The chord-tangent group law and elliptic uniformization

(The transported operation.) Define $\oplus :C_{\mathrm{\Lambda }}\times C_{\mathrm{\Lambda }}\to C_{\mathrm{\Lambda }}$ by $P\oplus Q:=\mathrm{\Phi }({\mathrm{\Phi }}^{-1}(P)+{\mathrm{\Phi }}^{-1}(Q))$. This is well defined because $\mathrm{\Phi }$ is a bijection [F7]; transport along a bijection carries the abelian group laws of $T_{\mathrm{\Lambda }}$ [F1] to $C_{\mathrm{\Lambda }}$, so $\oplus$ is commutative and associative, its identity is $\mathrm{\Phi }([0])=O$, and the inverse of $P$ is $\ominus P:=\mathrm{\Phi }(-{\mathrm{\Phi }}^{-1}(P))$. By the very definition $\mathrm{\Phi }([z]+[w])=\mathrm{\Phi }([z])\oplus \mathrm{\Phi }([w])$ for all $z,w\in \mathbb{C}$, and $\ominus \mathrm{\Phi }([s])=\mathrm{\Phi }([-s])=[\mathrm{\wp }(-s):{\mathrm{\wp }}^{\mathrm{\prime }}(-s):1]=[\mathrm{\wp }(s):-{\mathrm{\wp }}^{\mathrm{\prime }}(s):1]$ for $s\mathrm{\notin }\mathrm{\Lambda }$ by the parity of $\mathrm{\wp }$ and the oddness of ${\mathrm{\wp }}^{\mathrm{\prime }}$ [F2]; in the chart $\{Z\mathrm{\ne }0\}$ this reads $\ominus (x,y)=(x,-y)$, and $\ominus O=O$.

(The differentiated differential equation.) On $\mathbb{C}\setminus \mathrm{\Lambda }$ the functions $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ are holomorphic [F2] and $({\mathrm{\wp }}^{\mathrm{\prime }}{)}^2=4{\mathrm{\wp }}^3-g_2\mathrm{\wp }-g_3$ [F4]. Differentiating this identity with the sum, product and chain rules [F8] gives $2{\mathrm{\wp }}^{\mathrm{\prime }}{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=(12{\mathrm{\wp }}^2-g_2){\mathrm{\wp }}^{\mathrm{\prime }}$ on $\mathbb{C}\setminus \mathrm{\Lambda }$. At every point with ${\mathrm{\wp }}^{\mathrm{\prime }}\mathrm{\ne }0$ division gives ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$. If $z_0\in \mathbb{C}\setminus \mathrm{\Lambda }$ has ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)=0$, then $z_0$ is a $\mathrm{\Lambda }$-translate of one of $h_1,h_2,h_3$ by [F3], and [F3] also says each such zero is of order one; hence ${\mathrm{\wp }}^{\mathrm{\prime }}\mathrm{\ne }0$ on a punctured disc $D\setminus \{z_0\}$ around $z_0$ with $D\subseteq \mathbb{C}\setminus \mathrm{\Lambda }$ [F14], so the identity holds on $D\setminus \{z_0\}$. Both ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}$ and $6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$ are holomorphic on $D$, since ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}$ is the derivative of the holomorphic function ${\mathrm{\wp }}^{\mathrm{\prime }}$ and a holomorphic function has complex derivatives of every order [F9]; hence both are continuous on $D$ [F9], and the limit $z\to z_0$ along $D\setminus \{z_0\}$ gives ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z_0)=6\mathrm{\wp }(z_0{)}^2-\frac{1}{2}{g}_2$. Therefore ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}=6{\mathrm{\wp }}^2-\frac{1}{2}{g}_2$ on all of $\mathbb{C}\setminus \mathrm{\Lambda }$.

