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