TY - JOUR
T1 - The polylog quotient and the Goncharov quotient in computational Chabauty-Kim Theory i
AU - Corwin, David
AU - Dan-Cohen, Ishai
N1 - Funding Information:
The first author would like to thank his thesis advisor Bjorn Poonen for numerous comments and corrections on this paper. He would like to thank Amnon Besser for various pieces of advice, especially for a trick for going between complex and p-adic identities in Proposition A.4. He would like to thank Rob de Jeu, for useful discussion, eventually leading him to the paper [58], as well as Don Zagier, for subsequent discussions. He would like to thank Herbert Gangl for the idea of proving Lemma A.3 by substituting x = −1 and y = 1/3 into the Kummer–Spence identity. Both authors would like to thank Francis Brown for discussions about the subject and especially for the idea behind Proposition 3.10. The first author was supported by NSF grants DMS-1069236 and DMS-160194, Simons Foundation grant #402472 , and NSF RTG Grant #1646385 at various points during the work on this project.
Publisher Copyright:
© 2020 World Scientific Publishing Company.
PY - 2020/9/1
Y1 - 2020/9/1
N2 - 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 Spec 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.
AB - 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 Spec 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.
KW - Polylogarithm
KW - motives
KW - non-abelian Chabauty
UR - http://www.scopus.com/inward/record.url?scp=85091077253&partnerID=8YFLogxK
U2 - 10.1142/S1793042120500967
DO - 10.1142/S1793042120500967
M3 - Article
AN - SCOPUS:85091077253
SN - 1793-0421
VL - 16
SP - 1859
EP - 1905
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 8
ER -