The field of elliptic functions is generated by $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$

(Even and odd parts.) Let $\sigma (z):=-z$. In the field of meromorphic functions on $\mathbb{C}$ put $h:=\frac{1}{2}(f+f\circ \sigma )$ and $k:=\frac{1}{2}(f-f\circ \sigma )$: since $\sigma$ is a biholomorphism of $\mathbb{C}$, the composite $f\circ \sigma$ is meromorphic by [F1], so $h$ and $k$ are meromorphic by the field property in [F1], and $f=h+k$. Both are $\mathrm{\Lambda }$-periodic because $f$ is: for $\lambda \in \mathrm{\Lambda }$ one has $h(z+\lambda )=\frac{1}{2}(f(z+\lambda )+f(-z-\lambda ))=\frac{1}{2}(f(z)+f(-z))=h(z)$ and likewise for $k$. Finally $h\circ \sigma =h$ and $k\circ \sigma =-k$, that is, $h$ is even and $k$ is odd.

(Even meromorphic functions near $0$ are functions of the square.) Let $G$ be meromorphic on a disc $\mathrm{\mid }u\mathrm{\mid }<\rho$ and even, $G(-u)=G(u)$ for all $u\mathrm{\ne }0$ in that disc. Then there is a meromorphic $\mathrm{\Gamma }$ near $0$ with $G(u)=\mathrm{\Gamma }(u^2)$ for all small $u\mathrm{\ne }0$. Indeed, by [F8] the pole set of $G$ is discrete and a pole at $0$ has a finite order, so there are $0<{\rho }^{\mathrm{\prime }}\le \rho$ and an integer $m\ge 0$ such that $G$ has no poles in $0<\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$ and $u^{m}G(u)$ is holomorphic on $\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$; by the Taylor expansion in [F8] there are $b_n\in \mathbb{C}$ with $u^{m}G(u)={\sum }_{n\ge 0}{b}_n{u}^n$ for $\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$, and dividing by $u^m$ exhibits $G(u)={\sum }_{n\ge -m}{a}_n{u}^n$ on the annulus $0<\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$, with $a_n:=b_{m+n}$ for $n\ge -m$ and $a_n:=0$ for $n<-m$; this is the locally uniformly convergent Laurent expansion of $G$ on that annulus by [F8]. Evenness says ${\sum }_n{a}_n(-u{)}^n={\sum }_n{a}_n{u}^n$ for $0<\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$, so $a_n=(-1{)}^n{a}_n$ for every $n$ by uniqueness of Laurent coefficients [F8], and $a_n=0$ for every odd $n$. Hence $G(u)={\sum }_i{a}_{2i}{u}^{2i}=\mathrm{\Gamma }(u^2)$ for $0<\mathrm{\mid }u\mathrm{\mid }<{\rho }^{\mathrm{\prime }}$, where $\mathrm{\Gamma }(w):={\sum }_{i\ge \lceil -m\mathrm{/}2\rceil }{a}_{2i}{w}^i$ is a Laurent series with finitely many negative powers converging for $0<\mathrm{\mid }w\mathrm{\mid }<{\rho }^{\mathrm{\prime }2}$, thus a meromorphic function of $w$ near $0$.

