The modular symbol and continued fractions in higher dimensions

Abstract

The modular symbol has proved a very useful tool for dealing with holomorphic automorphic forms on the upper half-plane. It serves, for instance, to compute the action of Hecke operators on automorphic forms, to give the value of twisted L-functions of certain elliptic curves in terms of a finite sum of modular symbols, and to prove that divisors of degree zero supported on the cusps of a modular curve have finite order in the Jacobian of that curve. For these matters, see [1], Chaps. IV and V, and the references cited there. Here is a definition of the modular symbol. Let H be the upper half-plane, that is, the set of all complex numbers with positive imaginary part. A cusp of H is defined to be any rational number (considered as a point of the real line) or oc. The cusps belong to the boundary of H in the extended complex plane. Let SL(2,1R) act on H in the usual way. For any subgroup F of finite index in SL(2, 9 ) we can form the quotient H/F. There is a closed, compact Riemann surface X(F) such that H/F is biholomorphic to the complement of some finite set of points x 1 . . . . ,x~ in X(F). These points x i are called the cusps of X(F). Let ~?(F) denote the holomorphic differential l-forms on X(F). Integration of forms over cycles yields a canonical N-linear map from H~(X(F);IR) to the complex dual space of ~(F). The theory of Riemann surfaces tells us that this map is an isomorphism. Now choose any two rational cusps a and b on the boundary of H. Let l be any reasonable arc beginning at a, ending at b, and staying in H in the interim for instance, l could be the geodesic in H from a to b. Then one easily shows that integration over the image of l in X(F) gives a linear map from ~?(F) to I12. This linear map gives an element of HI(X(F); IR) in the canonical way described above. We denote this homology class by [a, hi. We call [ ] the modular symbol. In summary, the "classical" modular symbol is a map from the set of pairs of cusps of H into H~(X(F);IR). This symbol enjoys properties which are the