Elliptic Functions and Complex Tori

A full complex lattice (briefly, a lattice in this pair) is a subgroup $\mathrm{\Lambda }\subseteq \mathbb{C}$ of the form

$\mathrm{\Lambda }=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2=\{m{\omega }_1+n{\omega }_2:m,n\in \mathbb{Z}\},$

where ${\omega }_1,{\omega }_2\in \mathbb{C}$ are real-linearly independent: the only $(a,b)\in {\mathbb{R}}^2$ with $a{\omega }_1+b{\omega }_2=0$ is $(a,b)=(0,0)$. The pair $({\omega }_1,{\omega }_2)$ is then a lattice basis of $\mathrm{\Lambda }$, and it is oriented when

$\mathrm{Im}\text{\hspace{0.17em}⁣}\left(\frac{{\omega }_2}{{\omega }_1}\right)>0.$

The complex torus of $\mathrm{\Lambda }$ is the quotient

$T_{\mathrm{\Lambda }}:=\mathbb{C}\mathrm{/}\mathrm{\Lambda }=\{[z]:=z+\mathrm{\Lambda }:z\in \mathbb{C}\}$

carrying the quotient topology of the class map ${\pi }_{\mathrm{\Lambda }}:\mathbb{C}\to T_{\mathrm{\Lambda }}$, $z\mapsto [z]$ (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection): a subset $W\subseteq T_{\mathrm{\Lambda }}$ is open exactly when ${\pi }_{\mathrm{\Lambda }}^{-1}(W)$ is open in $\mathbb{C}$. Since $\mathrm{\Lambda }$ is a subgroup, the formula

$[z]+[w]:=[z+w]$

is well defined — if $z^{\mathrm{\prime }}=z+\lambda$ and $w^{\mathrm{\prime }}=w+\mu$ with $\lambda ,\mu \in \mathrm{\Lambda }$, then $z^{\mathrm{\prime }}+w^{\mathrm{\prime }}=z+w+(\lambda +\mu )$ with $\lambda +\mu \in \mathrm{\Lambda }$ — and makes $T_{\mathrm{\Lambda }}$ an abelian group with identity $[0]$ and inverse $-[z]=[-z]$. The class map is then a surjective group homomorphism with kernel $\mathrm{\Lambda }$.

Real-linear independence of ${\omega }_1,{\omega }_2$ is equivalent to $\mathrm{Im}({\omega }_2\mathrm{/}{\omega }_1)\mathrm{\ne }0$: if $a{\omega }_1+b{\omega }_2=0$ with real $(a,b)\mathrm{\ne }(0,0)$, then $b\mathrm{\ne }0$ (otherwise $a{\omega }_1=0$ forces $a=0$) and ${\omega }_2\mathrm{/}{\omega }_1=-a\mathrm{/}b\in \mathbb{R}$; conversely ${\omega }_2\mathrm{/}{\omega }_1=r\in \mathbb{R}$ gives ${\omega }_2-r{\omega }_1=0$. Consequently every lattice admits an oriented basis: if $\mathrm{Im}({\omega }_2\mathrm{/}{\omega }_1)<0$ one exchanges the two basis vectors and uses $\mathrm{Im}({\omega }_1\mathrm{/}{\omega }_2)>0$.

Change of basis. If $({\omega }_1,{\omega }_2)$ and $({\omega }_1^{\mathrm{\prime }},{\omega }_2^{\mathrm{\prime }})$ are two bases of the same lattice $\mathrm{\Lambda }$, then writing ${\omega }_j^{\mathrm{\prime }}={\sum }_k{a}_{kj}{\omega }_k$ exhibits the transition matrix $A=(a_{kj})\in M_2(\mathbb{Z})$, and the same argument applied to the inverse change of basis returns the inverse matrix, so $A\in {\mathrm{G}\mathrm{L}}_2(\mathbb{Z})$, that is, $\det A=\pm 1$. Thus two oriented bases of one lattice differ by a matrix in ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})=\{A\in M_2(\mathbb{Z}):\det A=1\}$: this is what makes the orientation condition, and not the particular basis, a property of the pair $(\mathrm{\Lambda },\text{orientation})$.

