论文标题
几何二次chabauty和$ p $ -Adic Heights
Geometric quadratic Chabauty and $p$-adic heights
论文作者
论文摘要
令$ x $为$ g> 1 $的曲线，超过$ \ mathbb {q} $，其jacobian $ j $具有mordell-weil Rank $ r $和néron-néron-SeveriRank $ρ$。当$ r <g+ρ-1 $ $时，几何二次chabauty方法确定了有限的$ p $ - adic点，其中包含$ x $的合理点。我们描述了几何二次chabauty的算法，该算法将几何二次chabauty方法转化为$ p $ - adic高度和$ p $ adic（Coleman）积分的语言。这种翻译还使我们可以与（原始）二次chabauty进行比较。我们表明，由几何方法产生的$ p $ - 亚种的有限集包含在同一个学方法产生的有限集中，并描述了它们的差异。
Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and Néron--Severi rank $ρ$. When $r < g+ ρ- 1$, 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.
