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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 Spec 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. [ABSTRACT FROM AUTHOR]

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL2(...). In (2010), Greicius developed necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius's methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over ℚ(α) with maximal image, where α is the real root of x³ + x + 1. Next, we extend Greicius's tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups. [ABSTRACT FROM AUTHOR]