0$. We also introduce $\omega_2=\omega_1+\omega_3$ and $\omega_4=0$. The numbers $\omega_1,\dots,\omega_4$ are called half-periods. The fundamental period parallelogram $\Delta$ is the half-open region consisting of the line segments $[0,2\omega_1)$, $[0,2\omega_3)$ and the interior of the parallelogram with vertices $0$, $2\omega_1$, $2\omega_2$, and $2\omega_3$. The function \begin{equation} \wp(z;\omega_1,\omega_3)=\frac1{z^2}+ \sum_{\substack{(m,n)\in\bbZ^2 \\ (m,n)\neq(0,0)}} \left(\frac1{(z-2m\omega_1-2n\omega_3)^2} - \frac1{(2m\omega_1+2n\omega_3)^2}\right), \lb{B.1} \end{equation} or $\wp(z)$ for short, was introduced by Weierstrass. It is an even elliptic function of order 2 with fundamental periods $2\omega_1$ and $2\omega_3$. Its derivative $\wp'$ is an odd elliptic function of order 3 with fundamental periods $2\omega_1$ and $2\omega_3$. Every elliptic function may be written as $R_1(\wp(z))+R_2(\wp(z))\wp'(z)$ where $R_1$ and $R_2$ are rational functions of $\wp$. The numbers \begin{equation} g_2 = 60\sum_{\substack{(m,n)\in\bbZ^2\\(m,n)\neq(0,0)}} \frac1{(2m\omega_1+2n\omega_3)^4}, \quad g_3 = 140\sum_{\substack{(m,n)\in\bbZ^2\\(m,n)\neq(0,0)}} \frac1{(2m\omega_1+2n\omega_3)^6} \lb{B.2} \end{equation} are called the invariants of $\wp$. Since the coefficients of the Laurent expansions of $\wp(z)$ and $\wp'(z)$ at $z=0$ are polynomials of $g_2$ and $g_3$ with rational coefficients, the function $\wp(z;\omega_1,\omega_3)$ is also uniquely characterized by its invariants $g_2$ and $g_3$. One also frequently uses the notation $\wp(z|g_2,g_3)$. The function $\wp(z)$ satisfies the first-order differential equation \begin{equation} \label{B.3} \wp'(z)^2 = 4\wp(z)^3-g_2\wp(z)-g_3 \end{equation} and hence the equations \begin{equation} \wp''(z) = 6\wp(z)^2-g_2/2 \text { and } \wp'''(z)= 12\wp'(z)\wp(z). \lb{B.4} \end{equation} Thus, $-2\wp$ is a stationary solution of the first KdV equation, $\sKdV_1(q)=0$ in \eqref{A.7} with $c_1=0$. The function $\wp'$, being of order $3$, has three zeros in $\Delta$. Since $\wp'$ is odd and elliptic it is obvious that these zeros are the half-periods $\omega_1,\omega_2=\omega_1+\omega_3$ and $\omega_3$. Let $e_j=\wp(\omega_j)$, $j=1,2,3$. Then \eqref{B.3} implies that $4e^3_j-g_2e_j-g_3=0$ for $j=1,2,3$. Therefore \begin{align} 0 &= e_1+e_2+e_3, \lb{B.5} \\ g_2 &=-4(e_1e_2+e_1e_3+e_2e_3)=2(e^2_1+e^2_2+e^2_3), \lb{B.6} \\ g_3 &= 4e_1e_2e_3 =\frac43 (e^3_1+e^3_2+e^3_3). \lb{B.7} \end{align} Weierstrass also introduced two other functions denoted by $\zeta$ and $\sigma$. The Weierstrass $\zeta$-function is defined by \begin{equation} \frac{d}{dz}\zeta(z) = -\wp(z), \quad \lim_{z\to0} \bigg(\zeta(z)-\frac1z\bigg) = 0. \lb{B.8} \end{equation} It is a meromorphic function with simple poles at $2m\omega_1+2n\omega_3, m,n\in\bbZ$ having residues $1$. It is not periodic but quasi-periodic in the sense that \begin{equation} \zeta(z+2\omega_j)=\zeta(z)+2\eta_j, \quad 1\leq j\leq 4, \lb{B.9} \end{equation} where $\eta_j=\zeta(\omega_j)$ for $j=1,2,3$ and $\eta_4=0$. The Weierstrass $\sigma$-function is defined by \begin{equation} \frac{\sigma'(z)}{\sigma(z)}=\zeta(z), \quad \lim_{z\to0} \frac{\sigma(z)}{z}=1. \lb{B.10} \end{equation} $\sigma$ is an entire function with simple zeros at the points $2m\omega_1+2n\omega_3, m,n\in\bbZ$. Under translation by a period $\sigma$ behaves according to \begin{equation} \sigma(z+2\omega_j)=-\sigma(z)e^{2\eta_j(z+\omega_j)}, \quad j=1,2,3. \lb{B.11} \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \label{tb0} $($\cite{GW98a}$)$ Given numbers $\alpha_1,\dots,\alpha_m$ and $\beta_1,\dots,\beta_m$ such that $\beta_k\neq\beta_\ell\,({\rm mod}\,\Delta)$ for $k\neq\ell$, the following identity holds \begin{equation} \prod_{j=1}^m \f{\sigma(z-\alpha_j)}{\sigma(z-\beta_j)} =\sum_{j=1}^m \f{\prod_{k=1}^m \sigma(\beta_j-\alpha_k)} {\prod_{\ell=1,\ell\neq j}^m \sigma(\beta_j-\beta_\ell)} \f{\sigma(z-\beta_j+\beta-\alpha)} {\sigma(z-\beta_j)\sigma(\beta-\alpha)}, \lb{B.12} \end{equation} where \begin{equation} \alpha=\sum_{j=1}^m \alpha_j \text{ and }\beta= \sum_{j=1}^m \beta_j \lb{B.13} \end{equation} and $\sigma$ is constructed {}from the fundamental periods $2\omega_1$ and $2\omega_3$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \label{tb1} $($\cite[p.