Normal convergence, parity and periodicity of the Weierstrass p function

Put $A:=\mathrm{\mid }{\omega }_1{\mathrm{\mid }}^2$, $B:=\mathrm{Re}({\omega }_1\overline{{\omega }_2})$, $C:=\mathrm{\mid }{\omega }_2{\mathrm{\mid }}^2>0$; expanding with [F3] gives $\mathrm{\mid }s{\omega }_1+t{\omega }_2{\mathrm{\mid }}^2=A{s}^2+2Bst+C{t}^2$ for real $s,t$, and $AC-B^2=(\mathrm{Im}({\omega }_1\overline{{\omega }_2}){)}^2>0$ because real-linear independence forbids $\mathrm{Im}({\omega }_1\overline{{\omega }_2})=0$. Completing the square in each variable gives $\mathrm{\mid }s{\omega }_1+t{\omega }_2{\mathrm{\mid }}^2\ge (AC-B^2)\max (s^2,t^2)\mathrm{/}\max (A,C)$, so with $\delta :=\sqrt{(AC-B^2)\mathrm{/}\max (A,C)}>0$ one has $\mathrm{\mid }s{\omega }_1+t{\omega }_2\mathrm{\mid }\ge \delta \max (\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }t\mathrm{\mid })$; in particular distinct lattice points are at distance at least $\delta$.

For $R>0$, every $\lambda =m{\omega }_1+n{\omega }_2\in \mathrm{\Lambda }$ with $\mathrm{\mid }\lambda \mathrm{\mid }\le R$ has $\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }n\mathrm{\mid }\le R\mathrm{/}\delta$ by step 1.1, so $\mathrm{\#}(\mathrm{\Lambda }\cap \bar{B}(0,R))\le (2R\mathrm{/}\delta +1{)}^2$; hence $\mathrm{\Lambda }\cap \bar{B}(0,R)$ is finite and $\mathrm{\Lambda }$ has no accumulation point. The nonzero lattice points with $\mathrm{\mid }\omega \mathrm{\mid }<1$ are therefore finite. For each $k\ge 0$, the shell $2^k\le \mathrm{\mid }\omega \mathrm{\mid }<2^{k+1}$ has at most $(2^{k+2}\mathrm{/}\delta +1{)}^2$ points, so its contribution to $\sum \mathrm{\mid }\omega {\mathrm{\mid }}^{-3}$ is at most $(2^{k+2}\mathrm{/}\delta +1{)}^2{2}^{-3k}$; these bounds form a convergent series, with terms $O(2^{-k})$. Thus ${\sum }_{\omega \mathrm{\ne }0}\mathrm{\mid }\omega {\mathrm{\mid }}^{-3}<\mathrm{\infty }$, and its finite-subset sums have arbitrarily small tails.

For an empty compact $K$, the uniform convergence assertion is vacuous. Otherwise let $K\subseteq \mathbb{C}\setminus \mathrm{\Lambda }$ be nonempty and compact and choose $R\ge \max (1,{\sup }_{z\in K}\mathrm{\mid }z\mathrm{\mid })$. For $\mathrm{\mid }\omega \mathrm{\mid }\ge 2R$ and $z\in K$, the displayed formula for $h_{\omega }$ gives [F1, F3, F4, step 2.1] $\mathrm{\mid }{h}_{\omega }(z)\mathrm{\mid }=\frac{\mathrm{\mid }2z\omega -z^2\mathrm{\mid }}{\mathrm{\mid }z-\omega {\mathrm{\mid }}^2\mathrm{\mid }\omega {\mathrm{\mid }}^2}\le \frac{\frac{5}{2}R\mathrm{\mid }\omega \mathrm{\mid }}{\frac{1}{4}\mathrm{\mid }\omega {\mathrm{\mid }}^4}=\frac{10R}{\mathrm{\mid }\omega {\mathrm{\mid }}^3}\mathrm{.}$ Choose a finite $F_0$ containing all $\omega$ with $\mathrm{\mid }\omega \mathrm{\mid }<2R$ and with the remaining finite-subset tails of $\sum \mathrm{\mid }\omega {\mathrm{\mid }}^{-3}$ smaller than $\epsilon \mathrm{/}(10R)$. Then for finite $F^{\mathrm{\prime }\mathrm{\prime }}\supseteq F^{\mathrm{\prime }}\supseteq F_0$, ${\sup }_{z\in K}\mid {\sum }_{\omega \in F^{\mathrm{\prime }\mathrm{\prime }}\setminus F^{\mathrm{\prime }}}{h}_{\omega }(z)\mid \le 10R{\sum }_{\omega \in F^{\mathrm{\prime }\mathrm{\prime }}\setminus F^{\mathrm{\prime }}}\mathrm{\mid }\omega {\mathrm{\mid }}^{-3}<\epsilon \mathrm{.}$ Thus the finite-subset net is uniformly Cauchy on $K$ and pointwise absolutely convergent; its limit is independent of enumeration. Since this holds on every compact subset of $\mathbb{C}\setminus \mathrm{\Lambda }$, the sum is normally convergent there, and $\mathrm{\wp }$ is well defined by [F1].

