Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture

DOI10.2140/ANT.2020.14.2369zbMATH Open1479.11113arXiv1904.04622OpenAlexW3092867307MaRDI QIDQ2212125

Author name not available (Why is that?)

Publication date: 18 November 2020

Published in: (Search for Journal in Brave)

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

m a t h c a l X (m a t h b b Z_{p})_{2}

containing the integral points

m a t h c a l X (m a t h b b Z)

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

m a t h c a l X (m a t h b b Z_{p})_{2} s e t m i n u s m a t h c a l X (m a t h b b Z)

. We also consider some algorithmic questions arising from Balakrishnan--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.

Full work available at URL: https://arxiv.org/abs/1904.04622

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