Geometry and arithmetic cycles attached to [formula omitted]—I

Abstract

IN THE study of the arithmetics of modular curves and in the classical theory of automorphic forms on the upper half plane with respect to subgroups of SL,(Z), modular symbols have served as an indispensable tool linking geometry and arithmetic (e.g. [S], [l 11). There is a natural generalization of this notion to the case of an arithmetic subgroup r of a reductive algebraic Q-group G as, for example, suggested by Mazur [ 123. The construction produces families of relative cycles in the arithmetic quotient X/T, where X denotes the symmetric space attached to G. In analogy with the classical case, there then arise various problems concerning these cycles; but they have not yet been solved in any great generality. In particular, one may ask under what conditions one obtains non-vanishing homology classes for X/‘T in this way, or, what is the relation between modular symbols and automorphic forms with respect to r, especially those which contribute to the cohomology of r. Moreover, can one get some information on special values of L-functions associated to automorphic forms out of modular symbols? (cf. [S], [12]). Being modest in our aim, we will focus on the case G = SL3/Q and a torsion-free subgroup r of finite index in SL, (Z). T’hen r acts properly on the associated symmetric space X = SO(3)\SL,(Iw), and the quotient X/r is a five-dimensional, non-compact, complete Riemannian manifold of finite volume. Recall that X/r may be viewed as the interior of a compact manifold R/r with boundary and that the inclusion is a homotopy equivalence [4]. Given an admissible parabolic Q-subgroup P of SL3 (R) one has the associated modular symbol (cf. 92), which gives rise to relative cycles of degree 2 (3) if P is a Bore1 subgroup (resp. a maximal parabolic one). One knows that H2 (B/r, Z(J?/T),C) is generated by cycles of the first type [2]. This paper is mainly devoted to the study of the geometry of modular symbols in the second case and their non-trivial contribution to the cohomology of r (cf. 5.2, 5.3). This result will be achieved by showing that the modular cycles in question have a positive intersection number with certain associated compact cycles in X/r. These so-called hyperbolic cycles are two-dimensional submanifolds C(6) in X/r attached to indefinite ternary quadratic forms b over Q, and C(6) is compact if b does not represent zero over Q. This natural idea to show that the modular cycles are cohomologically non-zero was originally carried out by Millson and Raghunathan [13], [14] in the context of compact arithmetic quotients of, for example, SO(p,q). By analysing the intersection number of hyperbolic cycles with complementary hyperbolic ones they proved that there are uniform arithmetic subgroups, so that the hyperbolic cycles considered in the corresponding locally symmetric space are non-bounding. On the other hand, such hyperbolic cycles also occur naturally as fix point components of involutions (cf. [lo], [13], [15]). From the results of Rohlfs [ 151, the fix point components of an involution on X/r can be parametrized by the first non-abelian cohomology set H’ (9, r) attached to g = { 1, p) (cf. 94 for definition). Since a modular cycle in X/T may also be interpreted as a fix point component of a certain