论文标题
标量chabauty和$ s $单位方程的限制
Restriction of Scalars Chabauty and the $S$-unit equation
论文作者
论文摘要
鉴于在数字字段$ k $上具有平滑，正确的几何曲线$ x $ x $属$ g $，而jacobian $ j $，chabauty's方法是$ p $ - ad的技术，限制$ \＃x（k）$时，当$ \ mathrm {rankrm {rank}} \ j（k）<g $。我们研究了一种称为“标量限制chabauty”（ROS Chabauty）的变体的局限性。当$ \ mathrm {rank} \ j（k）\ leq [k：\ mathbb {q}]（g -1）$时，通常会限制$ \＃x（k）$，但在一个亚组障碍物（$ \ mathrm of $ \ mathrm {j j $ j n q of）的情况下失败，但在一个亚组障碍物的存在下失败。相交的图像在高于指望的尺寸中，$ \ mathrm {res} _ {k/\ mathbb {q}} x $。我们定义了BCP障碍物，这是$ x $的几何形状引起的某些亚组障碍物。 BCP障碍物解释了所有已知示例，其中Ros Chabauty无法约束$ \＃x（k）$。我们还将Ros Chabauty扩展到仿射曲线上的$ s $ integral点。 假设$ k $不包含CM-Subfield。我们提出了$ p $ - adic算法，该算法可以指出，该算法计算$ s $ unit方程$ x+y = 1 $ = 1 $ for $ x，y in \ inthcal {o} _ {o} _ {k，s}^{\ times^$ s $ s $ s $ s $ intege $ 0 $ 0 $ 0 $ 0 $ 0。作为算法成功的证据，我们证明除了这些曲线之一以外的所有曲线都没有亚组障碍物，其余曲线对ROS Chabauty没有BCP障碍。相比之下，在广义的Leopoldt猜想下，我们证明使用经典chabauty的类似方法不能将解决方案绑定到$ s $单位方程时，当$ [k：\ m athbb q] \ geq 3 $ 3 $和$ k $并非完全真实。
Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound $\# X(K)$ when $\mathrm{rank}\ J(K) < g$. We study limitations of a variant called `Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty typically bounds $\# X(K)$ when $\mathrm{rank}\ J(K) \leq [K:\mathbb{Q}] (g - 1)$, but fails in the presence of a subgroup obstruction, a high-rank subgroup scheme of $\mathrm{Res}_{K/\mathbb{Q}} J$ which intersects the image of $\mathrm{Res}_{K/\mathbb{Q}} X$ in higher-than-expected dimension. We define BCP obstructions, which are certain subgroup obstructions arising from the geometry of $X$. BCP obstructions explain all known examples where RoS Chabauty fails to bound $\# X(K)$. We also extend RoS Chabauty to compute $S$-integral points on affine curves. Suppose $K$ does not contain a CM-subfield. We present a $p$-adic algorithm which conjecturally computes solutions to the $S$-unit equation $x+y = 1$ for $x,y \in \mathcal{O}_{K,S}^{\times}$ by using RoS Chabauty to compute $S$-integral points on certain genus $0$ affine curves. As evidence the algorithm succeeds, we prove that all but one of these curves have no subgroup obstructions and that the remaining curve has no BCP obstructions to RoS Chabauty. In contrast, under a generalized Leopoldt conjecture, we prove that analogous methods using classical Chabauty cannot bound solutions to the $S$-unit equation when $[K:\mathbb Q] \geq 3$ and $K$ is not totally real.
