Degree two of ℘ and its four branch points

(The torus form is a continuous proper map.) By [F3] and [F4], $\mathrm{\wp }$ is a $\mathrm{\Lambda }$-elliptic function and $\mathrm{\wp }=\bar{\mathrm{\wp }}\circ \pi$ with $\bar{\mathrm{\wp }}:T_{\mathrm{\Lambda }}\to \widehat{\mathbb{C}}$ its torus form, a meromorphic function on the Riemann surface $T_{\mathrm{\Lambda }}$. By [F5] the meromorphic function $\bar{\mathrm{\wp }}$ is a holomorphic, hence continuous, map. Let $K\subseteq \widehat{\mathbb{C}}$ be compact; since $\widehat{\mathbb{C}}$ is Hausdorff by [F6], $K$ is closed in $\widehat{\mathbb{C}}$ by [F12], so ${\bar{\mathrm{\wp }}}^{-1}(K)$ is closed in $T_{\mathrm{\Lambda }}$ by continuity; and $T_{\mathrm{\Lambda }}$ is compact by [F2], so ${\bar{\mathrm{\wp }}}^{-1}(K)$ is compact by [F12]. Hence $\bar{\mathrm{\wp }}$ is proper.

($\bar{\mathrm{\wp }}$ is nonconstant, and $h_j\mathrm{\notin }\mathrm{\Lambda }$.) The point $0$ is a lattice point, so $\mathrm{\wp }$ has a pole at $0$ by [F3] and $\bar{\mathrm{\wp }}([0])=\mathrm{\infty }$; the points $h_1,h_2,h_3$ are not lattice points: if $h_1=m{\omega }_1+n{\omega }_2$ with $m,n\in \mathbb{Z}$, then $(m-\frac{1}{2}){\omega }_1+n{\omega }_2=0$ with $m-\frac{1}{2}\mathrm{\ne }0$, contradicting [F1], and the same computation with $(m,n-\frac{1}{2})$ and $(m-\frac{1}{2},n-\frac{1}{2})$ handles $h_2,h_3$. Since the poles of $\mathrm{\wp }$ are exactly the lattice points by [F3], the value $\mathrm{\wp }(h_1)$ is finite, so $\bar{\mathrm{\wp }}([h_1])=\mathrm{\wp }(h_1)\in \mathbb{C}$ differs from $\bar{\mathrm{\wp }}([0])=\mathrm{\infty }$: the map $\bar{\mathrm{\wp }}$ is nonconstant.

(The chart expression of $\bar{\mathrm{\wp }}$ at $[0]$.) By [F2] the covering map $\pi$ is injective on some open ball $U$ around $0$, and $\chi :=(\pi {\mathrm{\mid }}_U{)}^{-1}:\pi (U)\to U$ is one of the charts of the atlas of $T_{\mathrm{\Lambda }}$, with $\chi ([w])=w$ for $[w]\in \pi (U)$. Take the chart ${\varphi }_{\mathrm{\infty }}$ of [F6] at $\bar{\mathrm{\wp }}([0])=\mathrm{\infty }$; the chart expression is $F(z)={\varphi }_{\mathrm{\infty }}(\bar{\mathrm{\wp }}({\chi }^{-1}(z)))={\varphi }_{\mathrm{\infty }}(\bar{\mathrm{\wp }}([z]))={\varphi }_{\mathrm{\infty }}(\mathrm{\wp }(z))(z\in U),$ where $\bar{\mathrm{\wp }}([z])=\mathrm{\wp }(z)$ uses [F4]. By the principal-part clause of [F3] there is a holomorphic $Q$ on a disc around $0$ with $\mathrm{\wp }(z)=z^{-2}+Q(z)$ for $z\mathrm{\ne }0$ there; then $z^2\mathrm{\wp }(z)=1+z^{2}Q(z)$ tends to $1$ as $z\to 0$, so $u(z):=(1+z^{2}Q(z){)}^{-1}$ is holomorphic on a neighbourhood of $0$ by the reciprocal rule of [F11], with $u(0)=1$, and $1\mathrm{/}\mathrm{\wp }(z)=z^{2}u(z)$ for $0<\mathrm{\mid }z\mathrm{\mid }$ small. Since also ${\varphi }_{\mathrm{\infty }}(\mathrm{\infty })=0=0^{2}u(0)$ and ${\varphi }_{\mathrm{\infty }}(\mathrm{\wp }(z))=1\mathrm{/}\mathrm{\wp }(z)$ for $0<\mathrm{\mid }z\mathrm{\mid }$ small, the chart expression satisfies $F(z)=z^{2}u(z)$ on a neighbourhood of $0$.

