with $g$ components $\frac {\partial }{\partial u_i}$ , and $u$ the column vector with $g$ components $u_i$ s. The very last term comes from the commutation relation $\tfrac {\partial }{\partial u_i}u_j=u_j\tfrac {\partial }{\partial u_i}+\delta _{ij}$ . It is then straightforward to verify the following

Lemma 5.9. If we define a function $G_0(u,\varOmega )$ (this is a Green’s function) by

\begin{align*} G_0(u,\varOmega ) =\left(\frac {(2\pi )^g}{\mathrm {det}\,\omega ^{\prime}}\right)^{\frac 12} \exp \big ({-}\tfrac 12\,{}^{t}{u}\,\eta ^{\prime}{\omega ^{\prime}}^{-1}u\big ), \end{align*}

then the following equation (a heat equation ) holds :

(5.10) \begin{equation} (L - H^L) G_0 =0. \end{equation}

Proof. We have

(5.11) \begin{equation} \begin{aligned} L\,G_0 &=-\frac 12 \mathrm {Tr}(\omega ^{\prime-1} L(\omega ^{\prime}))G_0 -\frac 12 ({}^{t}{u}\,L(\eta ^{\prime}){\omega ^{\prime}}^{-1}u)G_0 +\frac 12 ({}^{t}{u}\,\eta ^{\prime}{\omega ^{\prime}}^{-1}L(\omega ^{\prime}){\omega ^{\prime}}^{-1}u)G_0\\[3pt] &=\frac 12\left [\mathrm {Tr}(\omega ^{\prime-1}(\beta \omega ^{\prime} - \alpha \eta ^{\prime})) +({}^{t}{u}\,((\gamma \omega ^{\prime} - {}^{t}{\beta } \eta ^{\prime})\omega ^{\prime-1} -\eta ^{\prime}{\omega ^{\prime}}^{-1}(\beta \omega ^{\prime} - \alpha \eta ^{\prime})\omega ^{\prime-1}) \,u)\right ]G_0\\[3pt] &=\frac 12\left [\mathrm {Tr}(\beta -\alpha \eta ^{\prime} \omega ^{\prime-1}) + ({}^{t}{u}\,(\gamma -{}^{t}{\beta }\eta ^{\prime}{\omega ^{\prime}}^{-1} -\eta ^{\prime}{\omega ^{\prime}}^{-1}\beta +\eta ^{\prime}{\omega ^{\prime}}^{-1}\alpha \eta ^{\prime}{\omega ^{\prime}}^{-1})\,u) \right ]G_0 \end{aligned} \end{equation}

by using

(5.12) \begin{equation} L(\varOmega )=\Gamma ^L \varOmega =\bigg [ \begin{array}{r@{\quad}r} -\beta & \ \alpha \\[3pt] -\gamma & {}^{t}{\beta } \end{array} \bigg ] \bigg [ \begin{array}{r@{\quad}r} \omega ^{\prime} & \ \omega ^{\prime\prime} \\[3pt] \eta ^{\prime} & \ \eta ^{\prime\prime} \end{array} \bigg ] =\bigg [ \begin{array}{c@{\quad}c} -\beta \omega ^{\prime}+\alpha \eta ^{\prime} & -\beta \omega ^{\prime\prime}+\alpha \eta ^{\prime\prime} \\[3pt] -\gamma \omega ^{\prime} +{}^{t}{\beta }\eta ^{\prime} & -\gamma \omega ^{\prime\prime} +{}^{t}{\beta } \eta ^{\prime\prime} \end{array} \bigg ]. \end{equation}

The result of (5.11) coincides with that of

\begin{align*} \begin{aligned} H^{L_0}G_0 &= \frac 12 \big [{}^{t}{\partial _u}\ {}^{t}{u}\big ] \bigg [\, \begin{matrix} \alpha & \ \beta \\[3pt] {}^{t}{\beta } & \ \gamma \end{matrix}\, \bigg ] \bigg [ \begin{matrix} \partial _u \\[3pt] u \end{matrix} \bigg ]G_0 =\frac 12 \big [{}^{t}{\partial _u} \ {}^{t}{u}\big ] \bigg [\, \begin{matrix} \alpha & \ \beta \\[3pt] {}^{t}{\beta } & \ \gamma \end{matrix}\, \bigg ] \bigg [\, \begin{matrix} -\eta ^{\prime}\omega ^{\prime-1}u \\[3pt] u \end{matrix}\, \bigg ]G_0 \\[3pt] &=\frac 12 \mathrm {Tr}({}^{t}{\beta } -\alpha \eta ^{\prime} \omega ^{\prime-1})\,G_0 +\frac 12 \big [{-}{}^{t}{u}\eta ^{\prime} \omega ^{\prime-1} \ \ u \,\big ] \bigg [ \begin{array}{r@{\quad}r} \alpha & \ \beta \\[3pt] {}^{t}{\beta } & \ \gamma \end{array} \bigg ] \bigg [\, \begin{matrix} -\eta ^{\prime}\omega ^{\prime-1}u \\[3pt] u \end{matrix}\, \bigg ]G_0. \end{aligned} \end{align*}

Here we have used the generalised Legendre relation (5.3) and the symmetry of $\eta ^{\prime} \omega ^{\prime-1}$ .

