Modular Functions and Special Cycles

Abstract

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.<br /> 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 <em>L</em>-functions.<br /> 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.<br /> 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].<br /> 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 <em>f = k(1 − k)f</em>, where k is a positive integer. Such functions were introduced in [35]. Also it was conjectured in [35] and [36] that these functions have “algebraic” values at CM points. A precise formulation of the conjecture is given in the introduction. thesis [60]. In Chapter 5 we prove this conjecture for any pair of CM points lying in the same quadratic imaginary field. Our proof has two main parts. First, we show that the regularized Petersson scalar product of a binary theta series with a weight one weakly holomorphic cusp form is the logarithm of the absolute value of an algebraic number. Second, we prove that the special values of weight k Green’s function occurring in the conjecture can be written as Petersson product of this type, where the form of weight one is the (<em>k</em> − 1)-st Rankin-Cohen bracket of an explicit weakly holomorphic modular form of weight 2 − 2<em>k</em> with a binary theta series. The algebraicity of regularized Petersson products was proved independently at about the same time and by different method by W. Duke and Y. Li [23]; however, our result is stronger since we also give a formula for the factorization of the algebraic number in the number field to which it belongs.