Geometric quadratic Chabauty and p-adic heights

Let X be a curve of genus g > 1 over Q whose Jacobian J has Mordell-Weil rank r and Neron-Severi rank rho. When r < g+ rho - 1, the geometric quadratic Chabauty method determines a finite set of p-adic points containing the rational points of X. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of p-adic heights and p-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of p-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.(c) 2023 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).