Marked-lattice model of a modular form
= Marked-lattice model of a modular form
{title2=$F_f(L,t)=\omega_2^{-k}f(\omega_1/\omega_2)$}
Choose an oriented lattice basis with $t=\omega_2/N$ modulo the lattice. Changing such a basis is exactly the <Gamma 1 congruence subgroup> action, so the weight law makes the displayed function independent of the choice. It satisfies $F(uL,ut)=u^{-k}F(L,t)$. Holomorphy on the half-plane and at all <modular cusp> degenerations identifies the functions coming from <modular forms>.