Barcelona Fall Workshop on Number Theory II

Uniform Kummer theory for elliptic curves over Q

Presenters

Davide Lombardo (Pisa)

Abstract

Let $K$ be a number field and $\alpha \in K^\times$. Kummer theory shows that the Galois group $G_n$ of $K(\zeta_n, \sqrt[n]{\alpha})$ over $K(\zeta_n)$ is canonically isomorphic to a subgroup of $\mathbb{Z}/n\mathbb{Z}$; moreover, if $\alpha$ is not a root of unity, the index $\left( \mathbb{Z}/n\mathbb{Z} : G_n \right)$ is uniformly bounded as $n$ varies. Equivalently, there exists a constant $d(K, \alpha) > 0$ such that for all $n \geq 1$ we have $[K(\zeta_n, \sqrt[n]{\alpha}) / K(\zeta_n)] \geq d(K, \alpha) \cdot n$. The nature of the constant $d(K, \alpha)$ is well-understood in terms of arithmetical properties of $\alpha$. A natural generalisation of Kummer theory can be obtained by considering a commutative algebraic group $\mathcal{A}$ defined over $K$, a rational point $\alpha \in \mathcal{A}(K)$, and the tower of extensions $K \subseteq K(\mathcal{A}[n]) \subseteq K(\mathcal{A}[n], \frac{1}{n}\alpha)$: this reduces to the classical case by taking $\mathcal{A}=\mathbb{G}_m$. In this talk I will discuss the case where $\mathcal{A}$ is an elliptic curve $E$, and show that if $E$ is defined over $\mathbb{Q}$ we have $[\mathbb{Q}(E[n], \frac{1}{n} \alpha) : \mathbb{Q}(E[n])] \geq cn^2,$ where $c$ is now an \textit{absolute} constant, independent of $E$ and $\alpha$, provided only that $\alpha$ is not divisible by any $k>1$ in the free abelian group $E(\mathbb{Q})/\text{torsion}$. This is joint work with Sebastiano Tronto (Université du Luxembourg).