(The affine chart and its local parameters.) In the chart $\{Z\mathrm{\ne }0\}$ with coordinates $(x,y)=(X\mathrm{/}Z,Y\mathrm{/}Z)$ [F15], the cubic $C_{\mathrm{\Lambda }}$ is the zero set of $f(x,y):=y^2-p(x)$, because the defining equation divided by $Z^3$ reads $y^2=4x^3-g_{2}x-g_3$. Its gradient is $\mathrm{\nabla }f=(-p^{\mathrm{\prime }}(x),2y)$: if $y\mathrm{\ne }0$ then $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }y=2y\mathrm{\ne }0$, while if $y=0$ then $p(x)=0$, so $x=e_j$ for some $j$ by [F5] and $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }x=-p^{\mathrm{\prime }}(e_j)\mathrm{\ne }0$. Hence the gradient is nonzero at every point of $C_{\mathrm{\Lambda }}\cap \{Z\mathrm{\ne }0\}$ and the hypothesis of the chart lemma [F16] holds there: at a point with $y\mathrm{\ne }0$ the coordinate $x$ is a local parameter and $C_{\mathrm{\Lambda }}$ agrees near the point with a graph $x\mapsto (x,g(x))$ for a holomorphic $g$ with $g(x{)}^2=p(x)$, while at the point $(e_j,0)$ the coordinate $y$ is a local parameter and $C_{\mathrm{\Lambda }}$ agrees near it with a graph $y\mapsto (h(y),y)$ for a holomorphic $h$ with $h(0)=e_j$ and $y^2=p(h(y))$. Differentiating the latter identity with the chain rule [F8] gives $2y=p^{\mathrm{\prime }}(h(y))h^{\mathrm{\prime }}(y)$, so $h^{\mathrm{\prime }}(0)=0$ because $p^{\mathrm{\prime }}(e_j)\mathrm{\ne }0$; comparing the $y^2$-coefficients in $y^2=p(e_j+(h(y)-e_j))=p^{\mathrm{\prime }}(e_j)(h(y)-e_j)+O((h(y)-e_j{)}^2)$ gives $h(y)-e_j=(1\mathrm{/}{p}^{\mathrm{\prime }}(e_j))y^2+O(y^3)$, so $h(y)-e_j$ vanishes at $y=0$ with order exactly $2$.

(The point at infinity and the vertical directions there.) In the chart $\{Y\mathrm{\ne }0\}$ with coordinates $(u,v)=(X\mathrm{/}Y,Z\mathrm{/}Y)$ [F15], the point $O=[0:1:0]$ is $(0,0)$ and the cubic reads $G(u,v)=0$ for $G(u,v):=v-4u^3+g_{2}u{v}^2+g_3{v}^3$. Here $G(0,0)=0$ and $\mathrm{\partial }G\mathrm{/}\mathrm{\partial }v(0,0)=1\mathrm{\ne }0$, so by the chart lemma [F16] the coordinate $u$ is a local parameter at $O$ and $C_{\mathrm{\Lambda }}$ agrees near $O$ with the graph $v=\phi (u)$ of a holomorphic $\phi$ near $0$ with $\phi (0)=0$ and $\phi (u)=4u^3-g_{2}u\phi (u{)}^2-g_3\phi (u{)}^3$. Differentiating the relation $G(u,\phi (u))=0$ with the chain rule [F8] gives $(-12u^2+g_2\phi (u{)}^2)+(1+2g_{2}u\phi (u)+3g_3\phi (u{)}^2){\phi }^{\mathrm{\prime }}(u)=0$, so ${\phi }^{\mathrm{\prime }}(0)=0$ and hence $\phi (u)=O(u^2)$. The relation excludes $\phi \equiv 0$, since it would give $4u^3=0$ near $0$. Writing $\phi (u)=u^n\psi (u)$ with $\psi (0)\mathrm{\ne }0$ and $n\ge 1$, the relation $u^n\psi =4u^3-g_2{u}^{2n+1}{\psi }^2-g_3{u}^{3n}{\psi }^3$ forces $n=3$: for $n<3$ every term on the right has order greater than $n$, and for $n>3$ the term $4u^3$ is the unique lowest-order term on the right, so its order there is exactly $3$. Hence the line at infinity $\{Z=0\}$, whose local equation in this chart is $v$ [F15], vanishes along $C_{\mathrm{\Lambda }}$ at $O$ with order $3$, and it meets $C_{\mathrm{\Lambda }}$ nowhere else, because setting $Z=0$ in the cubic gives $4X^3=0$, hence $X=0$ and the point $[0:Y:0]=O$. Likewise, for $c\in \mathbb{C}$ the vertical line $\{X=cZ\}$ has local equation $u-cv$ at $O$ [F15], which restricts to $u-c\phi (u)=u(1-c\phi (u)\mathrm{/}u)$; since $\phi (u)=O(u^2)$ the bracket tends to $1\mathrm{\ne }0$, so the order of vanishing at $O$ is $1$.

