Effective Kummer Theory for Elliptic Curves

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha $ be the set of $N$-division points of $\alpha $ in $E(\overline {K})$. We prove strong effective and uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}\alpha ): K(E[N])]$. When $K=\mathbb Q$, and under a minimal (necessary) assumption on $\alpha $, we show that the inequality $[\mathbb Q(E[N],N^{-1}\alpha ): \mathbb Q(E[N])] \geq cN^2$ holds for a positive constant $c$ independent of both $E$ and $\alpha $.

Explicit Kummer theory for the rational numbers

Let [Formula: see text] be a finitely generated multiplicative subgroup of [Formula: see text] having rank [Formula: see text]. The ratio between [Formula: see text] and the Kummer degree [Formula: see text], where [Formula: see text] divides [Formula: see text], is bounded independently of [Formula: see text] and [Formula: see text]. We prove that there exist integers [Formula: see text] such that the above ratio depends only on [Formula: see text], [Formula: see text], and [Formula: see text]. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

Kummer theory for number fields and the reductions of algebraic numbers

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

RAMIFICATION IN KUMMER EXTENSIONS ARISING FROM ALGEBRAIC TORI

We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions.