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Transactions of the American Mathematical Society

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The polylog quotient and the Goncharov quotient in computational Chabauty–Kim theory II


Authors: Ishai Dan-Cohen and David Corwin
Journal: Trans. Amer. Math. Soc. 373 (2020), 6835-6861
MSC (2010): Primary 11G55, 14F35, 14F42, 14G05; Secondary 14F30
DOI: https://doi.org/10.1090/tran/7964
Published electronically: August 6, 2020
MathSciNet review: 4155193
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Abstract: This is the second installment in a multi-part series starting with Corwin–Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen–Wewers, Dan-Cohen, and F. 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 $\operatorname {Spec}\mathbb {Z}$. To do so, we develop a refined version of the algorithm of Dan-Cohen–Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-$1$ part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin–Dan-Cohen [arXiv:1812.05707v3], where we verified Kim’s conjecture in an interesting new case.


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Additional Information

Ishai Dan-Cohen
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel
MR Author ID: 999259
Email: ishaidc@gmail.com

David Corwin
Affiliation: Department of Mathematics, 970 Evans Hall #3840, University of California, Berkeley, Berkeley, California 94720-3840
MR Author ID: 1073361
Email: corwind@alum.mit.edu

Received by editor(s): December 12, 2018
Received by editor(s) in revised form: April 23, 2019
Published electronically: August 6, 2020
Additional Notes: The first author was supported by ISF grant 87590021.
The second author was supported by NSF RTG grant 1646385, by NSF grants DMS-1069236 and DMS-1601946 (to Bjorn Poonen), and by Simons Foundation grant #402472 (to Bjorn Poonen).
Article copyright: © Copyright 2020 American Mathematical Society