Geometric quadratic Chabauty and $p$-adic heights

Abstract

Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and Néron--Severi rank $\rho$. When $r<g+\rho -1$, the geometric quadratic Chabauty method determines a finite set of $p$-adic points containing the rational points of $X$. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of $p$-adic heights and $p$-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of $p$-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.

arXiv commentUpdate definition of simple open. This clarifies that simple opens are everywhere locally soluble, which is not explicitly noted in the published version

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Cited by in corpus (1)

• p-adic adelic metrics and Quadratic Chabauty I