Motivic iterated integrals and integral points

Principal Investigator

Dr. I. Dan-Cohen (Essen/Ben Gurion)
 

Project description

Let $S$ be an open subscheme of $\mathrm{Spec} \; \mathbb{Z}$ and let $X$ be an $S$-model of a hyperbolic curve. In the last decade, Minhyong Kim has developed a new approach to the study of integral points which uses Deligne's theory of the unipotent fundamental group to construct certain subsets $X(\mathbb{Z}_p)_{S,n}$ of the set of $\mathbb{Z}_p$-points which contain $X(S)$ and are conjectured to converge to $X(S)$ as $n$ grows. In the special case of the punctured line, the unipotent fundamental group is known to be motivic, opening the door to motivic methods. Our main goal in this project is to use Goncharov's theory of motivic iterated integrals, as well as methods developed by Francis Brown, to construct an algorithm for computing the sets $X(\mathbb{Z}_p)_{S,n}$ for the thrice punctured line over $\mathbb{Q}$, and hence, conjecturally, for solving the $S$-unit equation. We will also study more general curves over more general bases.

Related publications

Ishai Dan-Cohen, and Stefan Wewers, Mixed Tate motives and the unit equation. International Mathematics Research Notices, Issue 17, 2016

Ishai Dan-Cohen and Tomer Schlank, Morphisms of rational motivic homotopy types. Appl. Categ. Structures 29 (2021), no. 2, 311–347

Preprints

Ishai Dan-Cohen, Explicit motivic Chabauty-Kim theory III: towards the polylogarithmic quotient over general number fields. 28 pages. arxiv.org/abs/1510.01362

Ishai Dan-Cohen, Rational motivic path spaces and Kim's relative unipotent section conjecture, arXiv:1703.10776 To appear, Rendiconti del Seminario Matematico della Università di Padova.