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 
.
