Arithmetic Billiards

in Recreational Mathematics Magazine (2022)

Kummer theory for commutative algebraic groups

Doctoral thesis (2022)

This thesis consists of four research articles that treat different aspects of Kummer theory for commutative algebraic groups, with particular emphasis on explicit and effective results.

Kummer theory for number fields via entanglement groups

in Manuscripta Mathematica (2021)

Some uniform bounds for elliptic curves over Q

E-print/Working paper (2021)

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image of ... [more ▼]

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of E, and the Kummer extensions generated by points of infinite order in E(Q). [less ▲]

The degree of Kummer extensions of number fields

in International Journal of Number Theory (2021)

Let K be a number field, and let \alpha_1, ... , \alpha_r be elements of K* which generate a subgroup of K* of rank r. Consider the cyclotomic-Kummer extensions of K given by K(\zeta_n, \sqrt[n_1]{\alpha ... [more ▼]

Let K be a number field, and let \alpha_1, ... , \alpha_r be elements of K* which generate a subgroup of K* of rank r. Consider the cyclotomic-Kummer extensions of K given by K(\zeta_n, \sqrt[n_1]{\alpha_1}, ... , \sqrt[n_r]{\alpha_r}), where n_i divides n for all i. There is an integer x such that these extensions have maximal degree over K(\zeta_g, \sqrt[g_1]{\alpha_1}, ... , \sqrt[g_r]{\alpha_r}), where g=\gcd(n,x) and g_i=\gcd(n_i,x). We prove that the constant x is computable. This result reduces to finitely many cases the computation of the degrees of the extensions K(\zeta_n, \sqrt[n_1]{\alpha_1}, ... , \sqrt[n_r]{\alpha_r}) over K. [less ▲]

Addendum to: Reductions of algebraic integers

in Journal of Number Theory (2020)

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. We consider Kummer extensions of G of the form K(\zeta_{2^m}, \sqrt[2^n]G)/K(\zeta_{2^m}), where n \leq m. In ... [more ▼]

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. We consider Kummer extensions of G of the form K(\zeta_{2^m}, \sqrt[2^n]G)/K(\zeta_{2^m}), where n \leq m. In the paper "Reductions of algebraic integers" (J. Number Theory, 2016) by Debry and Perucca, the degrees of those extensions have been evaluated in terms of divisibility parameters over K(\zeta_4). We prove how properties of G over K explicitly determine the divisibility parameters over K(\zeta_4). This result has a clear computational advantage, since no field extension is required. [less ▲]

Radical entanglement for elliptic curves

E-print/Working paper (2020)

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 ... [more ▼]

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 [less ▲]