Last Updated：2022/12/24
Serre constructed an example of an algebraic variety X over a number field K plus two embeddings of K into ℂ such that the geometric fundamental groups of X_1( ℂ) and X_2( ℂ) are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).
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Serre
constructed
an
example
of
an
algebraic
variety
X
over
a
number
field
K
plus
two
embeddings
of
K
into
ℂ
such
that
the
geometric
fundamental
groups
of
X_1(
ℂ)
and
X_2(
ℂ)
are
not
isomorphic,
while
the
algebraic
fundamental
groups
(i.e.
their
profinite
completions)
clearly
are
isomorphic,
see
(Serre).
