Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 29 / 1 / a / Solution

Let $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ be a complex lattice. Orient its basis so that $\mathrm{Im}({\omega }_1/{\omega }_2)>0$. Multiplication by ${\omega }_2^{-1}$ gives the normalized complex lattice $\mathbb{Z}\tau +\mathbb{Z}$ with $\tau ={\omega }_1/{\omega }_2\in \mathbb{H}$. Thus every homothety of complex lattices class has a normalized representative.

If $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, thenThe last equality follows because the two integer vectors are an integral basis: their determinant is one. Hence points in the same modular orbit give homothetic complex lattices.

Conversely, suppose ${\Lambda }_{{\tau }^{\prime }}=\alpha {\Lambda }_{\tau }$ for $\alpha \ne 0$. There is an integer basis-change matrix of determinant $\pm 1$ such thatBoth normalized bases have positive orientation, and multiplication by $\alpha$ preserves real orientation because its real determinant is $|\alpha {|}^2>0$. The basis-change determinant is therefore $+1$. Dividing the two equalities gives ${\tau }^{\prime }=(a\tau +b)/(c\tau +d)$. Thus the modular orbits and homothety of complex lattices classes correspond bijectively.