Computation of the Unipotent Albanese Map on Elliptic and Hyperelliptic Curves

TL;DR

This paper develops algorithms for explicitly computing the unipotent Albanese map on elliptic and hyperelliptic curves using $p$-adic de Rham periods, advancing the non-abelian Chabauty method with new examples and computational techniques.

Contribution

It introduces new algorithms for computing the unipotent Albanese map on elliptic and hyperelliptic curves, including the use of descent theory and explicit period calculations.

Findings

Algorithms for finite level unipotent Albanese maps are presented.

New examples of period map coordinates using $p$-adic Coleman integrals are provided.

Computations with tangential basepoints are demonstrated.

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 $\mathcal{U}$. 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 $\mathcal{U}$ 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 $\mathcal{U}$ 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.