(Intersection multiplicity convention.) For a projective line $L$ and a point $P_0\in C_{\mathrm{\Lambda }}\cap L$ call the multiplicity of $L$ at $P_0$ the order of vanishing at $P_0$ of the restriction of a local equation of $L$ to $C_{\mathrm{\Lambda }}$, computed in a local parameter of $C_{\mathrm{\Lambda }}$ at $P_0$; by [F16] the transition between two local parameters is holomorphic with holomorphic inverse, so its derivative never vanishes and the order does not depend on the local parameter, and multiplying a local equation of a line by a holomorphic function without zeros does not change the order. The intersection divisor $L\cdot C_{\mathrm{\Lambda }}$ is the formal sum of the points of the finite set $C_{\mathrm{\Lambda }}\cap L$ taken with these multiplicities; a multiplicity $1$, $2$ or $3$ is called a simple, double or triple intersection.

(The differentiated addition identity where all values are finite.) Let $z,w\in \mathbb{C}$ satisfy $z,w,z+w\mathrm{\notin }\mathrm{\Lambda }$ and $\mathrm{\wp }(z)\mathrm{\ne }\mathrm{\wp }(w)$, and put $x:=\mathrm{\wp }(z)$, $y:={\mathrm{\wp }}^{\mathrm{\prime }}(z)$, $u:=\mathrm{\wp }(w)$, $v:={\mathrm{\wp }}^{\mathrm{\prime }}(w)$, $d:=u-x\mathrm{\ne }0$, $m:=(v-y)\mathrm{/}d$ and $t:=\mathrm{\wp }(z+w)$. By [F3] the inequality $\mathrm{\wp }(z)\mathrm{\ne }\mathrm{\wp }(w)$ says $z\equiv ̸\pm w$ modulo $\mathrm{\Lambda }$, so $z\mathrm{\notin }\pm w+\mathrm{\Lambda }$. By [F6] the function ${\mathrm{\Phi }}_w(\zeta ):=\mathrm{\wp }(\zeta +w)+\mathrm{\wp }(\zeta )+\mathrm{\wp }(w)-\frac{1}{4}Q(\zeta {)}^2$ with $Q(\zeta ):=({\mathrm{\wp }}^{\mathrm{\prime }}(\zeta )-{\mathrm{\wp }}^{\mathrm{\prime }}(w))\mathrm{/}(\mathrm{\wp }(\zeta )-\mathrm{\wp }(w))$ is the zero meromorphic function of $\zeta$ on $\mathbb{C}$; on a small disc around $z$ each of the functions $\zeta \mapsto \mathrm{\wp }(\zeta +w)$, $\mathrm{\wp }(\zeta )$ and $Q(\zeta )$ is holomorphic (here $z+w\mathrm{\notin }\mathrm{\Lambda }$, $z\mathrm{\notin }\mathrm{\Lambda }$ and $\mathrm{\wp }(z)\mathrm{\ne }\mathrm{\wp }(w)$), so ${\mathrm{\Phi }}_w$ is holomorphic there and, being identically zero, has derivative $0$ there. By the sum, product and quotient rules [F8], at $\zeta =z$ one has $0={\mathrm{\wp }}^{\mathrm{\prime }}(z+w)+{\mathrm{\wp }}^{\mathrm{\prime }}(z)-\frac{1}{2}Q(z)Q^{\mathrm{\prime }}(z)$ with $Q(z)=m$ and $Q^{\mathrm{\prime }}(z)=({\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)(x-u)-(y-v)y)\mathrm{/}(x-u{)}^2=(my-{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z))\mathrm{/}(u-x)$, the last equality because $x-u=-d$ and $y-v=-md$; hence ${\mathrm{\wp }}^{\mathrm{\prime }}(z+w)=-y+\frac{1}{2}m{Q}^{\mathrm{\prime }}(z)$. Next [F6] gives $t=-x-u+\frac{1}{4}{m}^2$, that is $m^2=4(t+x+u)$. By [F4] at $z$ and at $w$, $v^2-y^2=4(u^3-x^3)-g_2(u-x)=(u-x)(4(u^2+ux+x^2)-g_2)$, while $v^2-y^2=(v-y)(v+y)=md(v+y)$; dividing by $d\mathrm{\ne }0$ gives $m(v+y)=4(u^2+ux+x^2)-g_2$, and substituting $v+y=2y+md$ gives $m^{2}d+2my=4(u^2+ux+x^2)-g_2$. Substituting $m^2=4(t+x+u)$ and $(t+x+u)(u-x)=t(u-x)+(u^2-x^2)$ yields $4t(u-x)+4(u^2-x^2)+2my=4(u^2+ux+x^2)-g_2$, that is $2my=4x(u+2x)-g_2-4t(u-x)$. By step 1.2, $g_2=12x^2-2{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)$, so $2my=4xu-4x^2+2{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)-4t(u-x)$, which says $my-{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)=2(u-x)(x-t)$; therefore $Q^{\mathrm{\prime }}(z)=2(x-t)$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z+w)=-y+m(x-t)$.

