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In this paper we exhibit the notion of (uniformly) good sections ofarithmetic fundamental groups. We introduce and investigate the problem ofcuspidalisation of sections of arithmetic fundamental groups, its ultimate aimis to reduce the solution of the Grothendieck anabelian section conjecture tothe solution of its birational version. We show that (uniformly) good sectionsof arithmetic fundamental groups of smooth, proper, and geometrically connectedhyperbolic curves over slim (and regular) fields can be lifted to sections ofcuspidally abelian absolute Galois groups. As an application we prove a (pro-p)p-adic version of the Grothendieck anabelian section conjecture for hyperboliccurves, under the assumption that the existence of sections of arithmeticfundamental groups, and cuspidally abelian Galois groups, implies the existenceof tame points. We also prove that the existence of uniformly good sections ofarithmetic fundamental groups for hyperbolic curves over number fields impliesthe existence of divisors of degree 1, under a finiteness condition of theTate-Shafarevich group of the jacobian of the curve.

Good Sections of Arithmetic Fundamental Groups