(The classes of order two.) If $z\in \mathbb{C}$ has $2z\in \mathrm{\Lambda }$, then $2z=m{\omega }_1+n{\omega }_2$ with $m,n\in \mathbb{Z}$, so $z=\frac{m}{2}{\omega }_1+\frac{n}{2}{\omega }_2$; replacing $m$ by $m+2$ and $n$ by $n+2$ changes $z$ by elements of $\mathrm{\Lambda }$, so $[z]$ is one of $[0]$, $[h_1]$, $[h_2]$, $[h_3]$. These four classes are pairwise distinct and $h_1,h_2,h_3\mathrm{\notin }\mathrm{\Lambda }$: the differences $h_1$, $h_2$, $h_3$, $h_1-h_2=-\frac{1}{2}({\omega }_2-{\omega }_1)$, $h_1-h_3=-\frac{1}{2}{\omega }_2$ and $h_2-h_3=-\frac{1}{2}{\omega }_1$ are all non-lattice, because an equation such as $h_1-h_2=m{\omega }_1+n{\omega }_2$ reads $(m-\frac{1}{2}){\omega }_1+(n+\frac{1}{2}){\omega }_2=0$, a nontrivial real-linear combination vanishing, contrary to [F1]; the other cases are identical with the non-integer coefficients $m-\frac{1}{2}$, $n-\frac{1}{2}$, $m+\frac{1}{2}$ in one of the two slots. Hence the only classes $x\in T_{\mathrm{\Lambda }}$ with $x=-x$ are $[0],[h_1],[h_2],[h_3]$, and the last three are distinct nonzero classes.

(Ramification index versus the derivative at finite points.) Let $z\in \mathbb{C}\setminus \mathrm{\Lambda }$. The chart expression of $\bar{\mathrm{\wp }}$ in a source chart inverse to $\pi$ near $z$ and the centered target chart ${\psi }_{\mathrm{\wp }(z)}(\xi ):=\xi -\mathrm{\wp }(z)$ at the finite value $\mathrm{\wp }(z)$ is $w\mapsto \mathrm{\wp }(w)-\mathrm{\wp }(z)$ for $w$ near $z$, because $\bar{\mathrm{\wp }}([w])=\mathrm{\wp }(w)$ by [F4]; hence by [F8], $e_{[z]}(\bar{\mathrm{\wp }})={\mathrm{ord}}_z(\mathrm{\wp }-\mathrm{\wp }(z))=:m$, a positive finite integer by [F8] and [F9]. If $m=1$, then by [F10] there is a holomorphic $g$ near $z$ with $g(z)\mathrm{\ne }0$ and $\mathrm{\wp }(w)-\mathrm{\wp }(z)=(w-z)g(w)$, so the product rule and the derivative of the identity in [F11] give ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=g(z)+0\cdot g^{\mathrm{\prime }}(z)=g(z)\mathrm{\ne }0$; this proves ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=0\Rightarrow m\ge 2$. Conversely, if $m\ge 2$, then by [F10] $\mathrm{\wp }(w)-\mathrm{\wp }(z)=(w-z{)}^{m}g(w)$ with $g(z)\mathrm{\ne }0$, and the product rule of [F11] gives ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=m\cdot 0^{m-1}g(z)+0^m{g}^{\mathrm{\prime }}(z)=0$, so ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0\Rightarrow m=1$. Thus for $z\in \mathbb{C}\setminus \mathrm{\Lambda }$, $e_{[z]}(\bar{\mathrm{\wp }})=1$ iff ${\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$, and $e_{[z]}(\bar{\mathrm{\wp }})\ge 2$ iff ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=0$.

(The equality criterion: $w\equiv \pm z$ implies $\mathrm{\wp }(w)=\mathrm{\wp }(z)$.) Suppose $w=z+\lambda$ or $w=-z+\lambda$ with $\lambda \in \mathrm{\Lambda }$. By the periodicity and evenness clauses of [F3], $\mathrm{\wp }(w)=\mathrm{\wp }(z+\lambda )=\mathrm{\wp }(z)$ in the first case and $\mathrm{\wp }(w)=\mathrm{\wp }(-z+\lambda )=\mathrm{\wp }(-z)=\mathrm{\wp }(z)$ in the second, both as values in $\widehat{\mathbb{C}}$ with poles matched.

