Closed geodesics, periods and arithmetic of modular forms

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

“…In § §5 and 6 we restrict to certain arithmetic groups f and verify that the rational structures on 82m+2(r) constructed by S. Katok [8] coincide with the usual ones coming from Eichler-Shimura theory. The rational structures coming from EichlerShimura theory are described in detail in §5 and compared to those of S. Katok in §6.…”

“…As a consequence the result of S. Katok from [8] may be restated as saying that the decomposable classes span H 1 (f, Eft) We observe that this homological reformulation does not require the existence of a quotient f \ H. It makes sense for arbitrary subgroups f of 8L2(R). Our reformulation is to some degree justified by the fact that Goldman and Millson [5] have proved this algebraic version for any finitely generated, Zariski-dense subgroup of 8 L2 (R).…”

“…The proof of this formula depends on a de Rham interpretation of Hl(f, E) and a simplicial interpretation of H 1 (f , E*). Now if 8 m +1,-y denotes the relative (or hyperbolic) Poincare series given in (1.3) of [8] we have that the Petersson inner product ((f,8m+1,-y)hm+2 is the above Eichler-Shimura period. We find as a consequence of the previous formula the following theorem-see Theorem 4.2 of the text.…”

“…In the final section of our paper we give a reformulation of the period formula, Theorem 3, of S. Katok [8] in terms of an intersection product of decomposable cycles "11 Q9 VI . "12 Q9 V2.…”

Eichler-Shimura Homology, Intersection Numbers and Rational Structures on Spaces of Modular Forms

“…In what follows we will need to choose a nonzero element ~, in E* which is invariant under, in r. If, = [~~] then we define s, in S2V* and ~, in E* following S. Katok [8] by…”