$\mathbb{C}\setminus \mathrm{\Lambda }$ is path-connected. Let $x,y\in \mathbb{C}\setminus \mathrm{\Lambda }$. By step 2.1, the segment $[x,y]$ meets $\mathrm{\Lambda }$ in finitely many points $p_1,\dots ,p_N$, in their order along the segment. If $N=0$, the segment is already a path in the complement. Otherwise choose $r>0$ smaller than $\delta \mathrm{/}3$ and than every distance from a $p_j$ to either endpoint. The discs $B(p_j,r)$ are disjoint, contain no other lattice points, and neither endpoint lies in them. Replace the subsegment through each $p_j$ by one of the two arcs on $\mathrm{\partial }B(p_j,r)$ joining its endpoints. Each arc avoids the lattice, and the remaining straight pieces contain no lattice point; the resulting finite path joins $x$ to $y$ in $\mathbb{C}\setminus \mathrm{\Lambda }$.

The lattice $\mathrm{\Lambda }$ is at most countable: the map $(m,n)\mapsto m{\omega }_1+n{\omega }_2$ from $\mathbb{Z}\times \mathbb{Z}$ onto $\mathrm{\Lambda }$ is surjective by [F2] and $\mathbb{Z}\times \mathbb{Z}$ is at most countable by [F6], so [F6] gives a surjection $s:\mathbb{N}\to \mathrm{\Lambda }\setminus \{0\}$. For each lattice point retain only its least preimage, $j(\omega )=\min \{k:s(k)=\omega \}$, as in [F6]. The image of $j$ is an infinite subset of $\mathbb{N}$ (the distinct points $n{\omega }_1$, $n\ge 1$, already form an infinite subset of the target); list that image in increasing order, recursively taking its least unused element. This list exhausts the image because every natural number has only finitely many predecessors. Applying $s$ gives a repetition-free enumeration $({\omega }_j{)}_{j\ge 0}$ of $\mathrm{\Lambda }\setminus \{0\}$. Every finite subset is contained in a sufficiently long initial segment of this enumeration; the partial sums $S_N(z):=z^{-2}+{\sum }_{j<N}{h}_{{\omega }_j}(z)$ are holomorphic on $\mathbb{C}\setminus \mathrm{\Lambda }$, and step 3.1 makes them converge locally uniformly to $\mathrm{\wp }$. By [F5] the limit $\mathrm{\wp }$ is holomorphic on $\mathbb{C}\setminus \mathrm{\Lambda }$ and ${\mathrm{\wp }}^{\mathrm{\prime }}(z)=-2z^{-3}+{\sum }_{j\ge 0}(-2)(z-{\omega }_j{)}^{-3}=-2{\sum }_{\omega \in \mathrm{\Lambda }}(z-\omega {)}^{-3}$, the last series converging locally uniformly on $\mathbb{C}\setminus \mathrm{\Lambda }$ because $\mathrm{\mid }z-\omega \mathrm{\mid }\ge \frac{1}{2}\mathrm{\mid }\omega \mathrm{\mid }$ for $\mathrm{\mid }\omega \mathrm{\mid }\ge 2{\max }_K\mathrm{\mid }z\mathrm{\mid }$ gives $\mathrm{\mid }z-\omega {\mathrm{\mid }}^{-3}\le 8\mathrm{\mid }\omega {\mathrm{\mid }}^{-3}$ and step 2.1 applies.

