An Enhanced Higher Green's Function on the Modular Curve

Abstract

The higher Green's functions $G_{k/2}(z,w)$ on the modular curve are real-valued eigenfunctions of the hyperbolic Laplacian in each variable, with logarithmic singularities along the diagonal. Gross and Zagier conjectured that their values at CM points, after multiplication by suitable powers of the discriminants, are logarithms of absolute values of algebraic numbers. The conjecture was proved by Li, Bruinier, and Yang. Mellit proposed a complex refinement of this conjecture, predicting that the normalized CM values of a complex lift of the Green function are logarithms of algebraic numbers modulo periods. For $\Gamma ={\mathrm{SL}}_2(\mathbb{Z})$ and $k\in 4,6,8,10,14$, we follow Zagier's construction of CM invariants from Eichler integrals of the Eisenstein series $E_4$. By applying Maass raising operators in both variables to a normalized iterated Eichler integral of a Bergman-type kernel, we obtain a complex-valued extension ${\mathcal{G}}_{k/2}^{\bullet }(z,w)$ whose real part is a constant multiple of $G_{k/2}(z,w)$. We study the normalized CM values of ${\mathcal{G}}_{k/2}^{\bullet }$ and describe their dependence on the choice of representatives of quadratic form classes. In particular, we show that replacing the quadratic form corresponding to the first variable by an equivalent one changes the exponential of the normalized value only by a root of unity. We also prove the algebraicity of these exponentials in the special case where the second CM point is $w_{\ast }=(1+i\sqrt{d})/2$, with $d>3$ and $d\equiv 3(\mathrm{mod}4)$, and the class of the quadratic form corresponding to the first variable is equal to its inverse.