($\widehat{\mathbb{C}}$ is a connected Riemann surface.) By F6 the two standard charts cover $\widehat{\mathbb{C}}$ and have holomorphic transition maps on their overlap, so they form a holomorphic atlas; $\widehat{\mathbb{C}}$ is nonempty; it is compact Hausdorff and second countable by F6; and it is connected because $\mathrm{\Sigma }$ is a homeomorphism onto the connected space $S^2$, so that ${\mathrm{\Sigma }}^{-1}:S^2\to \widehat{\mathbb{C}}$ is a continuous surjection and the continuous image of a connected space is connected. Therefore $\widehat{\mathbb{C}}$ satisfies the Riemann-surface axioms of F6.

($e_{[0]}(\bar{\mathrm{\wp }})=2$ and $\mathrm{deg}\bar{\mathrm{\wp }}=2$.) Here $F(0)=0$ and $F=z^{2}u$ with $u(0)=1\mathrm{\ne }0$, so the order of $F$ at $0$ is exactly $2$ by [F10]; hence by [F8] and [F9], $e_{[0]}(\bar{\mathrm{\wp }})={\mathrm{deg}}_{0}F={\mathrm{ord}}_0(F-F(0))={\mathrm{ord}}_{0}F=2$. The fibre of $\bar{\mathrm{\wp }}$ over $\mathrm{\infty }$ is the single class $[0]$: indeed $\bar{\mathrm{\wp }}([z])=\mathrm{\infty }$ iff $\mathrm{\wp }(z)=\mathrm{\infty }$ iff $z\in \mathrm{\Lambda }$ by the pole clause of [F3], iff $[z]=[0]$. By step 1.1 the map $\bar{\mathrm{\wp }}$ is proper and nonconstant with connected Riemann surfaces as source and target by [F2] and step 1.7, so [F7] applies and the degree $d={\sum }_{x\in {\bar{\mathrm{\wp }}}^{-1}(\mathrm{\infty })}{e}_x(\bar{\mathrm{\wp }})=e_{[0]}(\bar{\mathrm{\wp }})=2$ is independent of the value: ${\sum }_{x\in {\bar{\mathrm{\wp }}}^{-1}(a)}{e}_x(\bar{\mathrm{\wp }})=2$ for every $a\in \widehat{\mathbb{C}}$.

(${\mathrm{\wp }}^{\mathrm{\prime }}$ vanishes at every nonzero half-period.) Let $h\in \mathbb{C}\setminus \mathrm{\Lambda }$ with $2h\in \mathrm{\Lambda }$. For every $z$ with $h\pm z\mathrm{\notin }\mathrm{\Lambda }$ the periodicity clause of [F3] with $\lambda =-2h\in \mathrm{\Lambda }$ gives $\mathrm{\wp }(h+z)=\mathrm{\wp }(h+z-2h)=\mathrm{\wp }(z-h)$, and the evenness clause of [F3] gives $\mathrm{\wp }(z-h)=\mathrm{\wp }(h-z)$; hence $\mathrm{\wp }(h+z)=\mathrm{\wp }(h-z)$ on the open set where both sides are defined. Differentiating both sides at $z=0$ with the chain rule of [F11] (the two one-variable maps $z\mapsto h+z$ and $z\mapsto h-z$ have derivatives $1$ and $-1$) gives ${\mathrm{\wp }}^{\mathrm{\prime }}(h)=-{\mathrm{\wp }}^{\mathrm{\prime }}(h)$, so ${\mathrm{\wp }}^{\mathrm{\prime }}(h)=0$. In particular ${\mathrm{\wp }}^{\mathrm{\prime }}(h_j)=0$ for $j=1,2,3$.

(The three half-periods are critical points.) By steps 1.2 and 2.2, $h_j\mathrm{\notin }\mathrm{\Lambda }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(h_j)=0$; by step 1.5, $e_{[h_j]}(\bar{\mathrm{\wp }})\ge 2$. Hence each of the three distinct nonzero classes $[h_1],[h_2],[h_3]$ is a critical point of $\bar{\mathrm{\wp }}$ in the sense of [F8].

