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2020 Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture
Francesca Bianchi
Algebra Number Theory 14(9): 2369-2416 (2020). DOI: 10.2140/ant.2020.14.2369

Abstract

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets 𝒳(p)2 containing the integral points 𝒳() of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference 𝒳(p)2𝒳(). We also consider some algorithmic questions arising from Balakrishnan and Dogra’s explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell.

Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the p-adic sigma function in place of a double Coleman integral.

Citation

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Francesca Bianchi. "Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture." Algebra Number Theory 14 (9) 2369 - 2416, 2020. https://doi.org/10.2140/ant.2020.14.2369

Information

Received: 12 April 2019; Revised: 1 February 2020; Accepted: 23 April 2020; Published: 2020
First available in Project Euclid: 12 November 2020

MathSciNet: MR4172711
Digital Object Identifier: 10.2140/ant.2020.14.2369

Subjects:
Primary: 11D45
Secondary: 11G50, 11Y50, 14H52

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.14 • No. 9 • 2020
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