\ 182, Theorem 5.12]{Ma85}$)$ Given an elliptic function $f$ of order $n$ with fundamental periods $2\,\omega_1$ and $2\,\omega_3$, let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be the zeros and poles of $f$ in the fundamental period parallelogram $\Delta$ repeated according to their multiplicities. Then \begin{equation} f(z) = C \frac{\sigma(z-a_1)\cdots\sigma(z-a_n)} {\sigma(z-b_1)\cdots\sigma(z-b_{n-1}) \sigma(z-b_n')}, \lb{B.14} \end{equation} where $C\in\bbC$ is a suitable constant, $\sigma$ is constructed {}from the fundamental periods $2\,\omega_1$ and $2\,\omega_3$, and where \begin{equation} b_n'-b_n=(a_1+\cdots+a_n)-(b_1+\cdots+b_n) \lb{B.15} \end{equation} is a period of $f$. Conversely, every such function is an elliptic function. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \label{tb2} $($\,\cite[p.\ 182, Theorem 5.13]{Ma85}$)$ Given an elliptic function $f$ with fundamental periods $2\, \omega_1$ and $2\, \omega_3$, let $b_1,\dots,b_r$ be the distinct poles of $f$ in $\Delta$. Suppose the principal part of the Laurent expansion near $b_k$ is given by \begin{equation} \sum^{\beta_k}_{j=1} \frac{A_{j,k}}{(z-b_k)^j}, \quad 1\leq k\leq r. \lb{B.16} \end{equation} Then \begin{align} f(z) = A_0+\sum^r_{k=1} \sum^{\beta_k}_{j=1}(-1)^{j-1} \frac{A_{j,k}}{(j-1)!} \zeta^{(j-1)}(z-b_k), \lb{B.17} \end{align} where $A_0\in\bbC$ is a suitable constant and $\zeta$ is constructed {}from the fundamental periods $2\,\omega_1$ and $2\,\omega_3$. Conversely, every such function is an elliptic function if $\sum^{r}_{k=1} A_{1,k}=0$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% One notes that this theorem resembles the partial fraction expansions for rational functions. Finally, we turn to elliptic functions of the second kind, the central object in our analysis. A meromorphic function $\psi:\mathbb{C}\to\mathbb{C}\cup\{\infty\}$ for which there exist two complex constants $\omega_1$ and $\omega_3$ with non-real ratio and two complex constants $\rho_1$ and $\rho_3$ such that \begin{equation} \psi(z+2\, \omega_j)=\rho_j \psi(z), \quad j=1,3, \lb{B.18} \end{equation} is called elliptic of the second kind. We call $2\,\omega_1$ and $2\,\omega_3$ the quasi-periods of $\psi$. Together with $2\,\omega_1$ and $2\,\omega_3$, $2\,m_1\omega_1+2\,m_3\omega_3$ are also quasi-periods of $\psi$ if $m_1,m_3\in\bbZ$. If every quasi-period of $\psi$ can be written as an integer linear combination of $2\,\omega_1$ and $2\,\omega_3$, then these are called fundamental quasi-periods. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \label{tb3} A function $\psi$ which is elliptic of the second kind and has fundamental quasi-periods $2\,\omega_1$ and $2\,\omega_3$ can always be put into the form \begin{equation} \psi(z) = C \exp(\lambda z) \frac{\sigma(z-a_1) \cdots\sigma(z-a_n)} {\sigma(z-b_1)\cdots\sigma(z-b_{n})} \lb{B.19} \end{equation} for suitable constants $C$, $\lambda$, $a_1,\dots,a_n$ and $b_1,\dots,b_n$. Here $\sigma$ is constructed {}from the fundamental periods $2\,\omega_1$ and $2\,\omega_3$. Conversely, every such function is elliptic of the second kind. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Symmetric products} \label{sC} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\theequation}{C.