Computation of unipotent Albanese maps on elliptic and hyperelliptic curves
<p>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 <em>p</em>-adic de Rham period map <em>j<sub>n</sub><sup>dr</sup></em> on elliptic and h...
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2019
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| Summary: | <p>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 <em>p</em>-adic de Rham period map <em>j<sub>n</sub><sup>dr</sup></em> on elliptic and hyperelliptic curves over number fields via their universal unipotent connections <em>U</em>. </p> <p>Several algorithms forming part of the computation of finite level versions <em>j<sub>n</sub><sup>dr</sup></em> of the unipotent Albanese maps are presented. The computation of the logarithmic extension of <em>U</em> 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 result due to Vologodsky and Hadian on the computation of the Hodge filtration on <em>U</em> over affine elliptic and odd hyperelliptic curves. </p> <p>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 <em>p</em>-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. </p> <p>Finally, we present the output of implementations of the aforementioned algorithms in Magma for small level on both elliptic curves and hyperelliptic curves. </p> |
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