(Vertical lines.) Let $c\in \mathbb{C}$ and $L:=\{X=cZ\}$; then $O\in L$ [F15], $L\cap \{Z=0\}=\{O\}$, and by step 1.4 the multiplicity of $L$ at $O$ is $1$. If $p(c)\mathrm{\ne }0$, then by [F13] applied to $t^2-p(c)$ there is $y_0\mathrm{\ne }0$ with $y_0^2=p(c)$, and the affine part of $L\cap C_{\mathrm{\Lambda }}$ is exactly the two points $P_{+}=(c,y_0)$ and $P_{-}=(c,-y_0)$, since in the chart $\{Z\mathrm{\ne }0\}$ the curve meets $x=c$ in the solutions of $y^2=p(c)$. At each of them $y\mathrm{\ne }0$, so by step 1.3 the coordinate $x$ is a local parameter and the local equation $x-c$ of $L$ has order $1$ there; hence the divisor is $P_{+}+P_{-}+O$, of total multiplicity $3$. Choose $z$ with $\mathrm{\Phi }([z])=P_{+}$ [F7]; then $z\mathrm{\notin }\mathrm{\Lambda }$, $\mathrm{\wp }(z)=c$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=y_0$, so by parity [F2] $P_{-}=(c,-y_0)=[\mathrm{\wp }(-z):{\mathrm{\wp }}^{\mathrm{\prime }}(-z):1]=\mathrm{\Phi }([-z])$, and step 1.1 gives $P_{+}\oplus P_{-}\oplus O=\mathrm{\Phi }([z]+[-z]+[0])=\mathrm{\Phi }([0])=O$. If $p(c)=0$, then $c=e_j$ for a unique $j$ [F5], and the only affine intersection is $P:=(e_j,0)$; by step 1.3 the local equation $x-e_j$ of $L$ has order $2$ at $P$ in the local parameter $y$, while the multiplicity at $O$ is $1$ by step 1.4, so the divisor is $2P+O$, of total multiplicity $3$. By [F3], $\mathrm{\wp }(h_j)=e_j$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(h_j)=0$, so $P=\mathrm{\Phi }([h_j])$; also $-h_j\equiv h_j$ modulo $\mathrm{\Lambda }$ because $2h_1={\omega }_1$, $2h_2={\omega }_2$ and $2h_3={\omega }_1+{\omega }_2$ all lie in $\mathrm{\Lambda }$, so step 1.1 gives $P=\mathrm{\Phi }([-h_j])=\ominus \mathrm{\Phi }([h_j])=\ominus P$, and $2P\oplus O=\mathrm{\Phi }([h_j]+[h_j]+[0])=\mathrm{\Phi }([2h_j])=\mathrm{\Phi }([0])=O$ because $2h_j\in \mathrm{\Lambda }$. Thus the transported sum of the divisor $P,-P,O$ is $O$, with the finite point occurring with multiplicity two.

(The line at infinity.) Let $L:=\{Z=0\}$. It has no affine point, and by step 1.4 it meets $C_{\mathrm{\Lambda }}$ only at $O$, with multiplicity $3$, so $L\cdot C_{\mathrm{\Lambda }}=3O$ and, by step 1.1 and $\mathrm{\Phi }([0])=O$, $O\oplus O\oplus O=\mathrm{\Phi }([0]+[0]+[0])=\mathrm{\Phi }([0])=O$.