(Local structure of $\mathrm{\wp }$ at $0$ and at the half-periods, and at regular points.) (a) By the principal-part clause of [F2], $\mathrm{\wp }(z)=z^{-2}+E_0(z)$ with $E_0$ holomorphic near $0$; $E_0$ is even because $\mathrm{\wp }$ is, so by step 1.2 there is a holomorphic ${\tilde{E}}_0$ near $0$ with $E_0(z)={\tilde{E}}_0(z^2)$, and hence $\frac{1}{\mathrm{\wp }(z)}=\frac{z^2}{1+z^2{E}_0(z)}=z^2{r}_0(z^2)$ with $r_0(w):=1\mathrm{/}(1+w{\tilde{E}}_0(w))$ holomorphic near $0$ and $r_0(0)=1\mathrm{\ne }0$. (b) For each $j$ the function $u\mapsto \mathrm{\wp }(h_j+u)$ is even in $u$, because $\mathrm{\wp }$ is even and $2h_j\in \mathrm{\Lambda }$: $\mathrm{\wp }(h_j+u)=\mathrm{\wp }(-h_j-u)=\mathrm{\wp }(h_j-u)$; by [F6] and [F3] the order of the zero of $\mathrm{\wp }(h_j+u)-e_j$ at $u=0$ equals $e_{[h_j]}(\bar{\mathrm{\wp }})=2$, so by [F7] $\mathrm{\wp }(h_j+u)-e_j=u^{2}g(u)$ with $g$ holomorphic near $0$ and $g(0)\mathrm{\ne }0$, and evenness in $u$ forces $g(u)=\tilde{g}(u^2)$ with $\tilde{g}(0)=g(0)\mathrm{\ne }0$ by step 1.2; thus $\mathrm{\wp }(h_j+u)-e_j=u^2{r}_j(u^2)$ with $r_j$ holomorphic near $0$ and $r_j(0)\mathrm{\ne }0$. (c) If $z_0\mathrm{\notin }\mathrm{\Lambda }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$, then $\mathrm{\wp }$ is holomorphic near $z_0$ with ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$, so by [F6] it is biholomorphic between a neighbourhood of $z_0$ and a neighbourhood of $\mathrm{\wp }(z_0)$, with a holomorphic local inverse.

(An even elliptic function descends through $\mathrm{\wp }$.) Let $h$ be even and $\mathrm{\Lambda }$-elliptic. Define $F:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ by $F(\mathrm{\wp }(z)):=h(z)$: this is well defined because every $a\in \widehat{\mathbb{C}}$ is a value of $\mathrm{\wp }$ by [F3], and if $\mathrm{\wp }(z)=\mathrm{\wp }(w)$ then $w\equiv \pm z$ modulo $\mathrm{\Lambda }$ by [F3], so $h(w)=h(z)$ by evenness and periodicity. $F$ is not identically $\mathrm{\infty }$, since $h$ is meromorphic and hence not the constant map $\mathrm{\infty }$ by [F1]. $F$ is meromorphic: (a) if $a\mathrm{\notin }\{\mathrm{\infty },e_1,e_2,e_3\}$, choose $z_0$ with $\mathrm{\wp }(z_0)=a$; then $z_0\mathrm{\notin }\mathrm{\Lambda }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z_0)\mathrm{\ne }0$, because otherwise the zero set description in [F3] gives $z_0\equiv h_j$ and $a=e_j$, contrary to the choice of $a$; by step 2.1(c) let $\phi$ be a holomorphic local inverse of $\mathrm{\wp }$ near $a$; then $F=h\circ \phi$ near $a$, a composite of holomorphic maps to $\widehat{\mathbb{C}}$, which is meromorphic because $h$ is meromorphic and not constant $\mathrm{\infty }$; (b) if $a=e_j$, put $u=z-h_j$; by step 2.1(b) $\mathrm{\wp }(h_j+u)-e_j={\eta }_j(u^2)$ for ${\eta }_j(w):=w{r}_j(w)$ holomorphic near $0$ with ${\eta }_j(0)=0$ and ${\eta }_j^{\mathrm{\prime }}(0)=r_j(0)\mathrm{\ne }0$, hence biholomorphic near $0$ by [F6], and by step 1.2 applied to the even meromorphic function $H_j(u):=h(h_j+u)$ there is a meromorphic $G_j$ with $H_j(u)=G_j(u^2)$ near $0$; for small $s$ the equation $s=\mathrm{\wp }(h_j+u)-e_j={\eta }_j(u^2)$ is solved by $u^2={\eta }_j^{-1}(s)$, so $F(e_j+s)=h(h_j+u)=G_j(u^2)=G_j({\eta }_j^{-1}(s))$ is meromorphic in $s$; (c) if $a=\mathrm{\infty }$, then $\mathrm{\wp }(z)=\mathrm{\infty }$ exactly for $z\in \mathrm{\Lambda }$. By step 2.1(a), $w:=1\mathrm{/}\mathrm{\wp }(z)={\eta }_0(z^2)$ with ${\eta }_0(t):=t{r}_0(t)$ holomorphic near $0$ and ${\eta }_0^{\mathrm{\prime }}(0)=1\mathrm{\ne }0$, so ${\eta }_0$ has a holomorphic local inverse by [F6]. The even meromorphic function $h$ has, by step 1.2, a meromorphic germ $\mathrm{\Gamma }$ with $h(z)=\mathrm{\Gamma }(z^2)$ near $0$ (including the case $h\equiv 0$). Hence in the source chart $w=1\mathrm{/}a$ at $a=\mathrm{\infty }$, the descended function is $F(1\mathrm{/}w)=\mathrm{\Gamma }({\eta }_0^{-1}(w))$, meromorphic near $w=0$. Thus $F$ is meromorphic at $\mathrm{\infty }$ too.