Dependence only on the lattice. The quotient $T_{\mathrm{\Lambda }}$, its topology, its abelian group structure and the class map depend on $\mathrm{\Lambda }$ alone and not on a chosen basis: a change of basis leaves the set $\mathrm{\Lambda }$, hence the equivalence relation $z\sim w⟺z-w\in \mathrm{\Lambda }$, unchanged. The oriented basis in the definition is a bookkeeping device for the orientation convention $\mathrm{Im}({\omega }_2\mathrm{/}{\omega }_1)>0$ used later when roots, half-periods and signs are named. The complex structure that upgrades $T_{\mathrm{\Lambda }}$ from a group with a topology to a Riemann surface is constructed in the next item of this page, where the discreteness of $\mathrm{\Lambda }$ in $\mathbb{C}$ is also proved.

Let $\mathrm{\Lambda }=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2\subseteq \mathbb{C}$ be a full complex lattice with oriented basis $({\omega }_1,{\omega }_2)$, and let $T_{\mathrm{\Lambda }}=\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ carry the quotient topology of the class map $\pi :\mathbb{C}\to T_{\mathrm{\Lambda }}$, $\pi (z)=[z]$ (Complex lattice and quotient torus). Then:

1. the charts inverse to the injective restrictions of $\pi$ to small balls form a holomorphic atlas on $T_{\mathrm{\Lambda }}$: each is a homeomorphism onto an open subset of $\mathbb{C}$, and any two are compatible;
2. $T_{\mathrm{\Lambda }}$ is Hausdorff, second countable and compact, hence a compact Riemann surface;
3. $\pi$ is a holomorphic covering map.

The atlas depends only on $\mathrm{\Lambda }$ as a subset of $\mathbb{C}$: neither the choice of a representative of a class nor the choice of the oriented basis $({\omega }_1,{\omega }_2)$ enters its definition.

Let $\mathrm{\Lambda }\subseteq \mathbb{C}$ be a full complex lattice with oriented basis $({\omega }_1,{\omega }_2)$ (Complex lattice and quotient torus). The Weierstrass $\mathrm{\wp }$-function of $\mathrm{\Lambda }$ is the function

${\mathrm{\wp }}_{\mathrm{\Lambda }}(z):=\frac{1}{z^2}+{\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}),z\in \mathbb{C}\setminus \mathrm{\Lambda },$

where the sum over the lattice points different from $0$ is the unordered (finite-subset) sum: for the directed set of finite subsets $F\subseteq \mathrm{\Lambda }\setminus \{0\}$ ordered by inclusion, one forms the net of partial sums ${\sum }_{\omega \in F}((z-\omega {)}^{-2}-{\omega }^{-2})$, and ${\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(\cdots )$ denotes its limit when the net converges and the limit does not depend on the directed set — equivalently, when the family is absolutely summable, i.e. when ${\sup }_F{\sum }_{\omega \in F}\mid (z-\omega {)}^{-2}-{\omega }^{-2}\mid <\mathrm{\infty }$ and the partial sums converge. The following theorem of this page proves that for every $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ the family is absolutely summable, with normal (locally uniform, enumeration-free) convergence on $\mathbb{C}\setminus \mathrm{\Lambda }$; this is the sense in which ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ is well defined beyond the displayed formula. In particular no ordering of $\mathrm{\Lambda }$ is used and the value does not depend on one.

The two correction terms. The summand $\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}=\frac{2z\omega -z^2}{(z-\omega {)}^2{\omega }^2}$ is holomorphic in $z$ on the disc $\mathrm{\mid }z\mathrm{\mid }<\mathrm{\mid }\omega \mathrm{\mid }$, so each summand is holomorphic near the origin; at $z=0$ its value is $(-\omega {)}^{-2}-{\omega }^{-2}=0$. The single uncorrected term $z^{-2}$ therefore supplies the entire principal part at the lattice point $0$, and the subtractions make the remaining series vanish at $0$: the constant term of the Laurent expansion of ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ at $0$ is $0$. At a general lattice point $\lambda \in \mathrm{\Lambda }$ the same computation after the translation $z\mapsto z+\lambda$ shows that the principal part of ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ at $\lambda$ is $(z-\lambda {)}^{-2}$.

