takes values in Z q (X). Then for η ∈ Z (p−1)q c (X), the compactly supported closed (p − 1)qforms on X, the Kudla-Millson lift is defined byIt turns out that Λ KM (τ, η) is actually a holomorphic modular form of weight 2 − k, so that we have a mapwhich also factors through cohomology. Moreover, the Fourier coefficients of Λ KM are given by periods of η over the special cycles.Theorem 1.2. Assume D be Hermitian, i.e., q = 2, and let f ∈ H + k with constant coefficient a + (0). We then have the following identity of closed 2-forms on X:Therefore the maps Λ B and Λ KM are naturally adjoint via the standard pairing ( ,Furthermore, this duality factors through cohomology, and H + k /M ! k , respectively. (See also Theorem 6.3.) This is based on the fundamental relationship between the two theta series involved: Theorem 1.3. Let L 2−k be the lowering Maass operator of weight 2 − k on H. Then L 2−k Θ(τ, z, ϕ KM ) = −dd c Θ(τ, z, ϕ 0 ).We show this by switching to the Fock model of the Weil representation. Then the idea for the proof of Theorem 1.2 is given by the following formal (!) calculation:

Abstract. Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL 2 (Z) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

Abstract. In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle integrals of the modular j invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the 'central value' of the (non-existing) 'L-function' for j. The key to the proofs is the construction of some kind of a simultaneous Green function for both the CM points and the geodesic cycles, which is of independent interest.

The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.