The torus is biholomorphic to its Weierstrass cubic

(Uniform gap and finiteness in bounded sets.) Put $A:=\mathrm{\mid }{\omega }_1{\mathrm{\mid }}^2$, $B:=\mathrm{Re}({\omega }_1\overline{{\omega }_2})$, $C:=\mathrm{\mid }{\omega }_2{\mathrm{\mid }}^2$. Then $A,C>0$, and expanding $\mathrm{\mid }t{\omega }_1+s{\omega }_2{\mathrm{\mid }}^2$ with the modulus laws of [F14] gives $A{t}^2+2Bts+C{s}^2$ for real $s,t$. Completing the square in the two variables gives $A{t}^2+2Bts+C{s}^2=A(t+\frac{B}{A}s{)}^2+\frac{AC-B^2}{A}{s}^2\ge \frac{AC-B^2}{A}{s}^2$ and symmetrically for $t$, so with $\delta :=\sqrt{(AC-B^2)\mathrm{/}\max (A,C)}$ one has $\mathrm{\mid }t{\omega }_1+s{\omega }_2\mathrm{\mid }\ge \delta \max (\mathrm{\mid }t\mathrm{\mid },\mathrm{\mid }s\mathrm{\mid })$. Here $AC-B^2>0$: by [F14] it equals $(\mathrm{Im}({\omega }_1\overline{{\omega }_2}){)}^2$, and $\mathrm{Im}({\omega }_1\overline{{\omega }_2})=0$ would make ${\omega }_2$ a real multiple of ${\omega }_1$, contradicting real-linear independence in [F1]. Hence every nonzero $\lambda \in \mathrm{\Lambda }$ has $\mathrm{\mid }\lambda \mathrm{\mid }\ge \delta$, and every set $\{\lambda \in \mathrm{\Lambda }:\mathrm{\mid }\lambda \mathrm{\mid }\le R\}$ is finite, since it is contained in the image of the finite set of integer pairs with $\max (\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }n\mathrm{\mid })\le R\mathrm{/}\delta$.

(Well-definedness and holomorphic ambient coordinates.) For $z\mathrm{\notin }\mathrm{\Lambda }$, the triple $(\mathrm{\wp }(z),{\mathrm{\wp }}^{\mathrm{\prime }}(z),1)$ has nonzero third coordinate, so $\mathrm{\Psi }(z):=[\mathrm{\wp }(z):{\mathrm{\wp }}^{\mathrm{\prime }}(z):1]$ is a point of the affine chart $\{Z\mathrm{\ne }0\}$ of [F9]. Periodicity of both functions in [F4] gives $\mathrm{\Psi }(z+\lambda )=\mathrm{\Psi }(z)$, so $\mathrm{\Phi }([z]):=\mathrm{\Psi }(z)$ is well defined on $T_{\mathrm{\Lambda }}\setminus \{[0]\}$. In a torus chart obtained by lifting to a small ball ([F2]), its ambient coordinate functions are $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$, both holomorphic by [F4]. Once the image is placed in the curve, its local parameter from [F10] is one of these ambient coordinates, which verifies holomorphy as a map into the curve by [F12].

(The function $H:=\mathrm{\wp }-z^{-2}$ at $0$.) Work on the disc $\mathrm{\mid }z\mathrm{\mid }<\delta \mathrm{/}2$ from step 1.1, which contains no other lattice point. The double pole with principal part $z^{-2}$ in [F4] means that $H$ extends holomorphically across $0$. Evenness and [F13] give $H(z)=a_0+a_2{z}^2+O(z^4)$ and $H^{\mathrm{\prime }}(z)=2a_{2}z+O(z^3)$. Substituting $\mathrm{\wp }=z^{-2}+H$ and ${\mathrm{\wp }}^{\mathrm{\prime }}=-2z^{-3}+H^{\mathrm{\prime }}$ into [F6], the coefficient of $z^{-4}$ on the left is $0$, whereas on the right it is $12a_0$; equivalently, multiply by $z^6$ and compare the Taylor coefficient of $z^2$. Thus $a_0=0$, so $H(0)=0$, $H(z)=O(z^2)$ and $H^{\mathrm{\prime }}(z)=O(z)$.