Translation and parity. Reindexing the sum by $\omega \mapsto -\omega$ — a bijection of $\mathrm{\Lambda }\setminus \{0\}$ — replaces $(z-\omega {)}^{-2}-{\omega }^{-2}$ by $(z+\omega {)}^{-2}-{\omega }^{-2}$ and $z$ by $-z$, which is the same expression; consequently the absolutely convergent sum satisfies ${\mathrm{\wp }}_{\mathrm{\Lambda }}(-z)={\mathrm{\wp }}_{\mathrm{\Lambda }}(z)$ once its convergence is known, and ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ is an even function. The sum depends only on the lattice $\mathrm{\Lambda }$, not on the oriented basis chosen to describe it, since the underlying index set $\mathrm{\Lambda }$ and every summand depend on $\mathrm{\Lambda }$ alone.

Finite-subset convergence and absolute summability. For a complex family $(a_i{)}_{i\in I}$, use the real and imaginary parts and modulus of Real and imaginary parts, complex conjugation, and modulus. Absolute summability implies convergence of its finite-subset net by Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$. For the reverse direction, suppose the finite-subset sums $s_F:={\sum }_{i\in F}{a}_i$ converge to $s\in \mathbb{C}$. Choose a finite $F_0$ such that $\mathrm{\mid }{s}_F-s\mathrm{\mid }<1$ whenever $F\supseteq F_0$; then $\mathrm{\mid }{s}_F\mathrm{\mid }\le \mathrm{\mid }s\mathrm{\mid }+1$ for every such $F$. For any finite set $P\subseteq I\setminus F_0$ on which $\mathrm{Re}(a_i)>0$, ${\sum }_{i\in P}\mathrm{Re}(a_i)=\mathrm{Re}(s_{F_0\cup P}-s_{F_0})\le \mathrm{\mid }s\mathrm{\mid }+1+\mathrm{\mid }{s}_{F_0}\mathrm{\mid }\mathrm{.}$ The same bound holds for finite sums of $-\mathrm{Re}(a_i)$ over negative terms, and likewise for the imaginary parts. Adding the finitely many terms in $F_0$ shows that the finite subsums of $\mathrm{\mid }\mathrm{Re}(a_i)\mathrm{\mid }$ and $\mathrm{\mid }\mathrm{Im}(a_i)\mathrm{\mid }$ are bounded. Since $\mathrm{\mid }{a}_i\mathrm{\mid }\le \mathrm{\mid }\mathrm{Re}(a_i)\mathrm{\mid }+\mathrm{\mid }\mathrm{Im}(a_i)\mathrm{\mid }$, the finite subsums of $\mathrm{\mid }{a}_i\mathrm{\mid }$ are bounded, so the family is absolutely summable by the definition in Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$. This argument uses no enumeration or choice principle.

The cubic tail. For $\mathrm{\mid }z\mathrm{\mid }\le R$ and $\mathrm{\mid }\omega \mathrm{\mid }\ge 2R$ the displayed numerator is bounded by $2R\mathrm{\mid }\omega \mathrm{\mid }+R^2$ and the denominator is bounded below by $\frac{1}{4}\mathrm{\mid }\omega {\mathrm{\mid }}^4$, so the summand is $O_R(\mathrm{\mid }\omega {\mathrm{\mid }}^{-3})$. The lattice alone therefore controls the size of the terms, and the convergence proof only has to count how many lattice vectors occur at each scale; that count and the resulting normal convergence are proved in the next items of this page.

Let $\mathrm{\Lambda }\subseteq \mathbb{C}$ be a full complex lattice and let $\pi :\mathbb{C}\to T_{\mathrm{\Lambda }}=\mathbb{C}\mathrm{/}\mathrm{\Lambda }$, $\pi (z)=[z]$, be the quotient map (Complex lattice and quotient torus), so that $T_{\mathrm{\Lambda }}$ is a compact Riemann surface and $\pi$ is a holomorphic covering map (The quotient $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is a compact Riemann surface).

A $\mathrm{\Lambda }$-elliptic function is a meromorphic function $f:\mathbb{C}\to \widehat{\mathbb{C}}$ on the plane (Meromorphic functions on a plane domain, the Riemann-sphere convention of Holomorphic maps and meromorphic functions on Riemann surfaces) satisfying the periodicity condition

$f(z+\lambda )=f(z)\text{for all }z\in \mathbb{C}\text{ and all }\lambda \in \mathrm{\Lambda },$

where both sides are values in $\widehat{\mathbb{C}}$: the equation $f(z)=\mathrm{\infty }$ is allowed, and it is required that $z+\lambda$ is a pole exactly when $z$ is, with the same $f$-value $\mathrm{\infty }$. Equivalently, $f$ is the pullback

$f=g\circ \pi$

of a meromorphic function $g$ on the Riemann surface $T_{\mathrm{\Lambda }}$; since $\pi$ is surjective such a $g$ is unique, and changing a representative of a class changes $z$ by an element of $\mathrm{\Lambda }$, under which $f$ is invariant by periodicity. The functions $g$ and $f=g\circ \pi$ are called the torus form and the plane form of the same elliptic function.

