三次穿孔射影直线的动机塞尔默概型
The motivic Selmer scheme of the thrice-punctured line
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中文总结 AI 辅助
本文研究数域S整数环上三次穿孔射影直线的动机塞尔默概型，证明其同构于仿射空间并构造坐标，为一般情形下具体化动机Chabauty-Kim方法提供关键基础。
中文摘要 AI 辅助
设X=ℙ¹∖{0,1,∞}为数域K的S整数环𝒪_{K,S}上的三次穿孔射影直线。对Deligne-Goncharov动机基本群π₁ᵐᵒᵗ(X,0)的任意商群Π，存在与之关联的塞尔默概型，该概型参数化具有混合泰特动机结构的Π挠子。我们给出动机塞尔默概型的若干便于计算的刻画，利用代数群、李代数及完全霍普夫代数的𝔾ₘ等变上同调。我们证明该塞尔默概型同构于仿射空间𝔸^ℕ_ℚ，并构造了实现该同构的坐标。这是在一般情形下将动机Chabauty-Kim方法具体化的关键要素，且不限制基域或基本群商群的选择。
英文摘要
Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $π_1^{\mathrm{mot}}(X,0) \twoheadrightarrow Π$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $Π$-torsors with a mixed Tate motive structure. We give several descriptions of the motivic Selmer scheme which make it amenable to computations, using $\mathbb{G}_m$-equivariant cocycles of algebraic groups, Lie algebras, and complete Hopf algebras. We prove that the Selmer scheme is isomorphic to an affine space $\mathbb{A}^N_{\mathbb{Q}}$, and we construct coordinates realising this isomorphism. This is a key ingredient for making the motivic Chabauty--Kim method explicit in a general setting, without restrictions on the base field or the choice of fundamental group quotient.
