Davide Lombardo (Pisa)
Let K be a number field and α∈K×. Kummer theory shows that the Galois group Gn of K(ζn,n√α) over K(ζn) is canonically isomorphic to a subgroup of Z/nZ; moreover, if α is not a root of unity, the index (Z/nZ:Gn) is uniformly bounded as n varies. Equivalently, there exists a constant d(K,α)>0 such that for all n≥1 we have [K(ζn,n√α)/K(ζn)]≥d(K,α)⋅n. The nature of the constant d(K,α) is well-understood in terms of arithmetical properties of α. A natural generalisation of Kummer theory can be obtained by considering a commutative algebraic group A defined over K, a rational point α∈A(K), and the tower of extensions K⊆K(A[n])⊆K(A[n],1nα): this reduces to the classical case by taking A=Gm. In this talk I will discuss the case where A is an elliptic curve E, and show that if E is defined over Q we have [Q(E[n],1nα):Q(E[n])]≥cn2, where c is now an \textit{absolute} constant, independent of E and α, provided only that α is not divisible by any k>1 in the free abelian group E(Q)/torsion. This is joint work with Sebastiano Tronto (Université du Luxembourg).
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