The motivic Selmer scheme of the thrice-punctured line

Abstract

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 $\pi_1^{\mathrm{mot}}(X,0) \twoheadrightarrow \Pi$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $\Pi$-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.