Elliptic function for a lattice

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.