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Beschreibung:
In this thesis we study algebraic cycles on Shimura varieties of orthogonal type. Such varieties are a higher dimensional generalization of modular curves and their important feature is that they have natural families of algebraic cycles in all codimesions. We mostly concentrate on low-dimensional examples: Heegner points on modular curves, Hirzebruch- Zagier cycles on Hilbert surfaces, Humbert surfaces on Siegel modular threefolds. In Chapter 2 we compute the restriction of Siegel Eisenstein series of degree 2 and more generally of Saito-Kurokawa lifts of elliptic modular forms to Humbert varieties. Using these restriction formulas we obtain certain identities for special values of symmetric square L -functions. In Chapter 3 a more general formula for the restriction of Gritsenko lifts to Humbert varieties is obtained. Using this formula we complete an argument which was given in a conjectural form in [76] (assertion on p. 246) giving a much more elementary proof than the original one of [36] that the generating series of classes of Heegner points in the Jacobian of a modular curve is a modular form. In Chapter 4 we present computations that relate the heights of Heegner points on modular curves and Heegner cycles on Kugo-Sato varieties to the Fourier coefficients of Siegel Eisenstein series of degree 3. This was the problem originally suggested to me as a thesis topic, and I was able to obtain certain results which are described here. Some of the results of this chapter overlap some of those given in the recent book [53]. succeed in calculating all terms completely, and also, similar results appeared in the recent book [53]. The main result of the thesis is contained in Chapter 5. In this chapter we study CM values of higher Green’s functions. Higher Green’s functions are real-valued functions of two variables on the upper half-plane which are bi-invariant under the action of a congruence subgroup, have a logarithmic singularity along the diagonal and satisfy Δ f = k(1 − k)f , where k is a positive integer. ...