The period group of a meromorphic $f:\mathbb{C}\to \widehat{\mathbb{C}}$ is

$\mathrm{Per}(f):=\{\omega \in \mathbb{C}:f(z+\omega )=f(z)\text{ for all }z\in \mathbb{C}\};$

it is a subgroup of $\mathbb{C}$, and $f$ is $\mathrm{\Lambda }$-elliptic exactly when $\mathrm{\Lambda }\subseteq \mathrm{Per}(f)$. The period group need not equal $\mathrm{\Lambda }$: if ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ is a lattice containing $\mathrm{\Lambda }$ then every ${\mathrm{\Lambda }}^{\mathrm{\prime }}$-elliptic function is $\mathrm{\Lambda }$-elliptic, so the same function can be elliptic for several lattices.

Descent and compatibility. If $f$ is $\mathrm{\Lambda }$-elliptic, the formula $g([z]):=f(z)$ is well defined because a different representative is $z+\lambda$, and the local expressions of $g$ in the quotient charts are local expressions of $f$, which are holomorphic or have a pole; since $\pi$ is a covering map, every point of $T_{\mathrm{\Lambda }}$ has a chart inverse to a bijective restriction of $\pi$, so $g$ is holomorphic as a map $T_{\mathrm{\Lambda }}\to \widehat{\mathbb{C}}$ away from the image of the poles and has poles there. Conversely $g\circ \pi$ is $\mathrm{\Lambda }$-periodic, and $f\mapsto g$ and $g\mapsto g\circ \pi$ are mutually inverse, so the two descriptions coincide.

Field structure. Sums, products, quotients with denominator not identically zero and constant multiples of $\mathrm{\Lambda }$-elliptic functions are again $\mathrm{\Lambda }$-elliptic, and the $\mathrm{\Lambda }$-elliptic functions form a subfield of the field of all meromorphic functions on $\mathbb{C}$ (Meromorphic functions on a connected plane domain form a field); equivalently they are the meromorphic functions on the compact torus $T_{\mathrm{\Lambda }}$. Constants are elliptic, and they are the only $\mathrm{\Lambda }$-elliptic functions with no poles: a holomorphic (pole-free) $\mathrm{\Lambda }$-periodic function is bounded on the compact fundamental domain and hence constant, by Liouville's theorem. This last statement is proved with the divisor laws in the next items of the page.

Zeros and poles. For a $\mathrm{\Lambda }$-elliptic function $f\equiv ̸0$, its zeros and poles are $\mathrm{\Lambda }$-invariant: $f(z+\lambda )=f(z)$ shows that $z$ is a zero or pole of a given order exactly when $z+\lambda$ is. Since their classes form closed, isolated subsets of the compact torus $T_{\mathrm{\Lambda }}$, only finitely many classes of zeros and poles occur; this finiteness is used when the divisor of a nonzero elliptic function is formed. The allowed zero function has every point as a zero and has no divisor of isolated zeros.

Let $\mathrm{\Lambda }\subseteq \mathbb{C}$ be a full complex lattice, and let the sum over $\mathrm{\Lambda }\setminus \{0\}$ and the product over $\mathrm{\Lambda }\setminus \{0\}$ be the unordered finite-subset limits of Weierstrass p function: the net of partial sums, respectively partial products, over the finite subsets ordered by inclusion, when it converges independently of the exhaustion. The Weierstrass $\zeta$-function and $\sigma$-function of $\mathrm{\Lambda }$ are

${\zeta }_{\mathrm{\Lambda }}(z):=\frac{1}{z}+{\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(\frac{1}{z-\omega }+\frac{1}{\omega }+\frac{z}{{\omega }^2}),z\in \mathbb{C}\setminus \mathrm{\Lambda },$

