Rational motivic path spaces and Kim's relative unipotent section conjecture

We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim's relative unipotent section conjecture into several smaller conjectures with a homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that the path spaces of the punctured projective line over a number field are concentrated in degree zero with respect to Levine's t-structure for mixed Tate motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to exposition throughout. 52 page

A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.Comment: Improvements to the exposition and numerous minor corrections throughou

M0,5M_{0,5}: Towards the Chabauty-Kim method in higher dimensions

If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of a smooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim method leads to the construction of locally analytic functions on X(ZZ_p) which vanish on X(Z); we call such functions "Kim functions". At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_{0,5} should be easier than the previously studied curve M_{0,4} since its points are closely related to those of M_{0,4}, yet they face a further condition to integrality. This is mirrored by a certain "weight advantage" we encounter, because of which, M_{0,5} possesses new Kim functions not coming from M_{0,4}. Here we focus on the case "ZZ[1/6] in half-weight 4", where we provide a first nontrivial example of a Kim function on a surface. Central to our approach to Chabauty-Kim theory (as developed in works by S. Wewers, D. Corwin, and the first author) is the possibility of separating the geometric part of the computation from its arithmetic context. However, we find that in this case the geometric step grows beyond the bounds of standard algorithms running on current computers. Therefore, some ingenuity is needed to solve this seemingly straightforward problem, and our new Kim function is huge.Comment: Minor correction