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In this paper we reinterpret the main results of [8] using the intersection theory of cycles with coefficients. To this end we give a functorial interpretation of Eichler-Schimura periods. Introduction. In this paper we reformulate the main results of S. Katok [8] in terms of homology theory with local coefficients. In order to do so we Srst give an interpretation of the period of a cusp form f of weight 2m + 2 (m > 1) over a closed geodesic [^y] as the Kronecker index of the cohomoiogy class Sh f with local coefficients associated to f by Shimura [13] and a 1-cycle with dual local coefficients associated to [^y]. We will call this period the Eichler-Shimura period. There are other interpretations of Eichler-Shimura periods but these seem to work only for arithmetic fundamental groups whereas the defining integral of the period makes sense quite generally. There is a simple idea from group homology theory which is the basis of our reformulation that of a "decomposable" cycle. We now give the definition. Suppose a is a representation of a group r into GL(V) where V is a finite dimensional vector space. Then we may consider the Eilenberg-Mac Lane homology group Hl(r,). We observe that if ey is an element of r and v is an element of V satisfying a(^y)v = v (a ey-invariant) then the 1-chain oy 3 v satisfies a(ty X v) = a(ty)v-v = o. Consequently, ty X v is a 1-cycle and gives rise to an element of Hl(r, a) which we also denote oy Q3 v. DEFINITIONS. We will call such cycles decomposable cycles and we will say a homology class is decomposable if it has a decomposable representative. REMARK. A general 1-cycle is of the form E.1=1 aj X vj where n n , 5(tyj)vj , vj = O. j=l j=l For more details on homology and cohomology of groups see Brown [3]. We can now give our formulation of Eichler-Shimura periods. Let V be the space of the standard 2-dimensional complex representation of SL2(R) and V* be its dual. The corresponding real space we will denote by VR. We let SmV denote Received by the editors January 31, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary llF06, 20H10, 55N25; Secondary 22E40. The second author was partially supported by an NSF grant. t1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Eichler-Shimura homology, intersection numbers and rational structures on spaces of modular forms