Group cohomology for modular forms with singularities

For a nonzero divisor D :=Sigma(n)(t=1) pD(t)D(t) of X0(1) with pD(t )> 0, let M-k(!,D)(SL2(Z)) be the space of meromorphic modular forms f of integral weight k on SL2(Z) such that f is holomorphic except at {D-1, ... , D-n} and that the order of pole of f at each Q is an element of {D-1, ... , D-n} is less than or equal to pQ. In this paper, we give an isomorphism between M-!,M-D (k) (SL2(Z)) and the first cohomology group with a certain coefficient module PD 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(!)(SL2(Z)) and H-1(SL2(Z), P-k(weak)),where M-k(!)(SL2(Z)) denotes the space of weakly holomorphic modular forms of integral weight k on SL2(Z).