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On the distribution of the order and index for the reductions of algebraic numbers Sgobba, Pietro in Journal of Number Theory (2021) Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of (\alpha_i mod ... [more ▼] Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of (\alpha_i mod p) lies in a given arithmetic progression associated to (\alpha_i). We also study the primes p for which the index of (\alpha_i mod p) is a fixed integer or lies in a given set of integers for each i. An additional condition on the Frobenius conjugacy class of p may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case r=1 of this problem. [less ▲] Detailed reference viewed: 61 (7 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: 133 (25 UL)Reductions of algebraic integers ; Perucca, Antonella in Journal of Number Theory (2016) Detailed reference viewed: 103 (4 UL)The order of the reductions of an algebraic integer Perucca, Antonella in Journal of Number Theory (2015) Detailed reference viewed: 94 (2 UL)Tame Galois realizations of $ GL_2(\Bbb F_l)$ over $\Bbb Q$ Arias De Reyna Dominguez, Sara ; in Journal of Number Theory (2009), 129(5), 1056--1065 This paper concerns the tame inverse Galois problem. For each prime number l, we construct infinitely many semistable elliptic curves over Q with good supersingular reduction at l. The Galois action on ... [more ▼] This paper concerns the tame inverse Galois problem. For each prime number l, we construct infinitely many semistable elliptic curves over Q with good supersingular reduction at l. The Galois action on the l-torsion points of these elliptic curves provides tame Galois realizations of GL_2(F_l) over Q. [less ▲] Detailed reference viewed: 34 (2 UL)Sur certaines surfaces elliptiques et courbes elliptiques de Mordell de rang non-nul associées à des discriminants de polynômes cubiques ou quartiques Leprévost, Franck in Journal of Number Theory (1999), 78 Detailed reference viewed: 33 (0 UL)Appendice : quelques données statistiques ; Leprévost, Franck ; in Journal of Number Theory (1999), 78 Detailed reference viewed: 35 (0 UL)Revêtements de courbes elliptiques à multiplication complexe par des courbes hyperelliptiques et sommes de caractères Leprévost, Franck ; in Journal of Number Theory (1997), 64(2), 165-182 Detailed reference viewed: 76 (0 UL) |
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