(Dichotomy for the fibres over finite values.) Let $a\in \mathbb{C}$ and $S:={\bar{\mathrm{\wp }}}^{-1}(a)\subseteq T_{\mathrm{\Lambda }}$. By [F7] the set $S$ is nonempty and finite and ${\sum }_{x\in S}{e}_x(\bar{\mathrm{\wp }})=2$ by step 2.1, each $e_x(\bar{\mathrm{\wp }})$ being a positive integer. If some $x\in S$ satisfies $x\mathrm{\ne }-x$, then for a representative $x=[z]$ one has $\mathrm{\wp }(-z)=\mathrm{\wp }(z)=a$ by the evenness clause of [F3], so $-x=[-z]\in S$ as well; the two distinct elements $x,-x$ of $S$ contribute at least $1+1=2$ to the sum, so necessarily $S=\{x,-x\}$ and $e_x(\bar{\mathrm{\wp }})=e_{-x}(\bar{\mathrm{\wp }})=1$. Otherwise every $x\in S$ satisfies $x=-x$, so $x\in \{[h_1],[h_2],[h_3]\}$ by step 1.4, since $[0]\mathrm{\notin }S$ as $\bar{\mathrm{\wp }}([0])=\mathrm{\infty }\mathrm{\ne }a$ by step 1.2; each $x\in S$ has $e_x(\bar{\mathrm{\wp }})\ge 2$ by step 3.1, and $S\mathrm{\ne }\mathrm{\varnothing }$ with ${\sum }_{x\in S}{e}_x(\bar{\mathrm{\wp }})=2$ forces $S=\{[h]\}$ for a single class $[h]$ with $h\in \{h_1,h_2,h_3\}$ and $e_{[h]}(\bar{\mathrm{\wp }})=2$.

(The three branch values are distinct and their fibres are single points.) The values $e_j=\mathrm{\wp }(h_j)$ are finite by step 1.2 and the pole clause of [F3]. For each $j$ the class $[h_j]$ lies in ${\bar{\mathrm{\wp }}}^{-1}(e_j)$ and has $e_{[h_j]}(\bar{\mathrm{\wp }})\ge 2$ by step 3.1, so the first alternative of step 4.1 is impossible for $a=e_j$ (it would give $e=1$ there); hence the second alternative holds and ${\bar{\mathrm{\wp }}}^{-1}(e_j)=\{[h_j]\},e_{[h_j]}(\bar{\mathrm{\wp }})=2.$ If $e_j=e_k$ for indices $j\mathrm{\ne }k$, then $[h_j]$ and $[h_k]$ are two distinct elements of the fibre ${\bar{\mathrm{\wp }}}^{-1}(e_j)$ by step 1.4, contradicting the displayed equality; hence $e_1,e_2,e_3$ are three distinct finite values, and the fibre over each is a single class.

(The equality criterion: conversely.) Suppose $\mathrm{\wp }(z)=\mathrm{\wp }(w)=:a$ in $\widehat{\mathbb{C}}$. If $a=\mathrm{\infty }$, then $z,w\in \mathrm{\Lambda }$ by the pole clause of [F3], so $[w]=[z]$ and $w\equiv z\equiv -z$ modulo $\mathrm{\Lambda }$. If $a\in \mathbb{C}$, then $[z],[w]\in S={\bar{\mathrm{\wp }}}^{-1}(a)$ and step 4.1 gives two alternatives: either $S=\{x,-x\}$ for a class $x$ with $x\mathrm{\ne }-x$, in which case $[z],[w]\in \{x,-x\}$ and $w\equiv \pm z$ modulo $\mathrm{\Lambda }$; or $S=\{[h]\}$ for a single class with $2h\in \mathrm{\Lambda }$, in which case $[z]=[w]=[h]=[-h]$ and again $w\equiv \pm z$ modulo $\mathrm{\Lambda }$.

(The critical locus of $\bar{\mathrm{\wp }}$.) A point $x\in T_{\mathrm{\Lambda }}$ is critical precisely when $e_x(\bar{\mathrm{\wp }})\ge 2$ by [F8]. For $x=[0]$ this holds with $e_{[0]}(\bar{\mathrm{\wp }})=2$ by step 2.1, and for $x=[h_j]$ it holds with $e_{[h_j]}(\bar{\mathrm{\wp }})=2$ by step 5.1. Conversely let $x=[z]$ be critical and $x\mathrm{\ne }[0]$; then $z\mathrm{\notin }\mathrm{\Lambda }$, so by step 1.5 the inequality $e_{[z]}(\bar{\mathrm{\wp }})\ge 2$ gives ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=0$. Apply step 4.1 to the finite value $a=\mathrm{\wp }(z)$: the first alternative would give $e_{[z]}(\bar{\mathrm{\wp }})=1$, contrary to $e_{[z]}(\bar{\mathrm{\wp }})\ge 2$, so the second alternative holds and $[z]=[h]$ with $h\in \{h_1,h_2,h_3\}$. Hence the critical points of $\bar{\mathrm{\wp }}$ are exactly $[0],[h_1],[h_2],[h_3]$, four pairwise distinct classes by step 1.4.

