Geometric quadratic Chabauty and $p$-adic heights

TL;DR

This paper develops algorithms for geometric quadratic Chabauty using $p$-adic heights and integrals, providing a comparison with the cohomological approach and analyzing their differences.

Contribution

It translates geometric quadratic Chabauty into $p$-adic height and integral language, and compares it with the cohomological method for the first time.

Findings

The geometric method's finite set is contained in the cohomological method's set.

Algorithms for geometric quadratic Chabauty are established.

A comparison and description of differences between the two methods are provided.

Abstract

Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and N\'eron--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.