(The image lies in $C_{\mathrm{\Lambda }}$.) For $z\mathrm{\notin }\mathrm{\Lambda }$ the point $\mathrm{\Phi }([z])=[\mathrm{\wp }(z):{\mathrm{\wp }}^{\mathrm{\prime }}(z):1]$ satisfies, by [F6], $Y^{2}Z={\mathrm{\wp }}^{\mathrm{\prime }}(z{)}^2\cdot 1=4\mathrm{\wp }(z{)}^3-g_2\mathrm{\wp }(z)-g_3=4X^3-g_{2}X{Z}^2-g_3{Z}^3$ with $(X,Y,Z)=(\mathrm{\wp }(z),{\mathrm{\wp }}^{\mathrm{\prime }}(z),1)$; hence $\mathrm{\Phi }([z])\in C_{\mathrm{\Lambda }}$.

(Expansions at the origin.) From step 2.1, near $0$ one has $\mathrm{\wp }(z)=z^{-2}(1+O(z^2))$ and, differentiating $\mathrm{\wp }=z^{-2}+H$ with the rules of [F13], ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-2z^{-3}+H^{\mathrm{\prime }}(z)=-2z^{-3}(1-\frac{1}{2}{z}^3{H}^{\mathrm{\prime }}(z))$ with $z^3{H}^{\mathrm{\prime }}(z)=O(z^4)$, so ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-2z^{-3}(1+O(z^4))$. Hence on a punctured neighbourhood of $0$ the quotients $u:=\mathrm{\wp }\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}$ and $v:=1\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}$ satisfy $u=-\frac{z}{2}(1+O(z^2))(1+O(z^4){)}^{-1}=-\frac{z}{2}+O(z^3)$ and $v=-\frac{z^3}{2}(1+O(z^4){)}^{-1}=-\frac{z^3}{2}+O(z^7)$, so in particular $v=O(z^3)$; both are holomorphic and bounded on a punctured disc around $0$, so by [F13] they extend holomorphically to $0$ with $u(0)=0=v(0)$ and $u^{\mathrm{\prime }}(0)=-\frac{1}{2}\mathrm{\ne }0$.

(Extension to $[0]$.) Let $U\subseteq \mathbb{C}$ be a ball around $0$ on which $\pi$ is injective, and use the torus chart $\chi :=(\pi {\mathrm{\mid }}_U{)}^{-1}:\pi (U)\to U$ ([F2]) around $[0]$, so that $\chi ([w])=w$. On the target side use the chart $\{Y\mathrm{\ne }0\}$ of [F9] with coordinates $(u,v)=(X\mathrm{/}Y,Z\mathrm{/}Y)$; the point $O=[0:1:0]$ has coordinates $(0,0)$. For $[w]\in \pi (U)\setminus \{[0]\}$ the coordinate functions of $\mathrm{\Phi }$ are $u=\mathrm{\wp }(w)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(w)$ and $v=1\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(w)$ (reading the homogeneous coordinates of step 1.2 in the chart $\{Y\mathrm{\ne }0\}$), and by step 3.1 these extend holomorphically to $w=0$ with values $0$. Hence the formula $\mathrm{\Phi }([0]):=O$ extends $\mathrm{\Phi }$ to a map on all of $T_{\mathrm{\Lambda }}$, and by [F12] this extension is holomorphic at $[0]$: the chart expression $w\mapsto (u(w),v(w))$ is holomorphic at $0$.