${\sigma }_{\mathrm{\Lambda }}(z):=z{\prod }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}{E}_2\text{\hspace{0.17em}⁣}\left(\frac{z}{\omega }\right)=z{\prod }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(1-\frac{z}{\omega })\exp \text{\hspace{0.17em}⁣}(\frac{z}{\omega }+\frac{z^2}{2{\omega }^2}),z\in \mathbb{C},$

where $E_2(w)=(1-w)e^{w+w^2\mathrm{/}2}$ is the second Weierstrass elementary factor (Weierstrass elementary factors). The following theorem of this page proves that the defining net converges normally on $\mathbb{C}\setminus \mathrm{\Lambda }$ for ${\zeta }_{\mathrm{\Lambda }}$ and on $\mathbb{C}$ for ${\sigma }_{\mathrm{\Lambda }}$, so that both functions are well defined and depend only on the set $\mathrm{\Lambda }$.

A lattice element $\omega \in \mathrm{\Lambda }$ is primitive when it is part of a $\mathbb{Z}$-basis of $\mathrm{\Lambda }$, equivalently when $\omega \mathrm{/}n\mathrm{\notin }\mathrm{\Lambda }$ for every integer $n\ge 2$. For a primitive period $\omega$ put

${\eta }_{\omega }:=2{\zeta }_{\mathrm{\Lambda }}\text{\hspace{0.17em}⁣}\left(\frac{\omega }{2}\right)\mathrm{.}$

The quasi-period laws

${\zeta }_{\mathrm{\Lambda }}(z+\omega )={\zeta }_{\mathrm{\Lambda }}(z)+{\eta }_{\omega },{\sigma }_{\mathrm{\Lambda }}(z+\omega )=-\exp \text{\hspace{0.17em}⁣}\left({\eta }_{\omega }(z+\frac{\omega }{2})\right){\sigma }_{\mathrm{\Lambda }}(z)$

for every primitive $\omega \in \mathrm{\Lambda }$ are proved in Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions ↗, together with ${\zeta }_{\mathrm{\Lambda }}^{\mathrm{\prime }}=-{\mathrm{\wp }}_{\mathrm{\Lambda }}$, ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}\mathrm{/}{\sigma }_{\mathrm{\Lambda }}={\zeta }_{\mathrm{\Lambda }}$, ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}(0)=1$, the oddness of ${\zeta }_{\mathrm{\Lambda }}$ and ${\sigma }_{\mathrm{\Lambda }}$, and the fact that ${\sigma }_{\mathrm{\Lambda }}$ is entire with simple zeros exactly at the lattice points.

Why the corrections. The summand of ${\zeta }_{\mathrm{\Lambda }}$, $\frac{1}{z-\omega }+\frac{1}{\omega }+\frac{z}{{\omega }^2}=\frac{z^2}{(z-\omega ){\omega }^2},$ vanishes to second order at $z=0$, so the single term $1\mathrm{/}z$ carries the whole principal part there; at a general lattice point the translation $z\mapsto z+\lambda$ exhibits the principal part $(z-\lambda {)}^{-1}$. Similarly the factor $E_2(z\mathrm{/}\omega )$ has a simple zero at $z=\omega$ and no other zero, and its expansion $\log E_2(w)=-w^3\mathrm{/}3-w^4\mathrm{/}4-\cdots$ shows that the correction $e^{z\mathrm{/}\omega +z^2\mathrm{/}(2{\omega }^2)}$ is exactly what makes the product converge on compact sets. Both facts are proved in the items named above.

Normalisation. The factor $z$ in front of ${\sigma }_{\mathrm{\Lambda }}$ is chosen so that $z=0$ is a simple zero and ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}(0)=1$; the constants ${\eta }_{\omega }$ are the analogues of the half-period values $\mathrm{\wp }(\omega \mathrm{/}2)$, and for an oriented basis $({\omega }_1,{\omega }_2)$ the Legendre relation ${\eta }_{{\omega }_1}{\omega }_2-{\eta }_{{\omega }_2}{\omega }_1=2\pi i$ holds; it is proved in Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions ↗. Extend the quasi-period constants additively from the chosen basis by ${\eta }_{m{\omega }_1+n{\omega }_2}:=m{\eta }_1+n{\eta }_2$. For an arbitrary period $\omega =m{\omega }_1+n{\omega }_2$, iteration gives ${\zeta }_{\mathrm{\Lambda }}(z+\omega )={\zeta }_{\mathrm{\Lambda }}(z)+{\eta }_{\omega },{\sigma }_{\mathrm{\Lambda }}(z+\omega )=(-1{)}^{m+n+mn}\exp \text{\hspace{0.17em}⁣}\left({\eta }_{\omega }(z+\frac{\omega }{2})\right){\sigma }_{\mathrm{\Lambda }}(z)\mathrm{.}$ The sign is $-1$ when $\omega$ is primitive: then $\gcd (m,n)=1$, so $m,n$ are not both even and $m+n+mn$ is odd. For a primitive period, the additive constant agrees with the displayed definition ${\eta }_{\omega }=2{\zeta }_{\mathrm{\Lambda }}(\omega \mathrm{/}2)$, by applying the zeta translation law at $z=-\omega \mathrm{/}2$ and using oddness.

