Group cohomology for modular forms with singularities

Abstract

For a nonzero divisor $D:={\sum }_{t=1}^n{p}_{D_t}{D}_t$ of $X_0\left(1\right)$ with $p_{D_t}>0$, let $M_k^{!,D}\left({\mathrm{SL}}_2\right(Z\left)\right)$ be the space of meromorphic modular forms f of integral weight k on ${\mathrm{SL}}_2\left(Z\right)$ such that f is holomorphic except at $\{D_1,\dots ,D_n\}$ and that the order of pole of f at each $Q\in \{D_1,\dots ,D_n\}$ is less than or equal to $p_Q$. In this paper, we give an isomorphism between $M_k^{!,D}\left({\mathrm{SL}}_2\right(Z\left)\right)$ and the first cohomology group with a certain coefficient module $P_D$ when k is a negative even integer. More generally, by considering another coefficient module $P_k^{weak}$, we prove that there exists an isomorphism between $M_k^{!}\left({\mathrm{SL}}_2\right(Z\left)\right)$ and $H^1\left({\mathrm{SL}}_2\right(Z),P_k^{weak})$, where $M_k^{!}\left({\mathrm{SL}}_2\right(Z\left)\right)$ denotes the space of weakly holomorphic modular forms of integral weight k on ${\mathrm{SL}}_2\left(Z\right)$.