(Local biholomorphy at $[0]$.) Apply [F10] to $G(u,v):=v-4u^3+g_{2}u{v}^2+g_3{v}^3$: this is the equation of $C_{\mathrm{\Lambda }}$ in the chart $\{Y\mathrm{\ne }0\}$ by [F9], $G(0,0)=0$ and $\mathrm{\partial }G\mathrm{/}\mathrm{\partial }v(0,0)=1+2g_{2}uv+3g_3{v}^2{\mid }_{(0,0)}=1\mathrm{\ne }0$, so the curve is near $O$ the graph $v=\phi (u)$ of a holomorphic $\phi$ with $\phi (0)=0$, and $u$ is a local parameter on $C_{\mathrm{\Lambda }}$ at $O$, a homeomorphism of a neighbourhood of $O$ in $C_{\mathrm{\Lambda }}$ onto a plane domain. The chart expression of $\mathrm{\Phi }$ in the torus chart of step 4.1 and this local parameter is $w\mapsto u(w)$, which by step 3.1 equals $-\frac{w}{2}+O(w^3)$; it is holomorphic at $w=0$ with derivative $-\frac{1}{2}\mathrm{\ne }0$, so by F11 it restricts to a biholomorphism between complex domains contained in neighbourhoods of $0$ and of $u(0)=0$. Composing with the two charts, which are homeomorphisms by [F2] and [F10], the map $\mathrm{\Phi }$ carries a neighbourhood of $[0]$ homeomorphically onto an open subset of $C_{\mathrm{\Lambda }}$, and its inverse on that piece is holomorphic.

(Injectivity.) Let $z,w\in \mathbb{C}$ with $\mathrm{\Phi }([z])=\mathrm{\Phi }([w])$. If $z\in \mathrm{\Lambda }$ then $\mathrm{\Phi }([z])=O$ by step 4.1, and $O$ has $Z=0$ while every point $\mathrm{\Phi }([w^{\mathrm{\prime }}])$ with $w^{\mathrm{\prime }}\mathrm{\notin }\mathrm{\Lambda }$ has third homogeneous coordinate $1\mathrm{\ne }0$ by step 1.2; hence $w\in \mathrm{\Lambda }$ and $[w]=[z]$. If $z,w\mathrm{\notin }\mathrm{\Lambda }$, then comparing the chart $\{Z\mathrm{\ne }0\}$ coordinates of the common point gives $\mathrm{\wp }(z)=\mathrm{\wp }(w)$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z)={\mathrm{\wp }}^{\mathrm{\prime }}(w)$; by [F7] the first equality gives $w\equiv z$ or $w\equiv -z$ modulo $\mathrm{\Lambda }$, and in the second case ${\mathrm{\wp }}^{\mathrm{\prime }}(w)={\mathrm{\wp }}^{\mathrm{\prime }}(-z)=-{\mathrm{\wp }}^{\mathrm{\prime }}(z)$ by the oddness in [F4], so ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-{\mathrm{\wp }}^{\mathrm{\prime }}(z)$, that is ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=0$; then $[z]=[h_j]$ for some $j$ by [F7] and $w\equiv -z\equiv z$ modulo $\mathrm{\Lambda }$ because $2h_j\in \mathrm{\Lambda }$. Hence in all cases $[w]=[z]$, so $\mathrm{\Phi }$ is injective.

(Surjectivity.) Let $P=[X:Y:Z]\in C_{\mathrm{\Lambda }}$. If $Z\mathrm{\ne }0$, put $x:=X\mathrm{/}Z\in \mathbb{C}$ and $y:=Y\mathrm{/}Z\in \mathbb{C}$; the equation of $C_{\mathrm{\Lambda }}$ reads $y^2=4x^3-g_{2}x-g_3$. By [F7] the value $x$ is attained: choose $z\in \mathbb{C}$ with $\mathrm{\wp }(z)=x$; then $z\mathrm{\notin }\mathrm{\Lambda }$ by [F5], and by [F6] ${\mathrm{\wp }}^{\mathrm{\prime }}(z{)}^2=4x^3-g_{2}x-g_3=y^2$, so ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=y$ or ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-y$. In the first case $\mathrm{\Phi }([z])=[x:y:1]=P$; in the second case $\mathrm{\wp }(-z)=x$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(-z)=-(-y)=y$ by [F4], so $\mathrm{\Phi }([-z])=P$. If $Z=0$, then the equation gives $0=4X^3$, so $X=0$ and $P=[0:Y:0]=[0:1:0]=O=\mathrm{\Phi }([0])$ by step 4.1. Hence $\mathrm{\Phi }$ is surjective.