(The diagonal case of the differentiated identity.) Let $z_0\in \mathbb{C}\setminus \mathrm{\Lambda }$ satisfy ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$; then $2z_0\mathrm{\notin }\mathrm{\Lambda }$ by [F3], and by [F14] we may choose a disc $D$ centred at $z_0$ with $D\subseteq \mathbb{C}\setminus \mathrm{\Lambda }$ and $z_0+D\subseteq \mathbb{C}\setminus \mathrm{\Lambda }$. For $w\in D\setminus \{z_0\}$ the Taylor expansion of $\mathrm{\wp }$ at $z_0$ [F9] gives $\mathrm{\wp }(w)-\mathrm{\wp }(z_0)={\mathrm{\wp }}^{\mathrm{\prime }}(z_0)(w-z_0)+O((w-z_0{)}^2)\mathrm{\ne }0$ after shrinking $D$, so step 2.1 applies to the pair $(z_0,w)$ and $G(w):={\mathrm{\wp }}^{\mathrm{\prime }}(z_0+w)+{\mathrm{\wp }}^{\mathrm{\prime }}(z_0)-\widehat{m}(w)(\mathrm{\wp }(z_0)-\mathrm{\wp }(z_0+w))=0$ for $w\in D\setminus \{z_0\}$, where $\widehat{m}$ is the continuous extension to $w=z_0$ of $m(z_0,w)=({\mathrm{\wp }}^{\mathrm{\prime }}(w)-{\mathrm{\wp }}^{\mathrm{\prime }}(z_0))\mathrm{/}(\mathrm{\wp }(w)-\mathrm{\wp }(z_0))$: by [F10] applied to ${\mathrm{\wp }}^{\mathrm{\prime }}$ and to $\mathrm{\wp }$ on a disc around $z_0$ inside $\mathbb{C}\setminus \mathrm{\Lambda }$ the filled difference quotients $A(w)=({\mathrm{\wp }}^{\mathrm{\prime }}(w)-{\mathrm{\wp }}^{\mathrm{\prime }}(z_0))\mathrm{/}(w-z_0)$ (value ${\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z_0)$ at $w=z_0$) and $B(w)=(\mathrm{\wp }(w)-\mathrm{\wp }(z_0))\mathrm{/}(w-z_0)$ (value ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$ at $w=z_0$) are continuous at $z_0$, and since $B(z_0)\mathrm{\ne }0$ the quotient $\widehat{m}=A\mathrm{/}B$ is continuous at $z_0$ with $\widehat{m}(w)=m(z_0,w)$ for $w\mathrm{\ne }{z}_0$ and $\widehat{m}(z_0)={\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z_0)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z_0)$ [F11]. The functions $w\mapsto {\mathrm{\wp }}^{\mathrm{\prime }}(z_0+w)$ and $w\mapsto \mathrm{\wp }(z_0+w)$ are holomorphic on $D$, hence continuous there [F9], so $G$ is continuous at $z_0$ [F11]. Since $G$ vanishes on $D\setminus \{z_0\}$ it vanishes at $z_0$: given $\epsilon >0$ choose $\delta >0$ smaller than the radius of $D$ with $\mathrm{\mid }G(w)-G(z_0)\mathrm{\mid }<\epsilon$ for $\mathrm{\mid }w-z_0\mathrm{\mid }<\delta$, take $w=z_0+\delta \mathrm{/}2$ to get $\mathrm{\mid }G(z_0)\mathrm{\mid }=\mathrm{\mid }G(z_0)-G(w)\mathrm{\mid }<\epsilon$, and since this holds for every $\epsilon >0$ — take $\epsilon =\mathrm{\mid }G(z_0)\mathrm{\mid }$ if $\mathrm{\mid }G(z_0)\mathrm{\mid }>0$ — we get $\mathrm{\mid }G(z_0)\mathrm{\mid }=0$ [F12], hence $G(z_0)=0$ [F12], that is ${\mathrm{\wp }}^{\mathrm{\prime }}(2z_0)=-{\mathrm{\wp }}^{\mathrm{\prime }}(z_0)+({\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z_0)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z_0))(\mathrm{\wp }(z_0)-\mathrm{\wp }(2z_0))$.

