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 $Z^2$

Details 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 $Z^2$ 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 $Z^2$

2005-11-23

arxiv.org/abs/math/0511591v1

Mathematics

Number Theory

22 pages, part of the author's thesis, available at http://misha.uploads.net.ru

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

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 $Z^2$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with 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 $Z^2$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and 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 $Z^2$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-543071