(Local biholomorphy at the remaining points.) Let $z_0\in \mathbb{C}\setminus \mathrm{\Lambda }$ and $P:=\mathrm{\Phi }([z_0])\in C_{\mathrm{\Lambda }}$; write $f(x,y):=y^2-4x^3+g_{2}x+g_3$ for the defining polynomial in the chart $\{Z\mathrm{\ne }0\}$ of [F9]. If ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$, then $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }y(P)=2{\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$; by [F10] applied to $f$ (whose zero set is $C_{\mathrm{\Lambda }}$ in that chart), $x$ is a local parameter on $C_{\mathrm{\Lambda }}$ at $P$, and the chart expression of $\mathrm{\Phi }$ in the torus chart at $[z_0]$ and this parameter is $w\mapsto \mathrm{\wp }(w)$, holomorphic at $z_0$ with derivative ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$ by [F4]; so by F11 this chart expression restricts to a biholomorphism between neighbourhoods, and $\mathrm{\Phi }$ is a local biholomorphism at $[z_0]$. If instead ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)=0$, then $z_0\equiv h_j$ modulo $\mathrm{\Lambda }$ for some $j\in \{1,2,3\}$ by [F7], and $P=(e_j,0)$. By [F8] the roots $e_1,e_2,e_3$ of $p(x):=4x^3-g_{2}x-g_3$ are distinct, so $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }x(P)=-p^{\mathrm{\prime }}(e_j)\mathrm{\ne }0$; applying [F10] to $f$ near $P$ shows that $y$ is a local parameter on $C_{\mathrm{\Lambda }}$ at $P$. By [F7] the class $[h_j]$ is a critical point of $\bar{\mathrm{\wp }}$ with ramification index $2$, so $\mathrm{\wp }(w)-e_j$ has a zero of order $2$ at $h_j$; by [F13] it factors as $\mathrm{\wp }(w)-e_j=(w-h_j{)}^{2}g(w)$ with $g$ holomorphic near $h_j$ and $g(h_j)\mathrm{\ne }0$, so by the product rule of [F13] ${\mathrm{\wp }}^{\mathrm{\prime }}(w)=(w-h_j)(2g(w)+(w-h_j)g^{\mathrm{\prime }}(w))$ and ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(h_j)=2g(h_j)\mathrm{\ne }0$. The chart expression of $\mathrm{\Phi }$ in the torus chart at $[h_j]$ and the local parameter $y$ is $w\mapsto {\mathrm{\wp }}^{\mathrm{\prime }}(w)$, holomorphic with derivative ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(h_j)\mathrm{\ne }0$ at $w=h_j$; so by F11 this chart expression restricts to a biholomorphism between neighbourhoods, and $\mathrm{\Phi }$ is a local biholomorphism at $[h_j]$. Every class of $T_{\mathrm{\Lambda }}$ is $[0]$, a class $[z_0]$ with ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$, or some $[h_j]$, so $\mathrm{\Phi }$ carries a neighbourhood of every point of $T_{\mathrm{\Lambda }}$ homeomorphically onto an open subset of $C_{\mathrm{\Lambda }}$, with holomorphic inverse on that piece.

