The Chabauty-Kim method aims to provide a ‘non-abelian’ lift of the Chabauty-Coleman method, replacing abelian integrals with higher iterated Coleman integrals. Recent years have exhibited a several instances where the Chabauty-Kim method demonstrates finiteness of rational points: for example work by Ellenberg and Hast on curves geometrically dominating curves with CM Jacobians; Balakrishnan and Dogra on curves with Mordell-Weil rank exceeding genus; Balakrishan, Dogra, Muller, Tuitman and Vonk on the split Cartan modular curve of level 13.

The computation of the de Rham unipotent Albanese map is a key step towards making finiteness statements more explicit. When the Chabauty-Kim method applies, we use the explicit description of this map to describe a p-adic iterated integral on the curve the zero locus of which contains the rational points. This makes it possible to recover the rational points locally analytically. The computation is a multi-step process, which is outlined below.

Here we present several algorithms I have developed towards computation of the finite level versions of this map when the curve X is an elliptic curve punctured at the origin . In particular, we present an algorithm for the computation of the universal unipotent logarithmic connection on X, and the Hodge filtration which is the induced on the universal unipotent connection. This induces a Hodge filtration on the de Rham fundamental group of X, and we utilise this explicit description to compute the unipotent Albanese map. We also conjecture a closed form for the finite level versions of these maps.