I saw the statement in the question from the book Moonshine Beyond the Monster.
We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation.
I can't understand why the author said quosi-periodicity is an example of projective representation:
$\quad$ Let $\theta_3(\tau,z, u) = \underset{n \in \mathbb{Z}}{\sum}e^{\pi i\tau n^2 + 2\pi i zn + 2\pi i u}$ for $\tau, z,u\in \mathbb{C}$.Fix $\tau \in \mathbb{H}$ (upper-half plane) and $u \in \mathbb{C}$, $\quad$ and consider the function as a function of $z$ only. Then it has period $1$ and quasi-period $\tau$: $$\theta_3(\tau, z + m\tau + l, u + mz + \frac{1}{2}m^2\tau)= \theta_3(\tau, z, u)$$ $\quad$ Author states it (I think function only depending on $z$) thus lives (projectively) on the torus.
$\quad$ Moreover, on page $177$ author says above quasi-periodicity relation is an example projective $\quad$ representation of the abelian $\quad$ group $\mathbb{C}^2$ on the space of functions $f:\mathbb{C}\rightarrow \mathbb{C}$.
My problem is that, I can't understand how does the definition of projective rep. is used here. Could anyone give a hint?