Die Deutsche Bibliothek -CIP-Einheitsaufnahme Bruinier, Jan Hendrik: Borcherds products on 0(2,1) and Chern classes of Heegner divisors / Jan H.

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:

Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of L-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.

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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.