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

Dan-Cohen, Ishai; Corwin, David

doi:10.1090/tran/7964

The polylog quotient and the Goncharov quotient in computational Chabauty–Kim theory II
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by Ishai Dan-Cohen and David Corwin PDF

Trans. Amer. Math. Soc. 373 (2020), 6835-6861 Request permission

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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