Group cohomology for modular forms with singularities

Abstract

For a nonzero divisor D:=∑t=1npDDt of X0(1) with pD>0, let Mk!,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 {D1,…,Dn} and that the order of pole of f at each Q∈{D1,…,Dn} is less than or equal to pQ. In this paper, we give an isomorphism between Mk!,D(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 Pkweak, we prove that there exists an isomorphism between Mk!(SL2(Z)) and H1(SL2(Z),Pkweak), where Mk!(SL2(Z)) denotes the space of weakly holomorphic modular forms of integral weight k on SL2(Z). © 2025 Elsevier Inc.