(The even part is rational in $\mathrm{\wp }$.) By step 3.1 the map $F$ is meromorphic on $\widehat{\mathbb{C}}$ and not identically $\mathrm{\infty }$, so by [F5] there are coprime polynomials $P,Q$, not both zero, with $F=P\mathrm{/}Q$ on the finite chart. Let $R:=P\mathrm{/}Q\in \mathbb{C}(x)$, viewed as the meromorphic map $\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ it defines; $F$ and $R$ are continuous by [F1] and agree on the dense open set $\mathbb{C}\setminus \{s:Q(s)=0\}$, so $F=R$ on $\widehat{\mathbb{C}}$. Hence $h(z)=F(\mathrm{\wp }(z))=R(\mathrm{\wp }(z))$ for every $z\in \mathbb{C}$: every even $\mathrm{\Lambda }$-elliptic function is rational in $\mathrm{\wp }$.

(The odd part.) Let $k$ be odd and $\mathrm{\Lambda }$-elliptic. The quotient $k\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}$ is meromorphic on $\mathbb{C}$ by [F1], because $k$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ are meromorphic and ${\mathrm{\wp }}^{\mathrm{\prime }}\equiv ̸0$ by [F2]; it is even, $(k\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }})(-z)=k(-z)\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}(-z)=(-k(z))\mathrm{/}(-{\mathrm{\wp }}^{\mathrm{\prime }}(z))=(k\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }})(z)$, using that ${\mathrm{\wp }}^{\mathrm{\prime }}$ is odd by [F2], and it is $\mathrm{\Lambda }$-periodic because $k$ is $\mathrm{\Lambda }$-periodic and ${\mathrm{\wp }}^{\mathrm{\prime }}$ is by [F2]. Being even and $\mathrm{\Lambda }$-elliptic, $k\mathrm{/}{\mathrm{\wp }}^{\mathrm{\prime }}=S(\mathrm{\wp })$ for some $S\in \mathbb{C}(x)$ by step 4.1, hence $k={\mathrm{\wp }}^{\mathrm{\prime }}\cdot S(\mathrm{\wp })$: every odd $\mathrm{\Lambda }$-elliptic function is of this form.

(Assembly.) Let $f$ be any $\mathrm{\Lambda }$-elliptic function and write $f=h+k$ as in step 1.1 with $h$ even and $k$ odd. By step 4.1 $h=R(\mathrm{\wp })$ for some $R\in \mathbb{C}(x)$ and by step 5.1 $k={\mathrm{\wp }}^{\mathrm{\prime }}S(\mathrm{\wp })$ for some $S\in \mathbb{C}(x)$, so $f=R(\mathrm{\wp })+{\mathrm{\wp }}^{\mathrm{\prime }}S(\mathrm{\wp })$. Conversely, if $R,S\in \mathbb{C}(x)$, then $R\circ \mathrm{\wp }$ and $S\circ \mathrm{\wp }$ are meromorphic as composites of holomorphic maps $\mathbb{C}\to \widehat{\mathbb{C}}$ read in the extended sense at the poles of $\mathrm{\wp }$, ${\mathrm{\wp }}^{\mathrm{\prime }}$ is meromorphic by [F2], and sums and products of meromorphic functions are meromorphic by [F1]; the sum is $\mathrm{\Lambda }$-periodic because $\mathrm{\wp }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}$ are, so it is $\mathrm{\Lambda }$-elliptic. Therefore the field of $\mathrm{\Lambda }$-elliptic functions is exactly $\mathbb{C}(\mathrm{\wp },{\mathrm{\wp }}^{\mathrm{\prime }})$, which by [F1] is $K(T_{\mathrm{\Lambda }})$ under pullback, and [F4] gives the cubic relation $({\mathrm{\wp }}^{\mathrm{\prime }}{)}^2=4{\mathrm{\wp }}^3-g_2\mathrm{\wp }-g_3$. ∎