Tronto, Sebastiano[University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]

2020

No

[en] Elliptic curves ; Galois representations

[en] Let G be a commutative connected algebraic group over a number field K, let A be a finitely generated and torsion-free subgroup of G(K) of rank r>0 and, for n>1, let K(n^{−1}A) be the smallest extension of K inside an algebraic closure K¯ over which all the points P∈G(K¯) such that nP∈A are defined. We denote by s the unique non-negative integer such that G(K¯)[n]≅(Z/nZ)s for all n≥1. We prove that, under certain conditions, the ratio between nrs and the degree [K(n^{−1}A):K(G[n])] is bounded independently of n>1 by a constant that depends only on the ℓ-adic Galois representations associated with G and on some arithmetic properties of A as a subgroup of G(K) modulo torsion. In particular we extend the main theorems of [13] about elliptic curves to the case of arbitrary rank