Now we recall that the different terms in the expansion of the theta function are periodic translates of one another. Analogously, to construct the different terms appearing in the expansion of the sigma function, we act on $G_0$ by iterating an element of the Heisenberg group. For a variable $z\in \mathbb {Q}(\{{\omega ^{\prime}}_{ij}\},\,\{{\omega ^{\prime\prime}}_{ij}\})$ , and a two $g$ -component column vectors $p$ , $q$ whose components also belong to $\mathbb {Q}(\{{\omega ^{\prime}}_{ij}\},\,\{{\omega ^{\prime\prime}}_{ij}\})$ , we introduce

\begin{align*} F(z,p,q)=\exp (z)\exp ({}^{t}{p}u)\exp ({}^{t}{q}\partial _u). \end{align*}

We write its inverse operator as

\begin{align*} F^{-1}(z,p,q)=\exp ({-}{}^{t}{q}\partial _u)\exp ({-}{}^{t}{p}u)\exp ({-}z). \end{align*}

Lemma 5.13. Defining $F(z,p,q)$ for any $p$ , $q$ and $z$ , and $H^L$ for $L\in \boldsymbol {L}$ as in ( 5.8 ), the operator equality

(5.14) \begin{equation} F^{-1}(z,p,q)(L - H^L)F(z,p,q)=L - H^L \end{equation}

holds if and only if

(5.15) \begin{equation} L \bigg [\, \begin{matrix} q \\[3pt] p \end{matrix}\, \bigg ] =\Gamma ^L \bigg [\, \begin{matrix} q \\[3pt] p \end{matrix}\, \bigg ] \ \ \textrm {and} \ \ L(z)=\frac 12\,(\,{}^{t}{p}\alpha p - {}^{t}{q}\gamma q\,). \end{equation}

Here $\Gamma ^L$ is that of ( 5.6 ).

Proof. We calculate directly that

\begin{align*} \begin{aligned} F^{-1}(z,p,q)\,LF(z,p,q) &=F^{-1}(z,p,q) \big (\,L(z) +L({}^{t}{p})\,u+L({}^{t}{q}) \,\partial _u\,\big ) F(z,p,q)+L \\[3pt] &=L+ \big (\, L(z) +L({}^{t}{p})(u-q)+L({}^{t}{q})\,\partial _u\,\big ). \end{aligned} \end{align*}

Similarly, we find

\begin{align*} \begin{aligned} F^{-1}(z,p,q)&\,H^LF(z,p,q) =\frac 12 \big [{}^{t}{\partial _u}+{}^{t}{p} \ {}^{t}{u}-{}^{t}{q}\big ] \bigg [\, \begin{matrix} \alpha & \beta \\[3pt] {}^{t}{\beta }& \gamma \end{matrix}\, \bigg ] \bigg [\, \begin{matrix} \partial _u+p \\[3pt] u-q \end{matrix}\, \bigg ] \\[3pt] &=H^L +\big [{}^{t}{p}\ \ -{}^{t}{q}\big ] \bigg [\, \begin{matrix} \alpha & \beta \\[3pt] {}^{t}{\beta } & \gamma \end{matrix}\, \bigg ] \bigg [\, \begin{matrix} \partial _u \\[3pt] u \end{matrix}\, \bigg ] +\frac 12\big [{}^{t}{p}\ \ -{}^{t}{q}\big ] \bigg [\, \begin{matrix} \alpha & \beta \\[3pt] {}^{t}{\beta } &\gamma \end{matrix}\, \bigg ] \bigg [ \begin{matrix} p \\[3pt] -q \end{matrix} \bigg ]. \end{aligned} \end{align*}

Matching coefficients of $\partial _u$ , $u$ , and $1$ , and transposing and rearranging, we see that, respectively,

\begin{align*} L(q)=[{-}\beta \ \ \alpha \,] \bigg [\, \begin{matrix} q \\[3pt] p \end{matrix}\, \bigg ], \ \ \ L(p)=[\,-\gamma \ {}^{t}{\beta }\,] \bigg [\,\begin{matrix} q \\[3pt] p \end{matrix}\,\bigg ], \ \ \ L(z)=\frac 12\,(\,{}^{t}{p}\alpha p - {}^{t}{q}\gamma q\,) \end{align*}

as desired.

Corollary 5.16. The formula ( 5.15 ) holds if

(5.17) \begin{equation} \bigg [\, \begin{matrix} q \\[3pt] p \end{matrix}\, \bigg ] =\varOmega \bigg [ \begin{matrix} b^{\prime} \\[3pt] b^{\prime\prime} \end{matrix} \bigg ], \ \ z = z_0 + \tfrac 12{}^{t}{p}q, \end{equation}

where $b^{\prime}$ and $b^{\prime\prime}$ are arbitrary numerical constant vectors, and $z_0$ is an irrelevant numerical constant, which we set to zero below.

Denoting the constant vector ${}^{t}{[b^{\prime}\ b^{\prime\prime}]}$ simply by $b$ , we denote $p$ , $q$ , and $z$ with $z_0=0$ in (5.17) by $p(b)$ , $q(b)$ , and $z(b)$ , respectively. We define

\begin{align*} G(b,u,\varOmega )=F\big (z(b),p(b),q(b)\big )\,G_0(u,\varOmega ). \end{align*}

