Number of zero-modes on magnetized $T^4\mathrm{/}{Z}_N$ orbifolds analyzed by modular transformation

We study fermion zero-mode wavefunctions on $T^4\mathrm{/}{Z}_N$ orbifold with background magnetic fluxes. The number of zero-modes is analyzed by use of $Sp(4,\mathbb{Z})$ modular transformation. Conditions needed to realize three generation models are clarified. We also study parity transformation in the compact space which leads to better understanding of relationship between positive and negative chirality wavefunctions.