(Nonvertical lines with three distinct intersections.) Let $L$ be the projective line $\{Y=mX+bZ\}$ with $m,b\in \mathbb{C}$; then $O\mathrm{\notin }L$ by [F15], and $L\cap \{Z=0\}=\{[1:m:0]\}$ does not lie on $C_{\mathrm{\Lambda }}$ because $4\mathrm{\ne }0$. The affine points of $C_{\mathrm{\Lambda }}\cap L$ are the points $(x,mx+b)$ with $P_L(x)=0$, where $P_L(X):=(mX+b{)}^2-p(X)=-4X^3+m^2{X}^2+(g_2+2mb)X+(g_3+b^2)$ is a polynomial of degree $3$, and at such a point the multiplicity of $L$ in the sense of step 1.5 equals the multiplicity of $x$ as a root of $P_L$: if $y=mx+b\mathrm{\ne }0$, then $x$ is a local parameter and $C_{\mathrm{\Lambda }}$ is a graph $x^{\mathrm{\prime }}\mapsto (x^{\mathrm{\prime }},g(x^{\mathrm{\prime }}))$ near $x$ by step 1.3, $(g(x^{\mathrm{\prime }})-m{x}^{\mathrm{\prime }}-b)(g(x^{\mathrm{\prime }})+m{x}^{\mathrm{\prime }}+b)=p(x^{\mathrm{\prime }})-(m{x}^{\mathrm{\prime }}+b{)}^2=-P_L(x^{\mathrm{\prime }})$ and $g(x)+mx+b=2y\mathrm{\ne }0$, so the order of $g-m(\cdot )-b$ at $x$ equals the order of $P_L$ at $x$; and if $y=0$ (so $x=e_j$ and $b=-m{e}_j$), then both multiplicities equal $1$, because $P_L(e_j)=0$ and $P_L^{\mathrm{\prime }}(e_j)=2m(m{e}_j+b)-p^{\mathrm{\prime }}(e_j)=-p^{\mathrm{\prime }}(e_j)\mathrm{\ne }0$ by [F5], while in the local parameter $y$ of step 1.3 the line restricts to $y-m(h(y)-e_j)$ with derivative $1-m{h}^{\mathrm{\prime }}(0)=1\mathrm{\ne }0$ at $0$. Consequently the intersection divisor of a nonvertical line is the sum of its root points $(x,mx+b)$, each with the multiplicity of the root, a total multiplicity of $3=\mathrm{deg}{P}_L$ by [F13] and step 1.5. Now suppose $P_L$ has three distinct roots $x_1,x_2,x_3$, put $P_i:=(x_i,m{x}_i+b)$ and $y_i:=m{x}_i+b$, so that $L\cdot C_{\mathrm{\Lambda }}=P_1+P_2+P_3$; the $x_i$ are distinct, and $-P_L\mathrm{/}4$ is monic with $X^2$-coefficient $-m^2\mathrm{/}4$, so $x_1+x_2+x_3=m^2\mathrm{/}4$ by [F13]. By surjectivity of $\mathrm{\Phi }$ [F7] choose $z_1,z_2\in \mathbb{C}$ with $\mathrm{\Phi }([z_i])=P_i$; then $z_i\mathrm{\notin }\mathrm{\Lambda }$ (as $P_i\mathrm{\ne }O$), $\mathrm{\wp }(z_i)=x_i$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z_i)=y_i$, and $x_1\mathrm{\ne }{x}_2$ gives $\mathrm{\wp }(z_1)\mathrm{\ne }\mathrm{\wp }(z_2)$, hence $z_1\pm z_2\mathrm{\notin }\mathrm{\Lambda }$ by [F3]. Put $t:=\mathrm{\wp }(z_1+z_2)$; the secant slope $(y_2-y_1)\mathrm{/}(x_2-x_1)$ equals $m$. Step 2.1 applies to $(z_1,z_2)$ and gives ${\mathrm{\wp }}^{\mathrm{\prime }}(z_1+z_2)=-y_1+m(x_1-t)$, while [F6] gives $t=-x_1-x_2+\frac{1}{4}{m}^2$. By the sum relation above, $x_3=m^2\mathrm{/}4-x_1-x_2=t$, so $P_3$ has $x$-coordinate $t$, and its $y$-coordinate is $y_3=m{x}_3+b=mt+y_1-m{x}_1=y_1+m(t-x_1)=y_1-m(x_1-t)=y_1-({\mathrm{\wp }}^{\mathrm{\prime }}(z_1+z_2)+y_1)=-{\mathrm{\wp }}^{\mathrm{\prime }}(z_1+z_2)$. Thus $P_3=(\mathrm{\wp }(-z_1-z_2),{\mathrm{\wp }}^{\mathrm{\prime }}(-z_1-z_2))=\mathrm{\Phi }([-z_1-z_2])$ by the parity of $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ [F2], and by step 1.1 $P_1\oplus P_2\oplus P_3=\mathrm{\Phi }([z_1]+[z_2]+[-z_1-z_2])=\mathrm{\Phi }([0])=O$.