Using the Legendre relation (5.4), we note that ${}^{t}{p}\omega ^{\prime}-{}^{t}{q}\eta ^{\prime}= 2\pi i {}^{t}{b^{\prime\prime}}$ , and hence we obtain

\begin{align*} \begin{aligned} G&(b,u,\varOmega ) = F\big (z(b),p(b),q(b)\big )\,G_0(u,\varOmega )\\[3pt] &=\exp (\tfrac 12 {}^{t}{p}q)\,\exp ({}^{t}{p}u)\,\exp ({-}{}^{t}{q}\eta ^{\prime}\omega ^{\prime-1}u) \exp ({-}\tfrac 12 {}^{t}{q}\eta ^{\prime} \omega ^{\prime-1}q)\,G_0\\[3pt] &= \exp ({-}\tfrac 12 (2\pi i {}^{t}{b^{\prime\prime}} \omega ^{\prime-1}q) \exp (2\pi i {}^{t}{b}^{\prime\prime} \omega ^{\prime-1}u)\,G_0\\[3pt] &= \exp ({-}\tfrac 12 (2\pi i {}^{t}{b^{\prime\prime}} \omega ^{\prime-1}(\omega ^{\prime} b^{\prime} +\omega ^{\prime\prime}b^{\prime\prime}) \exp (2\pi i {}^{t}{b}^{\prime\prime} \omega ^{\prime-1}u)\,G_0 \\[3pt] &=\bigg (\frac {(2\pi )^g}{\mathrm {det}(\omega ^{\prime})}\bigg )^{\frac 12}\, \exp \big ({-}\tfrac 12 {}^{t}{u}\eta ^{\prime} \omega ^{\prime-1} u\big ) \,\exp \left(\,2\pi i\big (\,\tfrac 12 {}^{t}{b^{\prime\prime}} \omega ^{\prime-1} \omega ^{\prime\prime} b^{\prime\prime}+ {}^{t}{b^{\prime\prime}}(\omega ^{\prime-1}u+b^{\prime})\,\big )\right). \end{aligned} \end{align*}

Now, the following theorem, which is the foundation of BL theory, is obvious from (5.10) and (5.14).

Theorem 5.18. (The primary heat equation) For the function $G(b,u,\varOmega )$ above, one has

(5.19) \begin{equation} (L-H^L)\,G(b,u,\varOmega )=0 \end{equation}

for any $L\in \boldsymbol {L}$ .

5.3. The algebraic heat operators

For the coordinates of the space in which the sigma function is defined, we do not use $(u_1,\cdots, u_g)$ for subscripts of the variable $u$ , but denote instead, as in (4.3)

\begin{align*} u=(u_{w_g},\cdots, u_{w_1}). \end{align*}

That is, the components of $u$ are labelled by their weights, which are the Weierstrass gaps. $\Gamma ^L\in \mathfrak {sp} (2g,\mathbb {Q}[\mu ])$ . As a corollary to Theorem 5.1, we have

Corollary 5.20. Let

(5.21) \begin{equation} \rho (u)=\sum _{b}c_b\,G(b,u,\varOmega ), \end{equation}

where $b$ runs through the elements of any set $\subset$ $\mathbb {C}^{2g}$ and $c_b\in \mathbb {C}$ are constants such that the sum converges absolutely. Then we have, for any $L\in \boldsymbol {L}$ , that

\begin{align*} (L-H^L)\,\rho (u,\varOmega )=0. \end{align*}

Remark 5.22. Assume that $\mathscr {C}_{{\mu }}$ is non-singular. Let $\varOmega$ be the usual period matrix defined by ( 4.2 ), and $\delta$ be its Riemann constant. The function defined at ( 4.15 ) is written as

(5.23) \begin{equation} \tilde {\sigma }(u)=\tilde {\sigma }(u,\varOmega ) =\sum _{n\in \mathbb {Z}^g}G( {}^{t}{[ \delta ^{\prime}\ n+\delta ^{\prime\prime}]},u,\varOmega ). \end{equation}

Since the imaginary part of $\omega ^{\prime-1}\omega ^{\prime\prime}$ is positive definite, this series converges absolutely. This is a special case of $\rho (u)$ of ( 5.19 ).

Because both $L$ and $H^L$ are independent of $b$ , there are infinitely many linearly independent entire functions $\rho (u)$ on $\mathbb {C}^g$ satisfying $(L-H^L)\rho (u)=0$ . Moreover, since, for a fixed $b$ , the function $G(b,u,\varOmega )$ is independent of $L$ , we see that, by switching the choice of $L$ , there are infinitely many linearly independent operators of the form $(L-H^L)$ which satisfy $(L-H^L)\,\rho (u)=0$ for some fixed $\rho (u)$ .

However, because our aim is to find a method to calculate the power series expansion of the sigma function, we need a more detailed discussion. For our purpose, we require

1. (A1) for any (or any element of good generators) $L\in \boldsymbol {L}$ find a good mapping which associates $L$ to a (quadratic linear differential) operator which annihilate the sigma function (4.16) as an element in the ring $\mathbb {Q}[{\mu }][[u]]$ , and

2. (A2) to show that the sigma function and its absolute constant multiples are exactly the functions in $\mathbb {Q}[{\mu }][[u]]$ killed by the operators obtained in (A1)

Since $L$ is a derivation with respect to the $\mu _j$ s but $H^L$ is a differential operator with respect to the $u_{w_j}$ s, we see that, for some function $\varXi$ depending only on the $\mu _j$ s,

\begin{align*} L\,\varXi \tilde {\sigma }(u) =(L\varXi )\,\tilde {\sigma }(u)+\varXi (L\tilde {\sigma }(u)),\ \ \ \ H^L\,\varXi \tilde {\sigma }(u) =\varXi (H^L\tilde {\sigma }(u)). \end{align*}

Therefore, $\varXi \tilde {\sigma }(u)$ satisfies

(5.24) \begin{equation} (L-H^L)(\varXi \tilde {\sigma }(u)) =\tfrac {L\varXi }{\varXi }\,\varXi \,\tilde {\sigma }(u) =(L\log \varXi )\varXi \,\tilde {\sigma }(u). \end{equation}

If $\varXi \,\tilde {\sigma }(u)$ is the correct sigma function, the left-hand side of the above is in $\mathbb {Q}[\mu ][[u]]$ , So, in order to get the function $\hat {\sigma }(u)$ of (4.16), $L\log \varDelta \in \mathbb {Q}[\mu ]$ . We shall show that this indeed holds for any $L\in \boldsymbol {L}$ in the next section.

