Finite Hecke orbit criterion for holomorphy at a cusp
= Finite Hecke orbit criterion for holomorphy at a cusp
Let a level-one modular function be holomorphic on the upper half-plane. If the span of $f,T_pf,T_p^2f,\ldots$ is finite-dimensional, then $f$ is holomorphic at infinity. Indeed, a pole of order $N$ would make $T_p^rf$ have pole order $p^rN$, producing linearly independent functions.