Consistency condition of multiplier system

Question

I have recently started reading modular forms and while reading I got to know that we are interested in holomorphic functions F on the upper half plane which satisfies the transformation law $$ F(Mz)=v(M)j(M,z)^{k}F(z),\quad\forall z \in \mathbb{H}, $$ $M \in \gamma$ , where $\gamma$ is some subgroup of finite index in full modular group and $j(M,z)=(cz+d)$ with $$ M=\begin{bmatrix} a & b \\ c & d \end{bmatrix}. $$ After that I deduce that if $v$ is a multiplier associated with non-constant functions respecting the above transformation law then $v$ satisfies $$ v(M_1M_2)j(M_1M_2,z)^k= v(M_1)v(M_2)j(M_1,M_2z)^kj(M_2,z)^k, $$ for all $M_1,M_2 \in \gamma$ and $z\in \mathbb{H}$. Author defined this property as consistency condition of the multiplier v. My question is why its called consistency condition. What's consistency here of multiplier. Please explain.

Answer 1

(Using your notation): let's examine $$ f(M_1 M_2 z)$$ in two different ways. On the one hand, $$ f(M_1 M_2 z) = v(M_1 M_2) j(M_1 M_2, z)^k f(z).$$ On the other hand, we have that $$ f(M_1 M_2 z) = f\big( M_1 (M_2 z) \big) = v(M_1) j(M_1, M_2 z)^k f(M_2 z) = v(M_1) j(M_1, M_2 z)^k v(M_2) j(M_2, z)^k f(z),$$ and thus we must have that $$ v(M_1 M_2) j(M_1 M_2, z)^k = v(M_1) v(M_2) j(M_1, M_2z)^k j(M_2, z)^k,$$ which is your consistency condition.