数域上的多对数Chabauty–Kim轨迹
Polylogarithmic Chabauty--Kim loci over number fields
浏览论文内容
中文总结 AI 辅助
本研究针对数域上射影直线的S-整点，对比动机与平展上同调Selmer概形，推导相关局部化映射公式，确定二次域上多对数Chabauty–Kim轨迹，证明Kim猜想在新情形成立，还给出虚二次域上S-Selmer截面猜想的Chabauty–Kim理论证明。
中文摘要 AI 辅助
我们研究数域上射影直线\\(\mathbb{P}^1\smallsetminus\{0,1,\infty\}\\)的\\(S\\)-整点的多对数Chabauty–Kim轨迹。我们对比了动机Selmer概形与平展上同调Selmer概形，描述了Galois群在Selmer概形及周期环上的作用，给出了多对数Selmer概形上局部化映射的显式公式。针对虚二次域与实二次域，我们推导了方程并确定了若干多对数Chabauty–Kim轨迹及全Chabauty–Kim轨迹，证明Kim猜想在若干新情形下成立；在其他情形中，我们表明多对数Chabauty–Kim方法即便结合\\(S_3\\)-对称化，也不足以精确分离出\\(S\\)-整点，原因是存在来自\\(p\\)-adic单位根的额外点。作为进一步应用，我们给出了虚二次域上\\(S\\)-Selmer截面猜想的Chabauty–Kim理论证明，其中\\(S\\)为空集或由一个不被复共轭固定的素数构成。
英文摘要
We study polylogarithmic Chabauty--Kim loci for $S$-integral points on $\mathbb{P}^1\smallsetminus\{0,1,\infty\}$ over number fields. We compare motivic and étale Selmer schemes, describe Galois actions on Selmer schemes and period rings, and give an explicit formula for the localisation map on the polylogarithmic Selmer scheme. For imaginary and real quadratic fields, we derive equations and determine several polylogarithmic and full Chabauty--Kim loci and show that Kim's Conjecture holds in several new cases. In other cases, we show that the polylogarithmic Chabauty--Kim method is insufficient to cut out precisely the $S$-integral points, even when combined with $S_3$-symmetrisation, due to additional points arising from $p$-adic roots of unity. As a further application, we give a Chabauty--Kim theoretic proof of the $S$-Selmer Section Conjecture for imaginary quadratic fields when $S$ is empty or $S$ consists of a single prime not fixed by complex conjugation.
