A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions of the group of algebraic points of an elliptic curve, by $Z^2$

Abstract

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 $Λ\cong\Z2$, (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.