Arithmetic Orr Invariants of Absolute Galois Groups

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  • Hisatoshi Kodani
    Advanced Institute for Materials Research, Tohoku University , 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan
  • Yuji Terashima
    Graduate School of Science, Tohoku University , Aramaki-aza-Aoba 6-3, Aoba-ku, Sendai, 980-8578, Japan

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<jats:title>Abstract</jats:title> <jats:p>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.</jats:p>

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