5.4. The matrix $V$

Throughout this section, we suppose that all the $\mu _j$ ’s, $x$ , and $y$ are variables or indeterminates. In view of the approach in [Reference Buchstaber and Leykin9], we require that $L\log \varDelta$ belongs to $\mathbb {Q}[{\mu }]$ because of conditions (A1), (A2) and equation (5.24). We also explain suitable choices for $\Gamma$ for which $L$ satisfies the condition.

Now we explain another method, known in singularity theory, to calculate the discriminant $\varDelta$ and a basis of the space of the vector fields tangent to the variety defined by $\varDelta =0$ .

Let $(x,y)$ and $(z,w)$ are different generic points of $\mathscr {C}$ . Following p.112 of [Reference Buchstaber and Leykin8], we define a function $\mathrm {ph}\big ((x,y),(z,w)\big )$ (pre-hessian), which is defined by

\begin{align*} \mathrm {ph}\big ((x,y),(z,w)\big ) =\frac 12\left |\, \begin{matrix} \dfrac {f_1(x,y)-f_1(z,w)}{x-z} & \dfrac {f_2(x,y)-f_2(z,w)}{x-z} \\[8pt] \dfrac {f_1(z,y)-f_1(x,w)}{y-w} & \dfrac {f_2(z,y)-f_2(x,w)}{y-w} \end{matrix}\, \right |\in \mathbb {Z}[\mu, \,x,\,y,\,z,\,w]. \end{align*}

For any $F\in \mathbb {Q}[\mu ][x,y]$ , we define

(5.25) \begin{equation} \mathrm {Hess}\,F=\Bigg |\, \begin{matrix} \frac {\partial ^2}{\partial x^2}F & \frac {\partial ^2}{\partial x\partial y}F \\[4pt] \frac {\partial ^2}{\partial y\partial x}F & \frac {\partial ^2}{\partial y^2}F \end{matrix}\, \Bigg |. \end{equation}

Lemma 5.26. (Buchstaber-Leykin [8], p.64) Let $I$ be the ideal in $\mathbb {Q}[{\mu }][x,y,z,w]$ generated by $f_1(x,y)$ , $f_2(x,y)$ , $f_1(z,w)$ , and $f_2(z,w)$ . The determinant $\mathrm {ph}\big ((z,w),(x,y)\big )$ has the following properties. (1) $\mathrm {ph}\big ((x,y),(x,y)\big )=\mathrm {Hess}\,f(x,y)$ . (2) $\mathrm {ph}\big ((x,y),(z,w)\big )=\mathrm {ph}\big ((z,w),(x,y)\big )$ . (3) We have

\begin{align*} \mathrm {ph}\big ((z,w),(x,y)\big )\,F\big ((x,y),(z,w)\big ) \equiv \mathrm {ph}\big ((x,y),(z,w)\big )\,F\big ((z,w),(x,y)\big )\,\bmod {I} \end{align*}

for any $F\big ((x,y),(z,w)\big )\in \mathbb {Q}[{\mu }][x,y,z,w]$ .

Proof. (1) Taking the limit $z\to x$ after subtracting the second row times $\frac {y-w}{x-z}$ from the first row in $\mathrm {ph}\big ((z,w),(x,y)\big )\,F\big ((x,y),(z,w)\big )$ , we have

\begin{align*} \frac 12 \left | \begin{array}{c@{\quad}c} f_{11}(x,y)+f_{11}(x,w) & f_{21}(x,y)+f_{21}(x,w) \\[3pt] \dfrac {f_1(x,y)-f_1(x,w)}{y-w} & \dfrac {f_2(x,y)-f_2(x,w)}{y-w} \end{array} \right |, \end{align*}

where $f_{11}(x,y)=\frac {\partial ^2}{\partial x^2}(x,y)$ , etc. Then, by taking limit $y\to w$ we get $\mathrm {Hess}\,f(x,y)$ . (2) is trivial. (3) By expanding the matrix, we see that the numerator

(5.27) \begin{equation} \begin{aligned} &\ \ \ \ \big ( f_1(x,y)f_2(z,y) -f_1(x,y)f_2(x,w) -f_1(z,w)f_2(z,y) +f_1(z,w)f_2(x,w) \big )\\[3pt] &\hskip 5pt -\big ( f_1(z,y)f_2(x,y) -f_1(z,y)f_2(z,w) -f_1(x,w)f_2(x,y) +f_1(x,w)f_2(z,w) \big )\\[3pt] &=\big (f_1(x,y)f_2(z,y)-f_1(z,y)f_2(x,y)\big ) -\big (f_1(x,y)f_2(x,w)-f_1(z,y)f_2(z,w)\big )\\[3pt] &\hskip 5pt -\big (f_1(z,w)f_2(z,y)-f_1(x,w)f_2(x,y)\big ) +\big (f_1(z,w)f_2(x,w)-f_1(x,w)f_2(z,w)\big )\\[3pt] &=\big (f_1(x,y)f_2(z,y)-f_1(x,w)f_2(z,w)\big ) -\big (f_1(x,y)f_2(x,w)-f_1(x,w)f_2(x,y)\big )\\[3pt] &\hskip 5pt -\big (f_1(z,w)f_2(z,y)-f_1(z,y)f_2(z,w)\big ) +\big (f_1(z,w)f_2(x,w)-f_1(z,y)f_2(x,y)\big ) \end{aligned} \end{equation}

