Finding Integral Points of Elliptic Curves over Imaginary Quadratic Fields

Abstract

We determine the quadratic Chabauty set for integral points on elliptic curves of rank $2$ defined over imaginary quadratic fields using quadratic Chabauty. This builds on the work of Bianchi and Balakrishnan et al. We give the first instance of the implementation of anticyclotomic heights for curves which are not base changes, along with an implementation of a certain sieve for elliptic curves introduced by Balakrishnan et al. and used by Bianchi to determine integral points of rank $2$. We give the first example of the determination of the integral points of an elliptic curve of rank $2$ defined over an imaginary quadratic field, which is not a base change via quadratic Chabauty.