Closed geodesics, periods and arithmetic of modular forms

Abstract

is the upper half-plane and F is a discrete cocompact subgroup of SL(2,N) acting by fractional linear transformations. We demonstrate that in this case the periods over closed geodesics play a role somewhat similar to that of Fourier coefficients of modular forms on SL(2,Z) and its congruence subgroups. More specifically, those periods uniquely determine a modular form (Theorem 2). This result is valid for cusp forms on any Fuchsian group of the first kind with or without cusps and is closely related to the study of relative Poincar6 series associated to closed geodesics. For each integer k > 2 and each closed geodesic [7o] we define special cusp forms of weight 2k, called relative Poincar6 series Ok, t~ol and prove that they generate the whole space S2k(F ) of cusp forms (Theorem 1). In w 3 we give an expression for periods of a relative Poincar6 series over closed geodesics in purely geometrical terms through the intersection of the corresponding geodesics (Theorem 3). An application of Theorems 1 and 3 to arithmetic subgroups of SL(2,1R) gives two natural rational structures o n S2k(l N) (Theorem 4). The relative Poincar6 series have been studied for general F by Petersson [18, 19] and Hejhal [5] (g>2). Wolpert [23] gives a basis of S4(F ) for g>2. For SL(2, TZ) the relative Poincar6 series have been studied by Zagier [25], Kohnen [8], Kohnen and Zagier [9], and Kramer [13]. A related problem of constructing cusp forms of weight two associated to closed geodesics has been treated by Kudla and Millson in [143. In connection with the problem of