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.

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

References in corpus (1)

• Weight filtrations on Selmer schemes and the effective Chabauty--Kim method

Cited by in corpus (1)

• p-adic adelic metrics and Quadratic Chabauty I