The Eichler cohomology groups and automorphic forms

Abstract

Introduction. In his papers [5; 6] Eichler demonstrated the significance for the study of automorphic forms of Bol 's discovery [3 ] of some remarkably simple differential operators taking automorphic forms into automorphic forms. In [6] in particular Eichler discussed a relation between the automorphic forms associated to a transformation group 9 on a Riemann surface D and some purely algebraic constructions involving the group 9, the first cohomology groups of 9 with certain modules of polynomials as coefficients; the cocycles appeared as the periods of the automorphic forms under iterated indefinite integration, generalizing the classical interpretation of the periods of the abelian integrals on D/g (which can of course be considered as automorphic forms on D) as cocycles of the group 9 or alternatively of the space ID/9. The object of studying such a relation is the development of tools for calculating the dimensions of spaces of automorphic forms and the traces of the Hecke operators on automorphic forms. The aim of the present paper is the study of a more general form of this relation in somewhat greater detail for one complex variable, but in such a manner that the results can be extended to several complex variables; the actual extension to several complex variables, as well as the application to the study of the Hecke operators, will be discussed elsewhere. As for the contents of this paper, ? 1 is devoted to an exposition of Bol's differential operators in a form more useful in the present context than that of [3]. In ?2 these differential operators are applied to give an exact cohomology sequence containing, in a rather more transparent form, the relation of Eichler discussed above. The interpretations of the terms appearing in this exact sequence are discussed in ??3 through 5; the only point of difficulty arises in ?4, Theorem 3 of that section really being a form of the Serre duality theorem [12] appropriate to the occasion. These results are combined in ?6 to give a formal statement of the fundamental result of the paper. 1. Differential operators preserving automorphic forms. Let SC be the group of 2 X 2 real matrices