Limiting cycles and periods of Maass forms

The aim of this note is to introduce a new kind of a period for Maass forms. For outsiders of the automorphic world, we recall that in the theory of automorphic functions eigenfunctions of the Laplacian on a finite volume hyperbolic Riemann surface are called Maass forms (after H. Maass [Ma] who realized their importance in Number Theory). There are 3 types of closed cycles naturally appearing in the theory of automorphic functions on the group G = PGL₂ (R) with respect to a lattice Γ ⊂ G: 1) closed horocycles which are associated to closed orbits in the automorphic space X = Γ \ G of a unipotent subgroup N ⊂ G; 2) closed geodesics and geodesic rays starting and ending in a cusp (both of these types are associated to closed orbits on X of the diagonal subgroup A ⊂ G); and 3) closed geodesics circles which are associated to orbits on X of a maximal compact subgroup K ⊂ G. Periods of Maass forms along all these cycles play a crucial role in Number Theory. We propose to consider one more period along a special type of a non-closed orbit of the subgroup A, i.e., along a non-closed special geodesic on the corresponding Riemann surface. These geodesics will have closed geodesics as their limit sets. Our justification for introducing such cycles is that (generalized) periods of Maass forms along these geodesics satisfy nice analytic properties. Periods (with characters) of Hecke-Maass forms along “classical” cycles lead to Fourier coefficients of cusp forms and to L-functions (e.g., the Hecke L-function given by a period along the geodesic ray connecting two cusps and a special value of a quadratic base change L-function from a theorem of J.-L. Waldspurger [W] appearing as the square of a period along a closed geodesic), and hence play an important role in Number Theory. Admittedly, we do not know yet what is the arithmetic meaning of these new periods (although their residues are connected to periods along closed geodesics and hence to special values of L-functions).