\arabic{equation}} \setcounter{theorem}{0} \setcounter{equation}{0} Let $X$ be a Riemann surface. In addition to the cartesian product $X^N=X\times\cdots\times X$ ($N$ factors), $N\in\bbN$, we also introduce the $N$th symmetric product of $X$ defined as the quotient space \begin{equation} X^N/S_N. \lb{C.1} \end{equation} Here $S_N$ denotes the symmetric group on $N$ letters acting as the group of permutations of the factors in the cartesian product $X^N$, that is, \begin{equation} \pi(x_1,\dots,x_n)=(x_{\pi(1)},\dots, x_{\pi(N)}), \quad \pi\in S_N. \lb{C1.a} \end{equation} Thus, the points in $X^N/S_N$ can be considered as $N$-tuples of points of $X$ without regard to their order. $X^N/S_N$ inherits the topology from $X^N$ (the quotient topology) and the canonical projection (quotient map) \begin{align} %\begin{split} \nu\colon \begin{cases} X^N \to X^N/S_N \\ (x_1,\dots,x_N)\mapsto [x_1,\dots,x_N]= \{\pi(x_1,\dots,x_N)\in X^N \,|\,\pi\in S_N\} \end{cases} \lb{C.2} %\end{split} \end{align} defines a complex structure on $X^N/S_N$ as follows. Consider a point $[p_1,\dots,p_N]\in X^N/S_N$, let $x_j$ be a local coordinate in an open neighborhood $U_j$ of $p_j\in X$, assuming $U_j\cap U_k=\emptyset$ if $p_j\neq p_k$ and $x_j=x_k$ in $U_j=U_k$ for $p_j=p_k$. Denote by $\sigma_1,\dots,\sigma_N$ the elementary symmetric functions of $x_1,\dots,x_N$, then the map \begin{align} \begin{split} &\nu(U_1\times \cdots \times U_N) \to \bbC^N \\ &[q_1,\dots,q_N] \mapsto (\sigma_1(x_1(q_1),\dots,x_N(q_N)),\dots, \sigma_N(x_1(q_1),\dots,x_N(q_N))) \lb{C.3} \end{split} \end{align} provides a coordinate chart on $\nu(U_1\times \cdots \times U_N)$. In this manner, $X^N/S_N$ (like $X^N$) becomes an $N$-dimensional complex manifold with $X^N$ an $N!$-sheeted branched analytic covering of $X^N/S_N$. Away from the branch locus the map $\nu$ is a covering map and one can take \begin{equation} (x_1(q_1),\dots,x_N(q_N)) \lb{C.4} \end{equation} as coordinates on $X^N/S_N$ (here the points $p_j$, corresponding to the charts $(U_j,x_j)$, are mutually distinct). At the other extreme, where $p_1=p_2=\dots=p_N$, local coordinates are given by $(\sigma_1(x_1(q_1),\dots,x_N(q_N)),\dots, \sigma_N(x_1(q_1),\dots, x_N(q_N)))$, that is, by \begin{equation} \Bigg(\sum_{j=1}^N x_j(q_j),\dots,\prod_{j=1}^N x_j(q_j)\Bigg). \lb{C.5} \end{equation} Next, assume the topological space $(X^N,\tau)$ is generated by the metric $d$ on $X^N$. We then write $\tau=(d)$ and hence $(X^N,\tau)=(X^N,(d))$. In addition, let $(X^N/S_N,\tau_{S_N})$ denote the topological space equipped with the quotient topology of $X^N/S_N$ relative to $(X,\tau)$, \begin{equation} \tau_{S_N}=\{U\subseteq X^N/S_N \,|\, \nu^{-1}(U)\in\tau\}, \lb{C.6} \end{equation} We now investigate a case in which $(X^N/S_N,\tau_{S_N})$ is also generated by a metric $D$ on $X^N/S_N$. For this purpose we now assume that the metric $d$ is such that each permutation in $S_N$ is an isometry\footnote{This holds for $X=\bbC$, $X=\bbC/\Lambda_\omega$, and $X=\bbC/\Lambda_{2\omega_1,2\omega_3}$ and the usual metrics on them (cf.\ Remark \ref{rC.2}).}, that is, \begin{equation} \text{for all $\pi\in S_N$: } \, d(\pi(x),\pi(y))=d(x,y), \quad x,y \in X^N \lb{C.7} \end{equation} (here $x=(x_1,\dots,x_N)\in X^N$, etc.). A standard situation in which \eqref{C.7} can be verified is as follows: Suppose $\delta$ is a metric on $X$. Then for any fixed $r\in [1,\infty)$, $d_r \colon X^N\times X^N\to [0,\infty)$, defined as \begin{equation} d_r (x,y)=\bigg(\sum_{j=1}^N \delta (x_j,y_j)^r\bigg)^{1/r}, \quad x=(x_1,\dots,x_N), \, y=(y_1,\dots,y_N)\in X^N, \end{equation} defines a metric on $X^N$ satisfying \eqref{C.7} (and similarly in the case $r=\infty$ using the supremum over $j\in\{1,\dots,N\}$). Since $S_N$ is transitive, the expression ${\min}_{\sigma,\rho\in S_N}\{d(\sigma(x),\rho(y))\}$ does not change when $x$ and $y$ are replaced by other representatives in their respective equivalence classes, that is, it depends only on $[x]$ and $[y]$. Hence, we may define \begin{equation} D([x],[y])={\min}_{\sigma,\rho\in S_N}\{d(\sigma(x),\rho(y))\}, \quad [x], [y] \in X^N/S_N. \lb{C.8} \end{equation} The assumption that the permutations are isometries then yields \begin{equation} D([x],[y])={\min}_{\rho\in S_N}\{d(x,\rho(y))\}, \quad [x], [y] \in X^N/S_N. \lb{C.9} \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \lb{tC.1} Let $(X^N,d)$ be a metric space and suppose that every permutation in $S_N$ is an isometry on $X^N$. Define $D$ as in \eqref{C.9}. Then $(X^N/S_N,D)$ is a metric space and the topology $(D)$ induced by the metric $D$ on $X^N/S_N$ is the quotient topology $\tau_{S_N}$, $(X^N/S_N,(D))=(X^N/S_N,\tau_{S_N})$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Clearly $D$ assumes non-negative real values only and symmetry of $D$ follows immediately from \eqref{C.8}. If $[x]=[y]$ then there is a $\rho\in S_N$ such that $x=\rho(y)$. Hence $d(x,\rho(y))=0$ and thus $D([x],[y])=0$. Next, suppose that $D([x],[y])=0$. Then there exists a $\rho\in S_N$ such that $x=\rho(y)$, that is, $y$ is equivalent to $x$ and hence $[x]=[y]$. For the triangle inequality one notes that, given $z\in X^N$, \begin{align} D([x],[y])&={\min}_{\rho\in S_N}\{d(x,\rho(y))\} \no \\ &\leq {\min}_{\rho\in S_N}\{d(x,\sigma(z))+d(\sigma(z),\rho(y))\} \no \\ &=d(x,\sigma(z))+{\min}_{\rho\in S_N}\{d(\sigma(z),\rho(y))\} \no \\ &=d(x,\sigma(z))+D([z],[y]), \quad \sigma\in S_N. \lb{C.10} \end{align} In particular \eqref{C.10} holds for that $\sigma$ which yields the minimum of the right-hand side of \eqref{C.10} and hence $D$ is a metric on $X^N/S_N$. The metric $D$ induces a topology $\tilde\tau$ on $X^N/S_N$ and we denote the resulting topological space by $(X^N/S_N,\tilde\tau)$. Let $\nu:X^N\to X^N/S_N,\, x\mapsto [x]$ denote the canonical projection. We will next show that the map $\nu:(X^N,d)\to (X^N/S_N,\tilde\tau)$ is open and continuous. It is obviously surjective. By \cite[Theorem 6.5.1]{Wi83} we then conclude that $\tau_{S_N}=\tilde\tau$. To prove that $\nu$ is continuous, let $U$ be an open set in $(X^N/S_N,\tilde\tau)$. We want to show that $\nu^{-1}(U)$ is open. Let $x$ be a point in $\nu^{-1}(U)$. Then $[x]$ is in $U$ and there is an $\varepsilon>0$ such that $B([x],\varepsilon)$, the ball of radius $\varepsilon$ centered at $[x]$, is a subset of $U$. Pick $y\in B(x,\varepsilon)\subset X^N$. We note that \begin{equation} D([x],[y])\leq d(x,y)<\varepsilon, \lb{C.11} \end{equation} that is, $[y]\in B([x],\varepsilon)\subset U$ and thus $y\in\nu^{-1}(U)$. Since $y$ is arbitrary, one infers $B(x,\varepsilon)\subset\nu^{-1}(U)$. To prove that $\nu$ is open, let $V$ be an open set in $X^N$. We want to show that $\nu(V)$ is open. Let $[x]$ be a point in $\nu(V)$. Then there is a point in the equivalence class of $x$ which is in $V$. Without loss of generality we may assume that $x$ is that point. In addition, there is an $\varepsilon>0$ such that $B(x,\varepsilon)$ is a subset of $V$. Pick $[y]\in B([x],\varepsilon)\subset (X^N/S_N,\tilde\tau)$. Note that this is equivalent to $D([x],[y])<\varepsilon$, which in turn means that there is a $\rho$ in $S_N$ such that $d(x,\rho(y))<\varepsilon$. Hence $\rho(y)\in B(x,\varepsilon)\subset V$ and thus $[y]=\nu(y)=\nu(\rho(y))\in\nu(V)$. Since $[y]$ is arbitrary, one concludes $B([x],\varepsilon)\subset\nu(V)$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{remark} \lb{rC.2} The results of this appendix apply in the three cases $X=\bbC$, $X=\bbC/\Lambda_\omega$, $X=\bbC/\Lambda_{2\omega_1,2\omega_3}$ considered in Section \ref{s3}. For brevity we just take a quick look at the simply periodic case $X=\bbC/\Lambda_\omega$: Consider the equivalence classes $[x]=\{x+m\omega \,|\, x\in\bbC, \, m\in\bbZ\}\in \bbC/\Lambda_\omega$, then the quotient topology on $\bbC/\Lambda_\omega$ is seen to be generated by the following metric $\delta\colon \bbC/\Lambda_\omega\times \bbC/\Lambda_\omega\to [0,\infty)$ on $\bbC/\Lambda_\omega$, \begin{equation} \delta([x],[y])=\inf_{m,n\in\bbZ} |x+m\omega - (y+n\omega)|, \quad [x], [y]\in \bbC/\Lambda_\omega. \end{equation} Analogous considerations apply to the elliptic case $X=\bbC/\Lambda_{2\omega_1,2\omega_3}$. \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The proof of Theorem \ref{t2.13}} \label{sD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\theequation}{D.\arabic{equation}} \setcounter{theorem}{0} \setcounter{equation}{0} In this appendix we provide the proof of Theorem \ref{t2.13}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \lb{tD.1} Assume $M\in\bbN$, $s_\ell\in\bbN$, $1\leq\ell\leq M$, $q_0\in\bbC$, and suppose $\zeta_\ell\in\bbC$, $\ell=1,\dots,M$, are pairwise distinct. Consider \begin{equation} q(z)=q_0-\sum_{\ell=1}^M s_\ell(s_\ell+1)\cP(z-\zeta_\ell), \lb{2.66a} \end{equation} and suppose the DG locus conditions \begin{equation} \sum_{\substack{\ell^\prime=1\\ \ell^\prime\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP^{(2k-1)}(\zeta_{\ell} -\zeta_{\ell^\prime})=0 \text{ for $1\leq k\leq s_{\ell}$ and $1\leq \ell\leq M$} \lb{2.67a} \end{equation} are satisfied. Then \begin{equation} f_0=1, \quad f_j(z)=d_j+\sum_{\ell=1}^M \sum_{k=1}^{\min(j,s_\ell)} a_{j,\ell,k} \cP(z-\zeta_\ell)^k, \quad j\in\bbN \lb{2.68a} \end{equation} for some $\{a_{j,\ell,k}\}_{1\leq k \leq \min(j,s_\ell), 1\leq \ell\leq M} \subset\bbC$ and $d_j\in\bbC$, $j\in\bbN$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} By equation \eqref{WP} we can treat the rational, simply periodic, and elliptic cases simultaneously. \\ \noindent $(1)$ $j=1$: Then \begin{align} f_1(z) = c_1 +\frac{1}{2}\, q(z) =c_1 + \frac{1}{2}\, q_0 - \sum_{\ell=1}^M \frac{1}{2} s_\ell(s_\ell+1)\, \cP(z-\zeta_\ell) \label{} \end{align} is of the form \eqref{2.68a} with $d_1=c_1 + \frac{1}{2}q_0$ and $a_{1,\ell,1}= -\frac{1}{2} s_\ell(s_\ell+1) $. \\ \noindent $(2)$ We assume \eqref{2.68a} holds for some $j\in\bbN$, that is, \begin{equation} f_j(z) =d_j+\sum_{\ell=1}^M \sum_{k=1}^{\min(j,s_\ell)} a_{j,\ell,k} \cP(z-\zeta_\ell)^k, \end{equation} or equivalently, \begin{align} f_j'(z)= \sum_{\ell=1}^M \sum_{k=1}^{\min(j,s_\ell)} a_{j,\ell,k}\, k \, \cP(z-\zeta_\ell)^{k-1}\, \cP'(z-\zeta_\ell). \label{6aa} \end{align} We now start the proof of \eqref{2.68a} for $j+1$: First, we recall the recurrence relation \eqref{A.1}, \begin{align} f_{j+1}'(z) &=\frac{1}{4} f_j(z)'''+ q(z) f_j'(z)+\frac{1}{2} q'(z) f_j(z) \label{np1} \\ & =\frac{1}{4} f_j(z)'''+ (q(z) f_j (z))'- \frac{1}{2} q'(z) f_j(z) \lb{10A} \\ &=\frac{1}{4} f_j(z)'''+ \frac{1}{2} q(z) f_j'(z)+\frac{1}{2} (q(z) f_j(z))'. \label{10a} \end{align} Since $q$ is elliptic, so are $f_k$ for all $k\in\bbN$ by the recursion relation \eqref{A.1} as the latter implies that each $f_k$ is a differential polynomial in $q$. Equations \eqref{10A} and \eqref{10a} then imply that as $z\to \zeta_\ell$, none of the terms in \eqref{np1} can have a constant term or a term of the form $(z-\zeta_{\ell})^{-1}$ in the Laurent expansion around $\zeta_\ell$. This fact will be used repeatedly in the remainder of this proof. Next we separately investigate each of the three terms on the right-hand side of \eqref{np1}. For brevity we denote $\min(j,s_\ell)$ by $m$ in the following. \\ $(i)$ Considering $f'''_j$ one computes \begin{equation} f'''_j(z) = \sum_{\ell=1}^M \sum_{k=1}^{m} a_{j,\ell,k} \frac{d ^3}{d z^3} \cP(z-\zeta_\ell)^k \end{equation} and \begin{align} \frac{d ^3}{d z^3} \cP(z-\zeta_\ell)^k = & \big[k (2 k+1) (2k+2) \cP(z-\zeta_\ell)^{k} \cP' (z-\zeta_\ell) \no \\ & -g_2 k (k- \frac{1}{2}) (k-1)\cP(z-\zeta_\ell)^{k-2} \cP' (z-\zeta_\ell) \no \\ & -g_3 k (k-1) (k-2)\cP(z-\zeta_\ell)^{k-3} \cP'(z-\zeta_\ell)\big], \lb{2.76} \end{align} using \eqref{B.3} and \eqref{B.4}. (For $k=2$ the term $\cP(z-\zeta_\ell)^{k-3}$ does not occur in \eqref{2.76}, for $k=1$ the terms $\cP(z-\zeta_\ell)^{k-3}$ and $\cP(z-\zeta_\ell)^{k-2}$ do not occur in \eqref{2.76}.) Thus, $\tfrac{1}{4}f'''_j$ is of the expected form \eqref{2.68a}, \begin{align} \frac{1}{4}f'''_j(z)=\frac{1}{4}\sum_{\ell=1}^M \sum_{k=1}^{m+1} \tilde a_{j+1,\ell,k} \cP(z-\zeta_\ell)^{k-1} \cP'(z-\zeta_\ell). \label{6a} \end{align} Moreover, the highest-order pole of $\tfrac{1}{4} f_j'''$ at $\zeta_\ell$ reads \begin{align} \frac{1}{4} \, m ( 4 m +2) (m+1)\, \frac{ (-2) a_{j,\ell,m}}{(z-\zeta_\ell)^{2 m +3}} . \label{11a} \end{align} $(ii)$ Considering $qf_j'$ one obtains \begin{align} &q(z) f_j'(z) \no \\ & = \bigg( q_0-\sum_{\ell=1}^M s_\ell(s_\ell+1)\cP(z-\zeta_\ell) \bigg) \bigg(\sum_{\ell=1}^M \sum_{k=1}^{m} a_{j,\ell,k} k \cP(z-\zeta_\ell)^{k-1} \cP'(z-\zeta_\ell) \bigg) \no \\ & = q_0 \sum_{\ell=1}^M \sum_{k=1}^{m} a_{j,\ell,k} k \cP(z-\zeta_\ell)^{k-1} \cP'(z-\zeta_\ell) \no \\ & \quad - \sum_{\ell=1}^M \sum_{k=1}^{m} s_\ell(s_\ell+1) a_{j,\ell,k} k \cP(z-\zeta_\ell)^{k} \cP'(z-\zeta_\ell) \label{2.79} \\ &\quad -\sum_{\ell=1}^M \bigg[\bigg( \sum_{k=1}^{m} a_{j,\ell,k} k \cP(z-\zeta_\ell)^{k-1} \cP'(z-\zeta_\ell) \bigg) \bigg( \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1)\cP(z-\zeta_{\ell'}) \bigg) \bigg]. \no \end{align} The first two terms on the right-hand side of \eqref{2.79} are already of the expected form \eqref{2.68a}. Next, we investigate the third term in \eqref{2.79}. Let \begin{equation} g_{1, \ell}(z) = \sum_{k=1}^{m} a_{j,\ell,k}\, k \, \cP(z-\zeta_\ell)^{k-1}\, \cP'(z-\zeta_\ell), \;\; h_{1, \ell}(z) = \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP(z-\zeta_{\ell'}). \end{equation} Then the third term in \eqref{2.79} equals $-\sum_{\ell=1}^M g_{1, \ell} h_{1, \ell}$. Next we recall (cf.\ \eqref{B.17}) that any elliptic function $f$ can be written in the form \begin{align} f(z) & = A_0 +\sum^M_{\ell=1} \sum^{ s}_{k=1}(-1)^{k-1} \frac{A_{\ell,k}}{(k-1)!} \zeta^{(k-1)}(z-\zeta_\ell), \ \ s \in \bbN \end{align} for appropriate $M,s\in\bbN$, $A_0, A_{\ell,k}\in\bbC$, $1\leq \ell\leq M$, $1\leq k\leq s$. Here $\zeta(\cdot)=\zeta(\cdot\,|g_2,g_3)$ abbreviates the Weierstrass $\zeta$-function in the elliptic case associated with the invariants $g_2$ and $g_3$ (see \cite[Sect.\ 18.1]{AS72}) and \begin{equation} \zeta(z)=\begin{cases} \zeta(z|0,0)=1/z & \text{in the rational case,} \\ \zeta\big(z|[2\pi^2/\omega^2]^2/3,[2\pi^2/\omega^2]^3/27\big) \\ =[\pi^2z/(3\omega^2)]+ (\pi/\omega)\cot(\pi z/\omega) & \text{in the simply periodic case} \end{cases} \end{equation} (cf.\ \cite[p.\ 652]{AS72}). Since $g_{1,\ell}$ and $ h_{1,\ell}$ are elliptic, we thus have \begin{align} \sum_{\ell=1}^M g_{1, \ell}(z) & = G_{1,0}+\sum^M_{\ell=1} \sum^{2 m +1}_{k=1}(-1)^{k-1} \frac{G_{1,\ell,k}}{(k-1)!} \zeta^{(k-1)}(z-\zeta_\ell), \\ \sum_{\ell=1}^M g_{1, \ell}(z) h_{1, \ell}(z) & = B_0 + \sum^M_{\ell=1} \sum^{2 m+1}_{k=1}(-1)^{k-1} \frac{ B_{\ell,k}}{(k-1)!} \zeta^{(k-1)}(z-\zeta_\ell) . \end{align} To calculate $B_{\ell,k}$ we expand $g_{1, \ell}$ and $h_{1,\ell}$ at $z=\zeta_{\ell}$ using \eqref{2.67a}. First we recall (cf.\ \eqref{P} and \cite[Sect.