A Rigid Analytic Gross-Zagier Formula and Arithmetic Applications

Abstract

Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.