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 ${\pi }_1^{cc}(U)\simeq {\pi }_1(F_D)$.

v2 corresponds to the published, deeply revised and much shortened version. We think that v1, that contains the same results, could still be helpful to the reader preferring a self-contained text

References in corpus (4)

• The cuspidalisation of sections of arithmetic fundamental groups II
• Good Sections of Arithmetic Fundamental Groups
• Un critère d'épointage des sections $l$-adiques
• On the birational anabelian section conjecture