(The branch locus.) By [F8] the branch values of $\bar{\mathrm{\wp }}$ are the images of its critical points, so by step 6.1 they are $\bar{\mathrm{\wp }}([0])=\mathrm{\infty }$ (step 1.2) and $\mathrm{\wp }(h_j)=e_j$; by step 5.1 the three $e_j$ are distinct and differ from $\mathrm{\infty }$, so the branch locus is the four-element set $\{\mathrm{\infty },e_1,e_2,e_3\}$.

(The zeros of ${\mathrm{\wp }}^{\mathrm{\prime }}$.) Since ${\mathrm{\wp }}^{\mathrm{\prime }}$ is $\mathrm{\Lambda }$-periodic by [F3], ${\mathrm{\wp }}^{\mathrm{\prime }}(z+\lambda )={\mathrm{\wp }}^{\mathrm{\prime }}(z)$ for all $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ and $\lambda \in \mathrm{\Lambda }$, so the zero set of ${\mathrm{\wp }}^{\mathrm{\prime }}$ is $\mathrm{\Lambda }$-invariant. For $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ step 1.5 together with step 6.1 gives ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=0⟺e_{[z]}(\bar{\mathrm{\wp }})\ge 2⟺[z]\in \{[h_1],[h_2],[h_3]\}$. Hence the zeros of ${\mathrm{\wp }}^{\mathrm{\prime }}$ are exactly the $\mathrm{\Lambda }$-translates of $h_1,h_2,h_3$: each $h_j$ is a zero by step 2.2, every zero is $\mathrm{\Lambda }$-translates of some $h_j$ by the equivalence just displayed, and there are no other zeros modulo $\mathrm{\Lambda }$.

(Each zero of ${\mathrm{\wp }}^{\mathrm{\prime }}$ is simple.) Fix $j$ and put $G(w):=\mathrm{\wp }(w)-e_j$ near $w=h_j$. By step 5.1, ${\mathrm{ord}}_{h_j}(\mathrm{\wp }-e_j)=e_{[h_j]}(\bar{\mathrm{\wp }})=2$, the identification of order and index being that of step 1.5; so by [F10] there is a holomorphic $g$ near $h_j$ with $g(h_j)\mathrm{\ne }0$ and $\mathrm{\wp }(w)-e_j=(w-h_j{)}^{2}g(w)$. Differentiating with the product rule and linearity of [F11] gives ${\mathrm{\wp }}^{\mathrm{\prime }}(w)=2(w-h_j)g(w)+(w-h_j{)}^2{g}^{\mathrm{\prime }}(w)=(w-h_j)(2g(w)+(w-h_j)g^{\mathrm{\prime }}(w))$, and the second factor at $w=h_j$ equals $2g(h_j)\mathrm{\ne }0$; hence ${\mathrm{ord}}_{h_j}({\mathrm{\wp }}^{\mathrm{\prime }})=1$, a simple zero. Since ${\mathrm{\wp }}^{\mathrm{\prime }}$ is $\mathrm{\Lambda }$-periodic, for $\lambda \in \mathrm{\Lambda }$ one has ${\mathrm{\wp }}^{\mathrm{\prime }}(h_j+\lambda +u)={\mathrm{\wp }}^{\mathrm{\prime }}(h_j+u)=u\cdot (2g(h_j+u)+u{g}^{\mathrm{\prime }}(h_j+u))$ for $u$ near $0$, so the zero at $h_j+\lambda$ has order one as well. Thus every zero of ${\mathrm{\wp }}^{\mathrm{\prime }}$ is simple, and by step 7.2 there is exactly one such zero at each nonzero half-period modulo $\mathrm{\Lambda }$.

Collecting the claims: $\bar{\mathrm{\wp }}$ is a nonconstant proper holomorphic map of connected Riemann surfaces (steps 1.1 and 1.2) of degree two (step 2.1), proving (1); steps 1.6 and 5.2 prove (2); step 6.1 together with the distinctness in step 1.4 identifies the critical points, i.e. the branch points, as $[0]$ and the three distinct nonzero classes $[h_1],[h_2],[h_3]$, and step 7.1 gives the equivalent description by branch values, proving (3); and steps 7.2 and 8.1 prove (4), that ${\mathrm{\wp }}^{\mathrm{\prime }}$ vanishes exactly at the $\mathrm{\Lambda }$-translates of the three half-periods and that each such zero is simple. ∎