Lifting Galois sections along torsors

Abstract

The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve $X$ over a finitely generated field $k$ of characteristic $0$ and any non empty Zariski open $U \subset X$, every section of $\pi _1 (X, \bar x) \to \mathrm{Gal}_k$ lifts to a section of $\pi _1 (U,\bar x) \to \mathrm{Gal}_k$. We consider in this article the problem of lifting Galois sections to the intermediate quotient $ \pi_1^{cc}(U)$ introduced by Mochizuki. We show that when $k = \mathbb Q$ and $D=X\setminus U$ is an union of torsion sub-packets every Galois section actually lifts to $ \pi_1^{cc}(U)$. One of the main tools in the proof is the construction of torus torsors $F_D$ and $E_D$ over $X$ and the geometric interpretation $ \pi_1^{cc}(U) \simeq \pi _1 (F_D)$.