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Arithmetic Billiards Perucca, Antonella ; ; Tronto, Sebastiano in Recreational Mathematics Magazine (2022) Detailed reference viewed: 103 (9 UL)Explicit Kummer generators for cyclotomic extensions ; Perucca, Antonella ; Sgobba, Pietro et al in JP Journal of Algebra, Number Theory and Applications (2022) Detailed reference viewed: 129 (16 UL)Kummer theory for commutative algebraic groups Tronto, Sebastiano 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. Detailed reference viewed: 27 (4 UL)Division in modules and Kummer theory Tronto, Sebastiano E-print/Working paper (2021) In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field ... [more ▼] In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative algebraic groups. With these tools we provide an effective bound for the degree of the field extensions arising from division points of elliptic curves, extending previous results of Javan Peykar for CM curves and of Lombardo and the author for the non-CM case. [less ▲] Detailed reference viewed: 54 (2 UL)Kummer theory for number fields via entanglement groups Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Manuscripta Mathematica (2021) Detailed reference viewed: 153 (6 UL)How to become the world record holder for solving the Rubik's cube Perucca, Antonella ; Tronto, Sebastiano Speeches/Talks (2021) Detailed reference viewed: 57 (13 UL)Some uniform bounds for elliptic curves over Q ; Tronto, Sebastiano 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 ▲] Detailed reference viewed: 67 (1 UL)Explicit Kummer theory for quadratic fields ; Perucca, Antonella ; Sgobba, Pietro et al in JP Journal of Algebra, Number Theory and Applications (2021) Let K be a quadratic number field and let \alpha \in K*. We present an explicit finite procedure to compute at once all Kummer degrees [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)] for n,m \geq 1 with n|m ... [more ▼] Let K be a quadratic number field and let \alpha \in K*. We present an explicit finite procedure to compute at once all Kummer degrees [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)] for n,m \geq 1 with n|m, where \zeta_m denotes a primitive m-th root of unity. We can also replace \alpha by any finitely generated subgroup of K*. [less ▲] Detailed reference viewed: 267 (13 UL)The degree of Kummer extensions of number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano 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 ▲] Detailed reference viewed: 202 (12 UL)Kummer theory for number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 88 (10 UL)Addendum to: Reductions of algebraic integers Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano 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 ▲] Detailed reference viewed: 161 (25 UL)Explicit Kummer theory for the rational numbers Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in International Journal of Number Theory (2020) Let G be a finitely generated multiplicative subgroup of Q* having rank r. The ratio between n^r and the Kummer degree [Q(\zeta_m,\sqrt[n]{G}) : Q(\zeta_m)], where n divides m, is bounded independently of ... [more ▼] Let G be a finitely generated multiplicative subgroup of Q* having rank r. The ratio between n^r and the Kummer degree [Q(\zeta_m,\sqrt[n]{G}) : Q(\zeta_m)], where n divides m, is bounded independently of n and m. We prove that there exist integers m_0, n_0 such that the above ratio depends only on G, \gcd(m,m_0), and \gcd(n,n_0). 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). [less ▲] Detailed reference viewed: 242 (31 UL)Radical entanglement for elliptic curves Tronto, Sebastiano 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 ▲] Detailed reference viewed: 35 (0 UL)Effective Kummer Theory for Elliptic Curves ; Tronto, Sebastiano E-print/Working paper (n.d.) Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform ... [more ▼] Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N ], N −1 α) : K(E[N ])]. When K = Q, and under a minimal (necessary) assumption on α, we show that the inequality [Q(E[N ], N −1 α) : Q(E[N ])] ≥ cN 2 holds with a constant c independent of both E and α. [less ▲] Detailed reference viewed: 71 (9 UL) |
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