2005/11/23 by Misha Gavrilovich, Gavrilovich, Misha
Mathematics · #03C45 #11G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/0511591
openalex publication_date 2005/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Galois action on \Ext1(E( \Q),\Z2) 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(\Q) of algebraic points of an elliptic curve, by Λ≅\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 (Pi)i>0,jPij=Pi,i,j>0 of points Pi∈ Ek(\Q) is as large as possible with respect to linear relations between the coordinates of the points Pi's. Our original motivations come from model theory; this paper presents results from the author's thesis.