= Double-coset operator on modular forms
{title2=$[A\alpha B]f=\sum_j f[\alpha_j]_k$}
Decompose $A\alpha B=\coprod_j A\alpha_j$ for finite-index congruence subgroups $A,B$. The sum is independent of representative choices because $f$ is invariant under $A$. Right multiplication by $B$ permutes these left cosets, so the sum is invariant under $B$. Each summand is holomorphic in the half-plane; <cusp holomorphy under rational slash operators> proves holomorphy at every new cusp. The same reasoning preserves vanishing at cusps. With the <Hecke-normalized rational slash operator>, no extra scalar is needed in the defining sum.
Back to article page
