Complex lattice and quotient torus

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.