The polylog quotient and the Goncharov quotient in computational Chabauty–Kim Theory I

Abstract

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of [Formula: see text]. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim’s conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim’s conjecture.

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

“…Proposition 4.5 and its proof are similar to Proposition 3.10 of [CDC1]. The proof spans segments 4.6-4.8.…”

“…The methods of Dan-Cohen-Wewers [DCW1,DCW2,DC] and Corwin-Dan-Cohen [CDC1,CDC2], while so far limited to the simplest of all cases (X " P 1 zt0, 1, 8u), have been particularly successful in going beyond the quadratic level. These articles incorporate the methods of mixed Tate motives and motivic iterated integrals (see, for instance, [DG,Gon,Bro1,Bro3]).…”

“…In turn, this may lead to a better understanding of the ramification of motivic iterated integrals and hence to more precise S-integral refinements of Goncharov's conjectures. As explained in [DCW2,DC,CDC1,CDC2], our algorithms for P 1 zt0, 1, 8u rely on such statements for halting, and a better understanding will lead to faster and more elegant algorithms. As we explain in [DCJ], our methods with resultants also help to clarify and simplify the geometric step for P 1 zt0, 1, 8u and for punctured lines in general.…”

“…The reverse "lexicographic" order, it seems to us, leads to the systematic reversal of a vast swath of mathematics. However, for reasons we do not understand, there appears to be quite a tradition of using lexicographic order, and the authors of [CDC1] in particular, chose to follow this tradition. Thus, we're forced to resolve this conflict within the body of the article by using both orderings and spelling out where and how we transition between them.…”

“…Proving this would require proving that the formulas for decomposition of certain motivic polylogarithms in terms of shuffle coordinates on the mixed Tate Galois group obtained via computations of certain p-adic periods in §5 hold precisely (and not only to within ǫ). Some methods for doing so are demonstrated for instance in [DCW2,CDC1] (along with attribution to those who taught us these methods). But these are not needed for the application to integral points.…”