References of "Tronto, Sebastiano 50033276"
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See detailExplicit Kummer generators for cyclotomic extensions
Hoermann, Fritz; Perucca, Antonella UL; Sgobba, Pietro UL et al

in JP Journal of Algebra, Number Theory and Applications (in press)

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See detailDivision in modules and Kummer theory
Tronto, Sebastiano UL

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 ▲]

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See detailSome uniform bounds for elliptic curves over Q
Lombardo, Davide; Tronto, Sebastiano UL

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 ▲]

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See detailKummer theory for number fields via entanglement groups
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

in Manuscripta Mathematica (2021)

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See detailExplicit Kummer theory for quadratic fields
Hörmann, Fritz; Perucca, Antonella UL; Sgobba, Pietro UL 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 ▲]

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See detailThe degree of Kummer extensions of number fields
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailKummer theory for number fields
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

in Proceedings of the Roman Number Theory Association (2020)

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See detailAddendum to: Reductions of algebraic integers
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailRadical entanglement for elliptic curves
Tronto, Sebastiano UL

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 ▲]

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See detailExplicit Kummer theory for the rational numbers
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailEffective Kummer Theory for Elliptic Curves
Lombardo, Davide; Tronto, Sebastiano UL

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 ▲]

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See detailArithmetic Billiards
Perucca, Antonella UL; Reguengo da Sousa, Joe; Tronto, Sebastiano UL

E-print/Working paper (n.d.)

Detailed reference viewed: 72 (4 UL)