Arithmetic Orr invariants of absolute Galois groups

Abstract

Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell$ analogue of Orr invariants for a Galois element associated with Galois action on étale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell$ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as ${\mathbb{Z}}_{\ell }$-module following Igusa-Orr's computation. Moreover, we investigate its relation with Ellenberg's obstruction to ${\pi }_1$-sections associated with lower central series filtration in the context of Grothendieck's section conjecture.

arXiv comment65 pages; table of contents is inserted; several minor errors are corrected; minor modifications in p.26, p.29, p.56; to appear in Int. Math. Res. Not. IMRN