Mixed Tate Motives and the Unit Equation

Abstract

Abstract. This is the second installment in a sequence of articles devoted to "explicit ChabautyKim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via p-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm. We also work out an elaborate example, which goes beyond the cases that were understood before, and allows us to verify Kim's conjecture in a range of new cases.

Arithmetic gauge theory: A brief introduction

“…On the one hand, the gauge-theoretical perspective has the potential to be applicable to a very broad class of phenomena encompassing many of the central problems of current day number theory [38]. On the other, unlike Faltings's proof, which is widely regarded as ineffective, the gauge-theory proof conjecturally leads to a computational method for actually finding rational solutions [37], a theme that is currently under active investigation [8,9,22,23]. It should be remarked that the map A that associates gauge fields to points has been well-known since the 1950s when the variety V is an elliptic curve, an abelian variety, or generally, a commutative algebraic group.…”

“…Part of the motivation for the étale topology is to have a topology that is fine enough so that natural torsors become locally trivial. 23 Read 'Sha'.…”

“…This implies that the image is a constructible set for the Zariski topology, which therefore admits a finite polynomial description. There is an increasing collection of examples for which the image has a precise enough description for the rational points to be computed completely [8,9,22,23]. A spectacular recent result [10] carries this out for the modular curve…”

Principal bundles and reciprocity laws in number theory

“…The main applications so far of the theory of the previous section are to explicit reciprocity laws on hyperbolic curves [3,6,10], and we give a brief survey of illustrative examples. Some other works with further developments which we do not discuss include [4] and [5].…”

Explicit Chabauty–Kim for the split Cartan modular curve of level 13

“…As in the method of Chabauty and Coleman, one hopes to be able to translate Kim's approach into a practical explicit method for computing (a finite set of p-adic points containing) X(Q) in practice for a given curve X/ Q having r ≥ g. However, in part due to the technical nature of the objects involved, this is a rather delicate task. Kim's results [Kim05] on integral points on P 1 \ {0, 1, ∞} have been made explicit by Dan-Cohen and Wewers [DCW15] and used to develop an algorithm to solve the S-unit equation [DCW16,DC17] using iterated p-adic integrals. The work [BDCKW] of the first author with Dan-Cohen, Kim and Wewers contains explicit results for integral points on elliptic curves of ranks 0 and 1.…”