Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

Abstract

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map j^{dr}_n on elliptic and hyperelliptic curves over number fields via their universal unipotent connections {mathscr {U}}. Several algorithms forming part of the computation of finite level versions j^{dr}_n of the unipotent Albanese maps are presented. The computation of the logarithmic extension of {mathscr {U}} in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on {mathscr {U}} over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.