Geometric quadratic Chabauty and [formula omitted]-adic heights

Abstract

Let X be a curve of genus g>1 over Q whose Jacobian J has Mordell–Weil rank r and Néron–Severi rank ρ. When r<g+ρ−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.