Computation of the Unipotent Albanese Map on Elliptic and Hyperelliptic Curves

Beacom, Jamie

doi:10.48550/arxiv.1711.03932

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2021

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“…In §10.2, we explicitly describe the local Selmer variety and unipotent Kummer map for X = E ′ and n = 3. In particular, we review results of [Bea17]. In §10.3, we explain why Π is semisimple as a motive, which simplifies many calculations.…”

Section: When (1)mentioning

confidence: 99%

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Explicit Motivic Mixed Elliptic Chabauty-Kim

Corwin¹

2021

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The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen-Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of an element of the Chabauty-Kim ideal for Z[1/ℓ]-points on a punctured elliptic curve, and lay some groundwork for certain kinds of higher genus curves. For this purpose, we develop an "explicit Tannakian Chabauty-Kim method" using Q ℓ -Tannakian categories of Galois representations in place of Q-linear motives. In future work, we intend to use this method to explicitly apply the Chabauty-Kim method to a curve of positive genus in a situation where Quadratic Chabauty does not apply.

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Section: When (1)mentioning

confidence: 99%

“…We refer to results of [Bea17] for the coordinate ring. In the notation of loc.cit., we have α 0 = e 0 , α 1 = e 1 , F = y x , and λ = −2.…”

Section: Proof Setmentioning

confidence: 99%