Primitive-period criterion. Fix a $\mathbb{Z}$-basis $({\omega }_1,{\omega }_2)$ of $\mathrm{\Lambda }$ and write $\omega =a{\omega }_1+b{\omega }_2$ with $a,b\in \mathbb{Z}$. For $\omega =0$, both basis membership and the no-divisor condition fail. If $\omega \mathrm{\ne }0$, put $d=\gcd (a,b)>0$. Uniqueness of coordinates shows that $\omega \mathrm{/}n\in \mathrm{\Lambda }$ for an integer $n\ge 2$ exactly when $n$ divides both $a$ and $b$; since $d$ divides both coordinates and every positive common divisor is at most $d$ (Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$), no such $n$ exists exactly when $d=1$. By Bézout's identity: for integers $a,b$ not both zero, $\gcd (a,b)$ is the least positive element of $\{ax+by:x,y\in \mathbb{Z}\}$; in particular $ax+by=\gcd (a,b)$ has an integer solution, in that case there are integers $x,y$ with $ax+by=1$. Then $\nu =-y{\omega }_1+x{\omega }_2$ belongs to $\mathrm{\Lambda }$, and the coordinate matrix of $(\omega ,\nu )$ has determinant $ax+by=1$, so $(\omega ,\nu )$ is a $\mathbb{Z}$-basis. Conversely, if $\omega$ is a member of a $\mathbb{Z}$-basis and $\omega \mathrm{/}n\in \mathrm{\Lambda }$, writing $\omega \mathrm{/}n$ in that basis would make its coordinate on $\omega$ equal to $1\mathrm{/}n$, not an integer.

Let $\mathrm{\Lambda }=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2\subseteq \mathbb{C}$ be a full complex lattice with oriented basis $({\omega }_1,{\omega }_2)$, and let

${\mathrm{\wp }}_{\mathrm{\Lambda }}(z)=\frac{1}{z^2}+{\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2})$

be the Weierstrass $\mathrm{\wp }$-function of Weierstrass p function. Then:

1. the sum converges absolutely at every $z\in \mathbb{C}\setminus \mathrm{\Lambda }$ and uniformly on every compact subset of $\mathbb{C}\setminus \mathrm{\Lambda }$, so it is normally convergent there and independent of any enumeration of $\mathrm{\Lambda }$;
2. ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ is holomorphic on $\mathbb{C}\setminus \mathrm{\Lambda }$, is even (${\mathrm{\wp }}_{\mathrm{\Lambda }}(-z)={\mathrm{\wp }}_{\mathrm{\Lambda }}(z)$) and is $\mathrm{\Lambda }$-periodic (${\mathrm{\wp }}_{\mathrm{\Lambda }}(z+\lambda )={\mathrm{\wp }}_{\mathrm{\Lambda }}(z)$ for every $\lambda \in \mathrm{\Lambda }$ and every $z\in \mathbb{C}$, with poles matched), so it is a $\mathrm{\Lambda }$-elliptic function;
3. at each lattice point $\lambda \in \mathrm{\Lambda }$ the function ${\mathrm{\wp }}_{\mathrm{\Lambda }}$ has a double pole with principal part $(z-\lambda {)}^{-2}$, and it has no other poles;
4. ${\mathrm{\wp }}_{\mathrm{\Lambda }}^{\mathrm{\prime }}(z)=-2{\sum }_{\omega \in \mathrm{\Lambda }}(z-\omega {)}^{-3}$ on $\mathbb{C}\setminus \mathrm{\Lambda }$, this series being normally convergent there, and the derivative ${\mathrm{\wp }}_{\mathrm{\Lambda }}^{\mathrm{\prime }}$ is odd and $\mathrm{\Lambda }$-elliptic as well.