  We study Galois action on $\Ext^1(E(\bar \Q),\Z^2)$ and interpret our results
as partially showing that the notion of a path on a complex elliptic curve $E$
can be characterised algebraically. The proofs show that our results are just
concise reformulations of Kummer theory for $E$ as well as the description of
Galois action on the Tate module.
  Namely, we prove (a),(b) below by showing they are equivalent to (c) which is
well-known: (a) Absolute Galois group acts transitively on the set of uniquely
divisible abelian \EndE-module extensions of $E(\bar\Q)$ of algebraic points of
an elliptic curve, by $\Lambda\cong\Z2$, (b) natural algebraic properties
characterise uniquely the Poincare's fundamental groupoid of a complex elliptic
curve, restricted to the algebraic points, (c) (Kummer theory) up to finite
index, the image of the Galois action on the sequences
$(P_i)_{i>0},jP_{ij}=P_i,i,j>0$ of points $P_i\in E^k(\bar\Q)$ is as large as
possible with respect to linear relations between the coordinates of the points
$P_i$'s.
  Our original motivations come from model theory; this paper presents results
from the author's thesis.
