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

Abstract

Let $E$ be an elliptic curve parametrized by a newform $\phi \in S_2(\Gamma_0(N))$. Let $k$ be a quadratic field of discriminant $d_k$ prime to $N$. Given a character $\chi$ of the ideal class group of $k$ with theta series $\theta(\chi)$, we relate the central derivative values $\Lambda'(E/k, \chi, 1) = \Lambda'(1/2, \phi \times \theta(\chi))$ of the $\chi$-twisted $L$-function of $E/k$ to arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N)\times X_0(N)$ when $k$ is imaginary quadratic, and to sums of Green's functions of Hirzebruch-Zagier divisors along real geodesic cycles of $X_0(N) \times X_0(N)$ determined by ideal classes when $k$ is real quadratic. More generally, we refine the higher Gross-Zagier formulae for CM cycles on spin Shimura varieties of any dimension this way. This gives two arithmetic height formulae for $\Lambda'(E/k, \chi, 1)$ when $k$ is imaginary quadratic, as well as a distinct proof of the Gross-Zagier formula, with implied relations between the arithmetic heights of Heegner divisors on $X_0(N)$ and Hirzebruch-Zagier divisors on $X_0(N) \times X_0(N)$. We also explain connections to the refined conjecture of Birch-Swinnerton-Dyer, and to Fourier coefficients of half-integral weight forms.