The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel

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A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves (2012, background): “Incidentally, we remark that the said implication of motivic conjectures, in particular, L(n) 6= 0 for n >> 0, has been proved in a range of cases, including P(1) \ {0, 1,∞}, CM elliptic curves minus their origins, and curves of higher genus whose Jacobians have CM [7, 13, 16] .”
Diophantine problems and p-adic period mappings (2018, unknown): “Also discuss Chabauty’s method, Kim’s nonabelian generalisation of Chabauty’s method, and compare to the methods of [LV].”
Quadratic Chabauty and rational points, I: p-adic heights (2016, background): “be less than the genus of the curve, which is somewhat restrictive. As such, one would like to have a reﬁnement of the Jacobian which remembers more information about the set X(K). The insight of Kim [30] is that, rather than trying to generalise the Jacobian of X, it is easier to generalise its Gal”
Quadratic Chabauty and Rational Points II: Generalised Height Functions on Selmer Varieties (2017, methodology): “In a series of papers [24, 25, 27], Kim proposed a generalisation of the Chabauty– Coleman method, which gives a nested sequence X(Kp)1 ⊃ X(Kp)2 ⊃ · · · ⊃ X(K) of sets X(Kp)n of p-adic points, each containing the set X(K), such that the “depth 1” set X(Kp)1 is exactly the one arising from the Chabau”