QUADRATIC CHABAUTY AND p-ADIC GROSS-ZAGIER

Let X be a quotient of the modular curve X0(N) whose Jacobian JX is a simple factor of J0(N)new over Q. Let f be the newform of level N and weight 2 associated with JX; assume f has analytic rank 1. We give analytic methods for determining the rational points of X using quadratic Chabauty by computing two p-adic Gross-Zagier formulas for f. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033-1148], which lies outside of the range of interpolation.