Deligne's freeness conjecture for the graded Galois Lie algebra of the thrice-punctured projective line

Let P1{0,1,}\mathbb P^1-\{0,1,\infty\} be the thrice-punctured projective line over Q\mathbb Q, let Gal(Q/Q)Gal(\overline{\mathbb Q}/\mathbb Q) be its absolute Galois group, and let GrP1{0,1,},lGal(Q/Q)Gr^{\bullet}_{\mathbb P^1-\{0,1,\infty\},l}Gal(\overline{\mathbb Q}/\mathbb Q) denote the associated graded Lie algebra for the relevant ll-adic filtration. Let the Soule elements be the distinguished graded elements arising from the Galois action. Deligne's conjecture. The graded Lie algebra

GrP1{0,1,},lGal(Q/Q)QlGr^{\bullet}_{\mathbb P^1-\{0,1,\infty\},l}Gal(\overline{\mathbb Q}/\mathbb Q)\otimes \mathbb Q_l

is a free graded Lie algebra over Ql\mathbb Q_l generated by the Soule elements, and its Lie algebra structure is induced from a Lie algebra over Z\mathbb Z independent of ll. This predicts an ll-independent, freely generated structure for the graded Lie algebra governing the outer Galois representation of the fundamental group of the thrice-punctured projective line; the supplied text does not state a resolution.

Sources & referencesView supporting material

Primary source

Arash Rastegar, “Deformation of Outer Representations of Galois Group”, arXiv:math/0405351 (2006).

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