(Nonvertical lines with a repeated intersection: the tangent case.) Let $L=\{Y=mX+bZ\}$ and suppose $P_L$ has a repeated root $x$; put $y:=mx+b$ and $P:=(x,y)$. By step 3.2 the multiplicity of $L$ at $P$ equals the multiplicity of the root $x$, so it is at least $2$; in particular $y\mathrm{\ne }0$, since a point with $y=0$ has multiplicity $1$ by step 3.2. Choose $z\in \mathbb{C}$ with $\mathrm{\Phi }([z])=P$ [F7]; then $z\mathrm{\notin }\mathrm{\Lambda }$, $x=\mathrm{\wp }(z)$, $y={\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$ and $2z\mathrm{\notin }\mathrm{\Lambda }$ by [F3]. Because the root is repeated, $P_L^{\mathrm{\prime }}(x)=2m(mx+b)-p^{\mathrm{\prime }}(x)=0$, so $2my=p^{\mathrm{\prime }}(x)=12x^2-g_2=2{\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)$ by step 1.2, that is $m={\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z)$. Step 3.1 gives ${\mathrm{\wp }}^{\mathrm{\prime }}(2z)=-y+m(x-\mathrm{\wp }(2z))$, that is $m\mathrm{\wp }(2z)+b=m\mathrm{\wp }(2z)+y-mx=y-m(x-\mathrm{\wp }(2z))=y-({\mathrm{\wp }}^{\mathrm{\prime }}(2z)+y)=-{\mathrm{\wp }}^{\mathrm{\prime }}(2z)$. Hence the point $R:=(\mathrm{\wp }(2z),-{\mathrm{\wp }}^{\mathrm{\prime }}(2z))=\mathrm{\Phi }([-2z])$ (parity [F2]; both coordinates are finite because $2z\mathrm{\notin }\mathrm{\Lambda }$) lies on $L$ and on $C_{\mathrm{\Lambda }}$, so its $x$-coordinate $\mathrm{\wp }(2z)$ is a root of $P_L$ by step 3.2. Since $\mathrm{deg}{P}_L=3$ and $x$ is a root of multiplicity at least $2$ [F13], Vieta's formula of [F13] gives its residual root $t_3=m^2\mathrm{/}4-2x$, including when $t_3=x$. Taking the diagonal limit $w\to z$ in [F6] is legitimate because $y={\mathrm{\wp }}^{\mathrm{\prime }}(z)\mathrm{\ne }0$: Taylor expansion [F9] gives $({\mathrm{\wp }}^{\mathrm{\prime }}(z)-{\mathrm{\wp }}^{\mathrm{\prime }}(w))\mathrm{/}(\mathrm{\wp }(z)-\mathrm{\wp }(w))\to {\mathrm{\wp }}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(z)=m$, and $2z\mathrm{\notin }\mathrm{\Lambda }$ makes $\mathrm{\wp }$ continuous there. Hence $\mathrm{\wp }(2z)=-2x+m^2\mathrm{/}4=t_3$. Thus $R=\mathrm{\Phi }([-2z])$ is exactly the residual intersection; if $t_3=x$ the root is triple and the divisor is $3P=2P+R$, while otherwise it is $2P+R$. In both cases step 1.1 and $\mathrm{\Phi }([0])=O$ give $2P\oplus R=\mathrm{\Phi }([z]+[z]+[-2z])=\mathrm{\Phi }([0])=O$.