($\mathrm{\Phi }$ is a homeomorphism; the topology of $C_{\mathrm{\Lambda }}$.) By steps 5.1 and 6.1 every $x\in T_{\mathrm{\Lambda }}$ has an open neighbourhood $U_x$ such that $\mathrm{\Phi }(U_x)$ is open in $C_{\mathrm{\Lambda }}$ and $\mathrm{\Phi }{\mathrm{\mid }}_{U_x}:U_x\to \mathrm{\Phi }(U_x)$ is a homeomorphism. Hence $\mathrm{\Phi }$ is continuous, because it is continuous on each member of the open cover $\{U_x\}$ of its domain; and $\mathrm{\Phi }$ is open: for open $W\subseteq T_{\mathrm{\Lambda }}$ one has $\mathrm{\Phi }(W)={\bigcup }_{x\in W}\mathrm{\Phi }(U_x\cap W)$, and each piece $\mathrm{\Phi }(U_x\cap W)$ is open in $\mathrm{\Phi }(U_x)$, hence in $C_{\mathrm{\Lambda }}$, because $\mathrm{\Phi }{\mathrm{\mid }}_{U_x}$ is a homeomorphism onto the open set $\mathrm{\Phi }(U_x)$. By steps 5.2 and 5.3 the map $\mathrm{\Phi }$ is bijective, so by F11 it is a homeomorphism. Consequently $C_{\mathrm{\Lambda }}$ inherits the following properties from $T_{\mathrm{\Lambda }}$: it is compact and connected as a continuous image of the compact connected torus ([F2], [F14]); it is Hausdorff, because distinct points $P\mathrm{\ne }Q$ of $C_{\mathrm{\Lambda }}$ have distinct preimages ${\mathrm{\Phi }}^{-1}(P)\mathrm{\ne }{\mathrm{\Phi }}^{-1}(Q)$ by injectivity, the Hausdorff torus [F2] separates them by disjoint open sets, and their $\mathrm{\Phi }$-images are disjoint open sets separating $P$ and $Q$; and it is second countable, because for a countable basis $\mathcal{B}$ of the second-countable torus [F2] the images $\mathrm{\Phi }(B)$, $B\in \mathcal{B}$, are open and form a basis by [F15]: given open $W\subseteq C_{\mathrm{\Lambda }}$ and $P\in W$, the set ${\mathrm{\Phi }}^{-1}(W)$ is open and contains ${\mathrm{\Phi }}^{-1}(P)$, so some $B\in \mathcal{B}$ has ${\mathrm{\Phi }}^{-1}(P)\in B\subseteq {\mathrm{\Phi }}^{-1}(W)$, whence $P\in \mathrm{\Phi }(B)\subseteq W$. The local parameters of [F10] are charts on $C_{\mathrm{\Lambda }}$ with holomorphic transitions, so by [F15] the space $C_{\mathrm{\Lambda }}$, nonempty and homeomorphic to $T_{\mathrm{\Lambda }}$ through $\mathrm{\Phi }$, is a compact connected Riemann surface.

(The inverse is holomorphic, and conclusion.) By steps 1.2, 4.1 and 6.1 the map $\mathrm{\Phi }:T_{\mathrm{\Lambda }}\to C_{\mathrm{\Lambda }}$ is holomorphic and a local biholomorphism at every point; by steps 5.2 and 5.3 it is bijective. At a point $P\in C_{\mathrm{\Lambda }}$, let charts $\phi$ around ${\mathrm{\Phi }}^{-1}(P)$ and $\psi$ around $P$ be such that $\psi \circ \mathrm{\Phi }\circ {\phi }^{-1}$ is the identity (these exist because $\mathrm{\Phi }$ is a local biholomorphism at ${\mathrm{\Phi }}^{-1}(P)$, as recorded in steps 5.1 and 6.1); then the chart expression of ${\mathrm{\Phi }}^{-1}$ is the identity too, hence holomorphic at $P$; since every point of $C_{\mathrm{\Lambda }}$ carries such charts, ${\mathrm{\Phi }}^{-1}$ is holomorphic by the locality clause of [F12]. Thus $\mathrm{\Phi }$ is a biholomorphism. Finally, by step 7.1 the space $C_{\mathrm{\Lambda }}$ is a compact connected Riemann surface homeomorphic to $T_{\mathrm{\Lambda }}$ through $\mathrm{\Phi }$; clause (1) is exactly [F8]. ∎