Balakrishnan, Jennifer S.Hashimoto, Sachi2022-11-012022-11-012022https://hdl.handle.net/2144/45282Let X be an Atkin-Lehner quotient of the modular curve X_0(N) whose Jacobian J_f is a simple quotient of J_0(N)^{new} over Q. We give analytic methods for determining the rational points of X using quadratic Chabauty by explicitly computing two p-adic Gross--Zagier formulas for the newform f of level N and weight 2 associated with J_f when f has analytic rank 1. Combining results of Gross-Zagier and Waldspurger, one knows that for certain imaginary quadratic fields K, there exists a Heegner divisor in J_0(N)(K) whose image is finite index in J_f(Q) under the action of Hecke. We give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna, assuming some hypotheses on the prime p and on K. This value is proportional to the logarithm of the Heegner divisor on J_f with respect to the differential form f dq/q. We also compute the p-adic height of the Heegner divisor on J_f using a p-adic Gross-Zagier formula of Perrin-Riou. Additionally, we give algorithms for the geometric quadratic Chabauty method of Edixhoven and Lido. Our algorithms describe how to translate their algebro-geometric method into calculations involving Coleman-Gross heights, logarithms, and divisor arithmetic. We achieve this by leveraging a map from the Poincaré biextension to the trivial biextension.en-USMathematicsHeegner pointsHeightsModular curvesp-adic L-functionsQuadratic ChabautyRational pointsThesis/Dissertation2022-10-280000-0002-8936-5545