is divisible by $(z-x)(w-y)$ , because the second expression is clearly divisible by $(z-x)$ , while the third expression is divisible by $(w-y)$ . Hence, $\mathrm {ph}\big ((x,y),(x,y)\big )\in \mathbb {Q}[\mu ][x,y]$ . Moreover, the second expansion is equal to

\begin{align*} &\big (f_1(x,y)f_2(z,y)-f_1(x,y)f_2(x,y)+f_1(x,y)f_2(x,y)-f_1(z,y)f_2(x,y)\big )\\[3pt] -\,&\big (f_1(x,y)f_2(x,w)-f_1(x,y)f_2(z,w)+f_1(x,y)f_2(z,w)-f_1(z,y)f_2(z,w)\big )\\[3pt] -\,&\big (f_1(z,w)f_2(z,y)-f_1(z,w)f_2(x,y)+f_1(z,w)f_2(x,y)-f_1(x,w)f_2(x,y)\big )\\[3pt] +\,&\big (f_1(z,w)f_2(x,w)-f_1(z,w)f_2(z,w)+f_1(z,w)f_2(z,w)-f_1(x,w)f_2(z,w)\big )\\[3pt] =\,&f_1(x,y)\big (f_2(z,y)-f_2(x,y)\big )+\big (f_1(x,y)-f_1(z,y)\big )f_2(x,y)\\[3pt] -\,&f_1(x,y)\big (f_2(x,w)-f_2(z,w)\big )-\big (f_1(x,y)-f_1(z,y)\big )f_2(z,w)\\[3pt] -\,&f_1(z,w)\big (f_2(z,y)-f_2(x,y)\big )-\big (f_1(z,w)-f_1(x,w)\big )f_2(x,y)\\[3pt] +\,&f_1(z,w)\big (f_2(x,w)-f_2(z,w)\big )+\big (f_1(z,w)-f_1(x,w)\big )f_2(z,w), \end{align*}

which implies that $\mathrm {ph}\big ((x,y),(z,w)\big )(w-y)$ already belongs to $I$ . A similar calculation shows that $\mathrm {ph}\big ((x,y),(z,w)\big )(z-x)\in {I}$ . For $F\big ((z,w),(x,y)\big )=x^ay^b$ , by using (2), we see

\begin{align*} \begin{aligned} \mathrm {ph}&\big ((z,w),(x,y)\big )\,x^ay^b -\mathrm {ph}\big ((x,y),(z,w)\big )\,z^aw^b\\[3pt] & =\mathrm {ph}\big ((z,w),(x,y)\big )\,(x^ay^b-z^aw^b)\\[3pt] & =\mathrm {ph}\big ((z,w),(x,y)\big )\,(x^ay^b-x^aw^b+x^aw^b-z^aw^b)\\[3pt] & =\mathrm {ph}\big ((z,w),(x,y)\big )\,\big (x^a(y^b-w^b)+(x^a-z^a)w^b\big ) \in I. \end{aligned} \end{align*}

Hence, (3) has been proved.

Below, we will use, instead of $T$ , the symmetric $2g\times 2g$ matrix

\begin{align*} V=\underset {\substack {a\in \mathrm {wt}(M)\\[3pt]b\in \mathrm {wt}(\mu )}}{[\,V_{a,b}\,]} =\underset {\substack {1\leq i\leq 2g\\[3pt]1\leq j\leq 2g}} {[\,V_{v_i,\,eq-v_{\kern 0.5pt 2g-j+1}}\,]} \in \mathrm {Sym}(2g,\,\mathbb {Q}[\mu ]) \end{align*}

where all the indices run in increasing order, defined by the equation

(5.28) \begin{equation} {}^{t}{({}^{\mathrm {rev}}{M}}(x,y))\,\cdot {V}\cdot \,{}^{\mathrm {rev}}{M}(z,w)=f(x,y)\, \mathrm {ph}\big ((z,w),(x,y)\big )\,F\big ((x,y),(z,w)\big ) \end{equation}

in the ring

\begin{align*} \mathbb {Q}[\mu ][x,y,z,w]\,\big /\,I. \end{align*}

We note that the weight of any entry of $V$ is given by

(5.29) \begin{equation} \mathrm {wt}(V_{a,\,b})=-(a+b). \end{equation}

We define

\begin{align*} H=\underset {\substack {a\in \mathrm {wt}(M)\\[3pt]b\in \mathrm {wt}(\mu )}}{[\,H_{a-q+2,\,b-q-2}\,]} =\underset {\substack {1\leq i\leq 2g\\[3pt]1\leq j\leq 2g}}{[\,H_{v_i-q+2,\,eq-v_{\kern 0.5pt 2g-j+1}-q-2}\,]} \end{align*}

as the matrix given by

\begin{align*} \mathrm {ph}\big ((x,y),(z,w)\big )={}^{t}{({}^{\mathrm {rev}}{M}}(x,y))\,H\,{}^{\mathrm {rev}}{M}(z,w). \end{align*}

