Eichler cohomology and automorphic forms

Abstract

From now on we assume that r is a nonnegative integer and that v is a multiplier system on r with respect to the degree r. (Note that v is then a multiplier system with respect to the degree-r 2 and V is also a multiplier system with respect to the degrees r and -r 2.) A well-known result due to Bol [1] states that dr+ (2) dzr+ (cz -+d)rE (Vz) (cz + d)-r-2FCr+) (Vz), for any V ( ), with ad bc 1. This can be proved either by induc- tion on r or directly by the use of Cauchy's integral formula. It follows im- mediately from (2) that if F e {P, r, v/, then