Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions

We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear to Fourier coefficients of certain half-integral weight forms.