\ 18.5]{AS72}) \begin{align} \cP (z) = \frac{1}{z^2} + \sum_{r=2}^{\infty} c_{r} z^{ 2 r-2}. \label{} \end{align} Thus, $\cP^k$ admits the Laurent expansion \begin{align} (\cP (z))^k = \frac{1}{z^{2k}} + \frac{1}{z^{2k-4}}\sum_{s=0}^{\infty} d_{s} z^{ 2 s} \label{pk} \end{align} with only even orders of $z$ occurring in the expansion of $\cP^k$ since $\cP$ is an even function. For the derivative of $\cP^k$ one computes \begin{align} \frac{d}{dz}(\cP (z))^k = (-2k) \frac{1}{z^{2k+1}} + (-2k+4)\frac{1}{z^{2k-3}}\sum_{s=0}^{\infty} d_{s} z^{ 2 s} +\frac{1}{z^{2k-4}}\sum_{s=1}^{\infty} d_{s} 2 s z^{ 2 s-1} \label{} \no \end{align} and hence only odd orders of $z$ occur in the expansion of $\frac{d}{dz} (\cP (z) )^k$. Thus, one concludes that only odd orders of $z$ occur in the expansion of $g_{1,\ell}$ at $z=\zeta_\ell$. On the other hand any elliptic function $f$, whose residue at $\zeta_\ell$ vanishes and whose principal part of its Laurent expansion at $z=\zeta_\ell$ contains only odd terms, can be written in the form \begin{equation} f(z)=\sum_{k=1}^{n_\ell} \tilde d_k \frac{d}{dz} (\cP (z-\zeta_\ell) )^k + O(1) \lb{2.87} \end{equation} for $z$ in a neighborhood of $\zeta_\ell$. Here $n_\ell\in\bbN$ depends on the order of the pole of $f$ at $\zeta_\ell$. By \eqref{2.67a} the odd powers of $(z-\zeta_\ell)^j$ in the expansion of $h_{1, \ell}(z)$ at $z=\zeta_\ell$ up to order $(2 s_\ell -1)$ are zero and hence \begin{align} &h_{1, \ell}(z) = h_{1,\ell,0} + \sum_{k=1}^{\infty} \frac{h_{1,\ell}^{(k)}(\zeta_{\ell}) }{k!} (z-\zeta_{\ell})^k = \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP(\zeta_{\ell}-\zeta_{\ell'}) \no\\ & \quad + \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP'(\zeta_{\ell}-\zeta_{\ell'}) (z-\zeta_{\ell}) + \frac{1}{2} \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP''(\zeta_{\ell}-\zeta_{\ell'}) (z-\zeta_{\ell})^2 \no \\ & \quad + \frac{1}{6} \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP^{(3)}(\zeta_{\ell}-\zeta_{\ell'}) (z-\zeta_{\ell})^3 + \ldots\no\\ &= h_{1,\ell,0} + h_{1,\ell,2} (z-\zeta_{\ell})^{ 2 } + h_{1,\ell,4} (z-\zeta_{\ell})^{ 4 } + \ldots + h_{1,\ell,2 s_\ell} (z-\zeta_{\ell})^{ 2 s_\ell } \no \\ & \quad + O\big((z-\zeta_{\ell})^{2 s_\ell+1 }\big). \end{align} Expanding $ g_{1,\ell} h_{1,\ell}$ at $z= \zeta_\ell$ then yields \begin{align} g_{1,\ell}(z) h_{1,\ell}(z) &= b_{ -2 m-1} \frac{1}{(z-\zeta_{\ell})^{ 2 m+1}}+ b_{ -2 m+1} \frac{1}{(z-\zeta_{\ell})^{ 2 m-1}}+ \ldots \no\\ &\quad + b_{2 s_\ell- 2m -1} (z-\zeta_{\ell})^{2 s_\ell -2 m-1} +O \big((z-\zeta_{\ell})^{2 s_\ell -2 m }\big) . \label{14a} \end{align} By \eqref{2.87} we can write (\ref{14a}) as \begin{align} g_{1, \ell}(z) h_{1, \ell}(z)&= \sum_{k=1}^{m} e_{j,\ell,k} k \cP(z-\zeta_\ell)^{k-1}\, \cP'(z-\zeta_\ell) + \frac{c_{1,\ell}}{z-\zeta_\ell} + c_{0,\ell} \no \\ & \quad + \Oh\big((z-\zeta_{\ell})^{ 1 }\big). \label{2.90} \end{align} Since no terms of the form $(z-\zeta_\ell)^{-1}$ and no constant term can occur in $\sum_{\ell=1}^M g_{1, \ell} h_{1, \ell}$ by the comment following \eqref{10a}, the coefficients $c_{1,\ell}$ of $(z-\zeta_{\ell})^{-1}$, $\ell=1,\dots,M$, in \eqref{2.90}, as well as the constant term $\sum_{\ell=1}^{M} c_{0,\ell}$, must be zero and we arrive at the expected form \eqref{2.68a}, \begin{align} \sum_{\ell=1}^M g_{1, \ell}(z) h_{1, \ell}(z)= \sum_{\ell=1}^M \sum_{k=1}^{m} e_{j,\ell,k}\, k \, \cP(z-\zeta_\ell)^{k-1}\, \cP'(z-\zeta_\ell) \label{} \end{align} of the third term in \eqref{2.79}. The highest-order pole of $q f_j'$ at $\zeta_\ell$ reads \begin{align} - s_\ell(s_\ell+1)m\frac{(-2)\, a_{j,\ell,m}}{(z-\zeta_\ell)^{2m+3}}. \label{12a} \end{align} $(iii)$ Considering $\frac{1}{2}q'f_j$ one obtains \begin{align} \frac{1}{2} q'(z) f_j(z) &=-\frac{1}{2} \, \bigg( \sum_{\ell=1}^M s_\ell(s_\ell+1)\cP '(z-\zeta_\ell) \bigg) \bigg(d_j + \sum_{\ell=1}^M \sum_{k=1}^{m} a_{j,\ell,k}\, \cP(z-\zeta_\ell)^{k} \bigg) \no \\ & = -\frac{1}{2} \, d_j \sum_{\ell=1}^M s_\ell(s_\ell+1) \cP '(z-\zeta_\ell) \no \\ & \quad - \frac{1}{2} \, \sum_{\ell=1}^M \sum_{k=1}^{m} s_\ell(s_\ell+1) a_{j,\ell,k}\, \cP(z-\zeta_\ell)^{k}\, \cP'(z-\zeta_\ell) \lb{2.93} \\ & \quad - \frac{1}{2} \, \sum_{\ell=1}^M \bigg[\bigg( \sum_{k=1}^{m} a_{j,\ell,k}\, \cP(z-\zeta_\ell)^{k} \bigg) \bigg( \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP'(z-\zeta_{\ell'}) \bigg)\bigg]. \no \end{align} The first two terms in \eqref{2.93} are already of the expected form \eqref{2.68a}. Next we investigate the third term in \eqref{2.93}. Let \begin{align} g_{2,\ell}(z) = \sum_{k=1}^{m} a_{j,\ell,k}\, \cP(z-\zeta_\ell)^{k}, \quad h_{2,\ell}(z) = \bigg( \sum_{\substack{\ell'=1\\ \ell'\neq \ell}}^M s_{\ell^\prime}(s_{\ell^\prime}+1) \cP'(z-\zeta_{\ell'}) \bigg). \label{} \end{align} Then the third term in \eqref{2.93} equals $-\f{1}{2}\sum_{\ell=1}^M g_{2,\ell} h_{2,\ell}$. Since $g_{2,\ell}$ and $ h_{2,\ell}$ are elliptic, one has \begin{align} \sum^M_{\ell=1} g_{2,\ell}(z) & = G_{2,0}+\sum^M_{\ell=1} \sum^{2 m }_{k=1}(-1)^{k-1} \frac{G_{2,\ell,k}}{(k-1)!} \zeta^{(k-1)}(z-\zeta_\ell), \\ \sum^M_{\ell=1} g_{2,\ell} (z) h_{2,\ell} (z) & = D_0 +\sum^M_{\ell=1} \sum^{ 2 m-1}_{k=1}(-1)^{k-1} \frac{ D_{\ell,k}}{(k-1)!} \zeta^{(k-1)}(z-\zeta_\ell) . \end{align} From \eqref{pk} one concludes that only even orders in $z$ can occur in the expansion of $g_{2,\ell}$ at $z=\zeta_\ell$. Next we expand $h_{2,\ell}$ at $z=\zeta_{\ell}$. By \eqref{2.67a}, the even powers of $(z-\zeta_\ell)^k$ in the expansion of $h_{2, \ell}$ at $z=\zeta_\ell$ up to order $(2 s_\ell -2)$ are zero and hence, \begin{align} h_{2,\ell}(z)& = \sum_{k=0}^{\infty} \frac{h_{2,\ell}^{(k) }(\zeta_{\ell})}{k!} (z-\zeta_{\ell})^k \no\\ &= h_{1,\ell,1} (z-\zeta_{\ell}) + h_{1,\ell,3} (z-\zeta_{\ell})^{3} + \ldots + h_{1,\ell,2 s_\ell-1}(z-\zeta_{\ell})^{ 2 s_\ell -1} \no \\ & \quad + \Oh \big((z-\zeta_{\ell})^{2 s_\ell}\big). \end{align} Expanding $ g_{2,\ell} \, h_{2,\ell}$ at $z= \zeta_\ell$ then yields \begin{align} g_{2,\ell}(z) h_{2,\ell}(z) &= \tilde b_{ -2 m+1} \frac{1}{ (z-\zeta_{\ell})^{ 2 m-1}}+ \tilde b_{ -2 m+3} \frac{1}{(z-\zeta_{\ell})^{ 2 m-3}}+ \ldots \no\\ & \quad + \tilde b_{2 s_\ell- 2m -1} (z-\zeta_{\ell})^{2 s_\ell -2 m-1} +O \big((z-\zeta_{\ell})^{2 s_\ell -2 m }\big). \label{20a} \end{align} By \eqref{2.87} we can write \eqref{20a} as \begin{align} g_{2, \ell}(z) h_{2, \ell}(z)= \sum_{k=1}^{m-1} \tilde e_{j,\ell,k}\, k \, \cP(z-\zeta_\ell)^{k-1}\, \cP'(z-\zeta_\ell) + \frac{\tilde c_{1,\ell}}{z-\zeta_\ell} +\tilde c_{0,\ell} + O \big((z-\zeta_{\ell})^{ 1 }\big). \label{2.97} \end{align} Since no terms of the form $(z-\zeta_\ell)^{-1}$ and no constant term can occur in $\sum_{\ell=1}^M g_{2, \ell} h_{2, \ell}$ by the comment following \eqref{10a}, the coefficients $\ti c_{1,\ell}$ of $(z-\zeta_{\ell})^{-1}$, $1\leq \ell\leq M$, in \eqref{2.97}, as well as the constant term $\sum_{\ell=1}^{M} \ti c_{0,\ell}$, must vanish and we arrive at the expected form \eqref{2.68a}, \begin{align} \sum_{\ell=1}^M g_{2, \ell}(z) h_{2, \ell}(z)= \sum_{\ell=1}^M \sum_{k=1}^{m-1} \tilde e_{j,\ell,k} k \cP(z-\zeta_\ell)^{k-1} \cP'(z-\zeta_\ell) \no . \label{} \end{align} The highest-order pole of $\frac{1}{2}q'f_n$ at $\zeta_\ell$ reads \begin{align} - \frac{1}{2}\, s_\ell(s_\ell+1)\frac{ (-2)\, a_{j,\ell,m} }{(z-\zeta_\ell)^{2 m +3}}. \label{15a} \end{align} Summing up (\ref{11a}), (\ref{12a}), and (\ref{15a}) yields \begin{align} \bigg[ \frac{1}{4} \, m ( 4 m +2) (m+1) - s_\ell(s_\ell+1) m - \frac{1}{2} s_\ell(s_\ell+1) \bigg] \frac{-2 a_{j,\ell,m}}{(z-\zeta_\ell)^{2 m +3}}. \label{18a} \end{align} This term becomes zero as soon as $m= s_\ell$. 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