The etale fundamental group of a connected locally Noetherian scheme 
with a geometric base point is the profinite group of automorphisms of the fiber
functor on finite étale covers of 
. Finite sets with a continuous
action of this group classify finite étale covers.
For 
, it is the absolute
Galois group of 
. Changing the geometric base
point changes the identification only up to inner
automorphism.
For a geometrically connected finite-type scheme over a field 
, the arithmetic and geometric
groups fit into the exact sequence
 |
A rational point supplies a continuous section of the last map, well defined up to conjugation by the geometric group. The 
-adic section conjecture
asks whether this construction is bijective for smooth proper geometrically connected
curves of genus at least 2 over finite extensions of 
.