Evenness. For every finite $F\subseteq \mathrm{\Lambda }\setminus \{0\}$ one has ${\sum }_{\omega \in F}{h}_{\omega }(-z)={\sum }_{\omega \in F}((z+\omega {)}^{-2}-{\omega }^{-2})={\sum }_{{\omega }^{\mathrm{\prime }}\in -F}((z-{\omega }^{\mathrm{\prime }}{)}^{-2}-{\omega }^{\mathrm{\prime }-2})={\sum }_{{\omega }^{\mathrm{\prime }}\in -F}{h}_{{\omega }^{\mathrm{\prime }}}(z)$, and $F\mapsto -F$ is a bijection of the directed set of finite subsets; since the net converges by step 3.1, the two limits agree and $\mathrm{\wp }(-z)=\mathrm{\wp }(z)$ for every $z\in \mathbb{C}\setminus \mathrm{\Lambda }$, the case $z\in \mathrm{\Lambda }$ being the statement that poles correspond.

At each lattice point $\lambda$ the principal part is $(z-\lambda {)}^{-2}$ and there is no other pole. For $\lambda =0$, choose $R<\delta \mathrm{/}2$. Every $h_{\omega }$ is holomorphic on $\mathrm{\mid }z\mathrm{\mid }\le R$, and the bound from step 3.1 together with $\sum \mathrm{\mid }\omega {\mathrm{\mid }}^{-3}<\mathrm{\infty }$ gives uniform convergence on this disc. Since each $h_{\omega }(0)=0$, the sum is holomorphic near $0$ and vanishes at $0$, so $\mathrm{\wp }(z)-z^{-2}$ extends holomorphically there. For $\lambda \mathrm{\ne }0$, split off $h_{\lambda }$ to obtain [F8, F1, step 3.1, step 1.1] $\mathrm{\wp }(z)-(z-\lambda {)}^{-2}=z^{-2}-{\lambda }^{-2}+{\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0,\lambda \}}{h}_{\omega }(z)\mathrm{.}$ On $B(\lambda ,\delta \mathrm{/}2)$, $z^{-2}$ is holomorphic and every remaining summand is holomorphic, since distinct lattice points are at least $\delta$ apart. The same tail bound from step 3.1, applied on compact subdiscs of this ball after omitting the finitely many nearby terms, gives local uniform convergence of the remaining series there. Thus the right side extends holomorphically to $\lambda$. In both cases $(z-\lambda {)}^2\mathrm{\wp }(z)$ extends holomorphically with value $1$, so [F8] gives a double pole with principal part $(z-\lambda {)}^{-2}$.

The derivative ${\mathrm{\wp }}^{\mathrm{\prime }}$ is $\mathrm{\Lambda }$-periodic. For $\lambda \in \mathrm{\Lambda }$ and $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ one has $z+\lambda \mathrm{\notin }\mathrm{\Lambda }$, and for every finite $F\subseteq \mathrm{\Lambda }$ the substitution $\omega \mapsto \omega +\lambda$ turns ${\sum }_{\omega \in F}(z+\lambda -\omega {)}^{-3}$ into ${\sum }_{{\omega }^{\mathrm{\prime }}\in F-\lambda }(z-{\omega }^{\mathrm{\prime }}{)}^{-3}$; since $\omega \mapsto \omega +\lambda$ is a bijection of $\mathrm{\Lambda }$ and of the directed set of finite subsets, the normally convergent series of step 4.1 gives ${\mathrm{\wp }}^{\mathrm{\prime }}(z+\lambda )=-2{\sum }_{{\omega }^{\mathrm{\prime }}}(z-{\omega }^{\mathrm{\prime }}{)}^{-3}={\mathrm{\wp }}^{\mathrm{\prime }}(z)$.