We see $H\in \mathrm {Mat}(2g,\,\mathbb {Q}[\mu ])$ by consideration of 5.27 in the proof of 5.25. Moreover, as for $V$ , the weight of any entry of $H$ is given by

\begin{align*} \mathrm {wt}(H_{a-q+2,\,b-q-2})=-(a+b-2q). \end{align*}

Example 5.30. In the case $(e,q)=(2,2g+1)$ , for $a=v_i=2i-2$ and $b=eq-v_{2g-j+1}=4g+2-2(2g-j+1)+2=2j+2$ with $1\leq i\leq g$ and $1\leq j\leq g$ , we see that

\begin{align*} \begin{aligned} \mathrm {wt}(V_{a,b})&=\mathrm {wt}(V_{2i-2,2j+2})=-2(i+j), \\[3pt] \mathrm {wt}(H_{a-q+2,\,b-q-2})& \,(\!=a+b-2q)=\mathrm {wt}(H_{v_i-(2g+1)+2,\,v_{2g-j+1}-(2g+1)-2}) =2(i-j)-(2g+1) \end{aligned} \end{align*}

Lemma 5.31. The matrix $H$ is of the form

(5.32)

If $e=2$ , then we have explicitly

(5.33)

Proof. Setting all the $\mu _j$ to be $0$ , we have

\begin{align*} \frac 12\,\left |\, \begin{matrix} \frac {q({-}x^{q-1}+z^{q-1})}{x-z} & \frac {e(y^{e-1}-w^{e-1})}{x-z} \\[10pt] \frac {q(x^{q-1}-z^{q-1})}{y-w} & \frac {e(y^{e-1}-w^{e-1})}{y-w} \end{matrix}\, \right | =-eq(x^{q-2}+x^{q-3}z+\cdots +z^{q-2})(y^{e-2}+y^{e-3}w+\cdots +w^{e-2}). \end{align*}

It follows that the counter-diagonal entries of $H$ are $-eq$ . From the definitions, the weight of $H$ is $-2(eq-e-q)$ and $\mathrm {wt}(M_{2g}(x,y))=-2g-1+w_g=-2(eq-q-e)$ . Therefore the entries below the counter-diagonal must be $0$ . For the case $e=2$ , we have

\begin{align*} \begin{aligned} \frac 12\,& \left |\ \begin{matrix} -\frac {q(x^{q-1}-z^{q-1})+(q-2)\mu _4(x^{q-3}-z^{q-3})+\cdots +\mu _{eq-e}(x-z)}{x-z} & \frac {2y-2w}{x-z} \\[3pt] \ \ \frac {q(x^{q-1}-z^{q-1})+(q-2)\mu _4(x^{q-3}-z^{q-3})+\cdots +\mu _{eq-e}(x-z)}{y-w} & \frac {2y-2w}{y-w} \end{matrix}\ \right |\\[3pt] &=-2\,\frac {q(x^{q-1}-z^{q-1})+(q-2)\mu _4(x^{q-3}-z^{q-3})+\cdots +\mu _{eq-e}(x-z)}{x-z}\\[3pt] &=-2\,\big (q(x^{q-2}+x^{q-3}z+\cdots +z^{q-2}) +(q-2)\mu _4(x^{q-4}+x^{q-5}z+\cdots +z^{q-4})+\cdots +\mu _{eq-e}\big ), \end{aligned} \end{align*}

giving the desired form of $H$ .

Proof. Since

\begin{align*} \begin{aligned} f(x,y)\,\mathrm {ph}\big ((x,y),(z,w)\big ) &=f(x,y)\,{}^{t}{({}^{\mathrm {rev}}{M}(z,w))}\,H\cdot \,{}^{\mathrm {rev}}{M}(x,y)\\[3pt] &\equiv {}^{t}{({}^{\mathrm {rev}}{M}}(z,w))\,{}^{\mathrm {rev}}{\!({-}\tfrac {1}{eq}\,T)}\,\,H\cdot \,{}^{\mathrm {rev}}{M}(x,y)\bmod {I} \end{aligned} \end{align*}

by (1.18), and the entries in $M(x,y)$ form a basis of $\mathbb {Q}[\mu ][x,y]/(f_1(x,y),f_2(x,y))$ , we see

(5.35) \begin{equation} V=-\tfrac 1{\,eq\,}\,{}^{\mathrm {rev}}{T}\,H. \ \ \end{equation}

Since $H$ is a skew-upper-triangular matrix of the form (5.32), we have proved $\det (V)=({-}1)^g\det (T)$ as desired.

Proof. Since the determinants of $T$ and $V$ are of homogeneous weight, it suffices to check the sum of weights of the counter-diagonal entries, which is given by

\begin{align*} \sum _{q\in \mathrm {wt}(M)}\mathrm {wt}(T_{a,\,eq-a}) =\sum _{q\in \mathrm {wt}(M)}({-}eq)=-2geq=-eq(e-1)(q-1). \end{align*}

This is the same as $\mathrm {wt}(\det (V))$ and $\mathrm {wt}(\det (T))$ .

Lemma 5.37. We have $\mathrm {wt}(\varDelta )=-eq(e-1)(q-1)$ . If $\gcd (e-1,q-1)=1$ , then we have

\begin{align*} \det T=({-}1)^g\det V=c\cdot \varDelta \end{align*}

with $c\in \mathbb {Q}^{\times }$ .

Proof. Letting all the coefficients $\mu _j$ of $p_j(x)$ for $1\leq j\leq e-1$ to be zero, the discriminant $\varDelta$ becomes a power of the square of the difference of all the roots of $p_e(x)=0$ . Since the weight of any root is $-e$ , the weight of the square of the difference is

\begin{align*} 2\cdot \binom {q}{2}\cdot ({-}e)=-eq(q-1). \end{align*}

Similar arguments on $y$ shows that $\mathrm {wt}\,\varDelta$ is $-qe(e-1)$ times an positive integer. By the assumption $\gcd (e-1,q-1)=1$ , we have $\mathrm {wt}\,\varDelta$ is $-eq(e-1)(q-1)$ times a positive integer. The statement follows from 5.35 combined with 1.21 (1).

