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.

让$X$是属曲线$G>1$超过$\mathbb{问}$其雅可比行列式$J$具有 Mordell–Weil 等级$r$和 Néron-Severi 等级$\rho$。什么时候$r<G+\rho -1$，几何二次 Chabauty 方法确定有限集$p$-包含有理点的有理点$X$。我们描述了几何二次 Chabauty 的算法，将几何二次 Chabauty 方法翻译成以下语言：$p$-adic高度和$p$-adic（科尔曼）积分。这种翻译还使我们能够与二次 Chabauty 的（原始）上同调方法进行比较。我们证明有限集$p$-将几何方法产生的进分点包含在上同调方法产生的有限集中，并描述它们的差异。