In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are given in terms of cycle integrals of the modular j-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coefficient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coefficients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.

Abstract. To an ideal class of a real quadratic field we associate a certain surface. This surface, which is a new geometric invariant, has the usual modular closed geodesic as its boundary. Furthermore, its area is determined by the length of an associated backward continued fraction. We study the distribution properties of this surface on average over a genus. In the process we give an extension and refinement of the Katok-Sarnak formula.

No abstract

No abstract

In this paper we give some applications of weakly holomorphic forms and their cycle integrals to rational period functions for the modular group. In particular, we give a simple construction of the associated modular integrals that works for all weights.