From now on we assume the modality of $C$ to be $0$ . For any $a\in \mathrm {wt}(M)$ , the coefficient $\mu _{eq-a}$ appears in the Weierstrass form, which is the reason why the modality is so important. Let

(5.39) \begin{equation} L_k = \sum _{j\in \mathrm {wt}(\mu )}\,V_{kj}\, \frac {\partial }{\partial \mu _j} \ \ \ (k\in \mathrm {wt}(M)). \end{equation}

It is natural to define

\begin{align*} \mathrm {wt}\left(\frac {\partial }{\partial \mu _{j}}\right)=j, \ \ \ \mathrm {wt}({L}_{v_i})=v_i. \end{align*}

We recall the definition of the matrix $\Gamma ^{L_k}$ for each $L_k$ ( $\mathrm {wt}(L_k)=-k$ ) (see 5.1), which we denote asFootnote ¹

\begin{align*} \Gamma ^{L_k} = \bigg [ \begin{array}{r@{\quad}c} -\beta _k &\ \ \alpha _k \\[3pt] -\gamma _k &\ {}^{t}{\!\beta _k} \end{array} \bigg ], \end{align*}

where $\alpha _k$ , $\gamma _k\in \mathrm {Sym}(g,\mathbb {Q}[\mu ])$ , and

(5.40) \begin{equation} \alpha _k=\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\alpha _{k;\,-i,k-j}]}, \ \ \beta _k=\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\beta _{k;\,j,k-i}]}, \ \ \gamma _k=\underset {\substack {i\in \mathrm {wt}(u)\\[3pt] j\in \mathrm {wt}(u)}}{[\gamma _{k;\,-i,k+j}]} \end{equation}

with decreasing order in $\mathrm {wt}(u)$ . Then the sum of two indices of any entry gives its weight;

\begin{align*} \mathrm {wt}(\alpha _{k;\,-i,k-j})=-i+(k-j), \ \ \mathrm {wt}( \beta _{k;\,j,k-i})=j+(k-i), \ \ \mathrm {wt}(\gamma _{k;\,-i,k+j})=-i+(k+j). \end{align*}

Note once again that $\Gamma ^{L_k}\in \mathrm {Mat}(2g,\mathbb {Q}[\mu ])$ follows from 2.1. Just to be sure, we shall redefine, as defined in (5.8),

(5.41) \begin{align} H^{L_k} &= \frac 1{\,2\,}\,\big [\,\tfrac {\partial \ }{\partial u_{w_g}} \ \cdots \ \tfrac {\partial \ } {\partial u_{w_1}} \ \ u_{w_g} \ \cdots \ u_{w_1}\,\big ] \Bigg [ \begin{array}{r@{\quad}c} \alpha _k &\ \,\beta _k \\[5pt] {}^{t}{\!\beta _k} &\ \,\gamma _k \end{array} \Bigg ]\! \left [ \begin{array}{c} \tfrac {\partial \ }{\partial u_{w_g}} \\[3pt] \vdots \\[3pt] \tfrac {\partial \ }{\partial u_{w_1}}\\[3pt] u_{w_g} \\[-2pt] \vdots \\[-4pt] u_{w_1} \end{array} \right ] \\[3pt] & =\frac 12\,\sum _{i\in \mathrm {wt}(u)}\sum _{j\in \mathrm {wt}(u)} \left({\alpha }_{k;\,-i,\,k-j}\,\frac {\partial ^2}{\partial \,u_i\partial \,u_j} \right. \nonumber \\[3pt] & \left.\ \ \ \ \ \ \ +2{\beta }_{k;\,j,k-i}\,u_i\,\frac {\partial \ \ }{\partial u_j} +{\gamma }_{k;-i,k+j}\ u_iu_j\right) +\tfrac 12\,{\mathrm {Tr}\,\beta _k}.\nonumber \end{align}

5.5. The operators tangent to the discriminant

We prove the following proposition for which there is no proof in [Reference Buchstaber and Leykin9].

Proposition 5.42. (Buchstaber-Leykin) Let

\begin{align*} L={M(x,y)}\,{}^{t}{[\,L_{v_1}\ L_{v_2}\ \cdots \ L_{v_{2g}}\,]}. \end{align*}

Then, in the ring $\mathbb {Q}[\mu ][x,y]/(f_1, f_2)$ , we have

(5.43) \begin{equation} L(\varDelta )=-\mathrm {Hess}f\cdot \varDelta . \end{equation}

Proof. For the case $e=2$ , $f(x,y)$ is of the form $y^2-p_2(x)$ . We denote $p_2(x)=p(x)$ for simplicity. Moreover, we denote $p^{\prime}(x)=\frac {\partial }{\partial x}p(x)$ and $p^{\prime\prime}(x)=\frac {\partial ^2}{\partial x^2}p(x)$ . In this case, the ring $\mathbb {Q}[\mu ][x,y]/(f_1,f_2)$ is identified with $\mathbb {Q}[\mu ][x]/(p^{\prime}(x))$ since $f_1(x,y)=\frac {\partial }{\partial y}f(x,y)=2y$ . Let $F$ be a splitting field of $p(x)$ . We write the factorisation of $p(x)$ in $F$ as $p(x) = (x-a_1)\cdots (x-a_q)$ . Then $\mu _{2i}$ is $({-}1)^i$ times the fundamental symmetric function of $a_1$ , $\cdots$ , $a_q$ of degree $i$ . Of course the ring $\mathbb {Q}[\mu ]$ is a sub-ring of $\mathbb {Q}[a_1,\,\cdots, \,a_q]$ . The Hessian of $f(x,y)=y^2-p(x)$ is

\begin{align*} \bigg |\, \begin{matrix} -p^{\prime\prime}(x) & 0 \\[3pt] 0 & 2 \end{matrix}\, \bigg | &=-2\,p^{\prime\prime}(x). \end{align*}

