Papers › Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture
Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture
Francesca Bianchi
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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(ℤₚ)₂ containing the integral points 𝒳(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 𝒳(ℤₚ)₂∖𝒳(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.
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