The basis vectors are periods of $\mathrm{\wp }$. For $j\in \{1,2\}$ the function $g_j(z):=\mathrm{\wp }(z+{\omega }_j)-\mathrm{\wp }(z)$ is holomorphic on $\mathbb{C}\setminus \mathrm{\Lambda }$, because $z+{\omega }_j\in \mathrm{\Lambda }$ exactly when $z\in \mathrm{\Lambda }$ by [F2]; its derivative is $g_j^{\mathrm{\prime }}(z)={\mathrm{\wp }}^{\mathrm{\prime }}(z+{\omega }_j)-{\mathrm{\wp }}^{\mathrm{\prime }}(z)=0$ by step 5.1 and the chain rule [F7], and $\mathbb{C}\setminus \mathrm{\Lambda }$ is a domain by step 3.2, so [F7] makes $g_j$ constant. The point $-{\omega }_j\mathrm{/}2$ lies in $\mathbb{C}\setminus \mathrm{\Lambda }$: otherwise ${\omega }_j\mathrm{/}2=m{\omega }_1+n{\omega }_2$ with $m,n\in \mathbb{Z}$, which for $j=1$ reads ${\omega }_1=2m{\omega }_1+2n{\omega }_2$ and contradicts the uniqueness of the real coordinates in [F2], and similarly for $j=2$. Evaluating there with the evenness of step 4.2 gives $g_j(-{\omega }_j\mathrm{/}2)=\mathrm{\wp }({\omega }_j\mathrm{/}2)-\mathrm{\wp }(-{\omega }_j\mathrm{/}2)=0$, so $g_j\equiv 0$.

Oddness and ellipticity of ${\mathrm{\wp }}^{\mathrm{\prime }}$. The series of step 4.1 is normally convergent, so the substitution $\omega \mapsto -\omega$ may be made in its finite-subset net: ${\mathrm{\wp }}^{\mathrm{\prime }}(-z)=-2{\sum }_{\omega }(-z-\omega {)}^{-3}=-2{\sum }_{\omega }(-(z+\omega ){)}^{-3}=2{\sum }_{\omega }(z+\omega {)}^{-3}=2{\sum }_{{\omega }^{\mathrm{\prime }}}(z-{\omega }^{\mathrm{\prime }}{)}^{-3}=-{\mathrm{\wp }}^{\mathrm{\prime }}(z)$ for every $z\in \mathbb{C}\setminus \mathrm{\Lambda }$. By step 4.3 the poles of ${\mathrm{\wp }}^{\mathrm{\prime }}$ are exactly the lattice points, each of order $3$, so ${\mathrm{\wp }}^{\mathrm{\prime }}$ is meromorphic on $\mathbb{C}$, and it is $\mathrm{\Lambda }$-periodic by step 5.1; hence ${\mathrm{\wp }}^{\mathrm{\prime }}$ is again $\mathrm{\Lambda }$-elliptic by [F9].

Hence $\mathrm{\wp }$ is $\mathrm{\Lambda }$-periodic: the set $\{\lambda \in \mathrm{\Lambda }:\mathrm{\wp }(z+\lambda )=\mathrm{\wp }(z)\text{ for all }z\}$ is a subgroup of $\mathrm{\Lambda }$ containing ${\omega }_1,{\omega }_2$ by step 6.1, so it is all of $\mathrm{\Lambda }$; equivalently $\mathrm{\wp }(z+m{\omega }_1+n{\omega }_2)=\mathrm{\wp }(z)$ for all integers $m,n$ and all $z\in \mathbb{C}\setminus \mathrm{\Lambda }$. Since also $z+\lambda \in \mathrm{\Lambda }$ exactly when $z\in \mathrm{\Lambda }$, the function $\mathrm{\wp }$ is meromorphic on $\mathbb{C}$ with poles matching under translation, so by step 4.1, step 4.3 and [F9] it is a $\mathrm{\Lambda }$-elliptic function.

Collecting: step 3.1 gives the absolute and locally uniform (normal) convergence, independent of enumeration; step 4.1 gives holomorphy and the derivative formula; step 4.3 gives the double poles with principal part $(z-\lambda {)}^{-2}$ and no others; step 4.2 gives evenness and steps 6.1 and 7.1 give $\mathrm{\Lambda }$-periodicity and the elliptic property of $\mathrm{\wp }$; step 6.2 gives the oddness and ellipticity of ${\mathrm{\wp }}^{\mathrm{\prime }}$.