The main idea is to consider $\frac {\mathrm {Hess}\,f}{2f} = \frac {p^{\prime\prime}(x)}{p(x)}$ in the localised ring

\begin{align*} \big (F[x]/(p^{\prime}(x))\big )_{p(x)} \end{align*}

of $F[x]/(p^{\prime}(x))$ with respect to the multiplicative set $\{1,\,p(x),\,p(x)^2,\,\cdots \}$ (see [Reference Matsumura20], Section 4). The following calculation is done in the localised ring above. Since $\gcd (p^{\prime}(x),\,p(x))=1$ in this situation, we see $\big (F[x]/(p^{\prime}(x))\big )_{p(x)}=F[x]/(p^{\prime}(x))$ . Now, we have

\begin{align*} \frac {p^{\prime\prime}(x)}{p(x)} &=\sum _{(i,j),i\lt j}\frac {2}{(x-a_i)(x-a_j)} =\sum _{(i,j),i\lt j}\frac {2}{a_i-a_j}\left(\frac {1}{x-a_i} -\frac {1}{x-a_j}\right)\\[3pt] &=2\sum _{i=1}^q\left(\sum _{j\neq i}\frac {1}{a_i-a_j}\right)\frac {1}{x-a_i} =-2\sum _{i=1}^q\left(\sum _{j\neq i}\frac {1}{a_i-a_j}\right)\frac {1}{p^{\prime}(a_i)} \frac {-p^{\prime}(a_i)}{x-a_i}\\[3pt] &=-2\sum _{i=1}^q\left(\sum _{j\neq i}\frac {1}{a_i-a_j}\right)\frac {1}{p^{\prime}(a_i)} \frac {p^{\prime}(x)-p^{\prime}(a_i)}{x-a_i} = -2 \sum _{i=1}^q c_i\,\frac {p^{\prime}(x)-p^{\prime}(a_i)}{x-a_i}, \end{align*}

where

\begin{align*} c_i =\left(\sum _{j \neq i}\frac {1}{a_i-a_j}\right)\frac {1}{p^{\prime}(a_i)}. \end{align*}

Since $\mathrm {Hess}f(x,y)=-2p^{\prime\prime}(x)$ and $p(x)=-f(x,y)$ in the localised ring, it suffices to show that

\begin{align*} \frac {\partial }{\partial \mu _{2i}}\log \varDelta =-2\sum _{j=1}^q c_j {a_j}^{q-i} \end{align*}

up to a non-zero constant multiple. Indeed, if we have the formula above, we have

\begin{align*} \begin{aligned} \frac {L(x)\varDelta }{\varDelta } &=\sum _{i,k}M_i(x,y)V_{ik}\frac {\partial }{\partial \mu _{2i}}\log \varDelta =-2\sum _{i,k}M_i(x,y)V_{ik}\sum _{j=1}^qc_j{a_j}^{q-k}\\[3pt] &=-2\sum _{j=1}^qc_j\sum _{i,k}M_i(x,y)V_{ik}{a_j}^{q-k} =-2\sum _{j=1}^qc_j\,f(x,y)\,\mathrm {ph}\big ((x,y),(a_j,0)\big )\\[3pt] &=-2f(x,y)\,\sum _{j=1}^qc_j\,({-}2)\,\frac {p^{\prime}(x)-p^{\prime}(a_j)}{x-a_j} =-2f(x,y)\,\frac {p^{\prime\prime}(x)}{p(x)} =2p^{\prime\prime}(x) =-\mathrm {Hess}\,f(x,y). \end{aligned} \end{align*}

Here we have used

\begin{align*} \mathrm {ph}\big ((x,y),(a_j,0)\big ) =\frac 1{\,2\,} \left | \begin{array}{c@{\quad}c} \frac {-p^{\prime}(x)-p^{\prime}(a_j)\,}{x-a_j} & \frac {2y}{\,x-a_j\,}\\[3pt] \frac {-p^{\prime}(x)-p^{\prime}(a_j)\,}{y} & \frac {2y}{\,y\,} \end{array} \right | =-2\frac {\,p^{\prime}(x)-p^{\prime}(a_j)\,}{x-a_j}. \end{align*}

To calculate $\frac {\partial \log (\varDelta )}{\partial \mu _{2i}}$ , we remove the assumption $\mu _2=0$ . Since $\varDelta$ is some non-zero constant multiple of

\begin{align*} \prod _{i\lt j}(a_i-a_j)^2, \end{align*}

we easily get the $q\times q$ -matrix $\big [\frac {\partial \mu _{2i}}{\partial a_j}\big ]$ , and then we get $\frac {\partial \log (\varDelta )}{\partial \mu _{2i}}$ by using its inverse matrix. For $(e,q)=(3,4)$ , $(3,5)$ , we know only a proof by direct calculation with Maple by using the explicit form of $\varDelta$ and the operators $L_{v_j}$ s.