scholarly journals A local-global principle for isogenies of prime degree over number fields

2014 ◽  
Vol 89 (3) ◽  
pp. 745-761 ◽  
Author(s):  
Samuele Anni
Keyword(s):  
Number Fields ◽  
Prime Degree ◽  
Global Principle

Power Integral Bases in Composits of Number Fields

1998 ◽  
Vol 41 (2) ◽  
pp. 158-165 ◽  
Author(s):  
István Gaál
Keyword(s):  
Number Fields ◽  
Parametric Family ◽  
Prime Degree ◽  
Index Form ◽  
Form Equation ◽  
Integral Bases ◽  
Quadratic Extensions ◽  
Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

Trace form associated to cyclic number fields of ramified odd prime degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050080
Author(s):  
Robson R. Araujo ◽  
Ana C. M. M. Chagas ◽  
Antonio A. Andrade ◽  
Trajano P. Nóbrega Neto
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Prime Degree ◽  
Ring Of Integers ◽  
Algebraic Lattices ◽  
Center Density

In this work, we computate the trace form [Formula: see text] associated to a cyclic number field [Formula: see text] of odd prime degree [Formula: see text], where [Formula: see text] ramified in [Formula: see text] and [Formula: see text] belongs to the ring of integers of [Formula: see text]. Furthermore, we use this trace form to calculate the expression of the center density of algebraic lattices constructed via the Minkowski embedding from some ideals in the ring of integers of [Formula: see text].

scholarly journals ON THE NUMBER OF ELLIPTIC CURVES WITH PRESCRIBED ISOGENY OR TORSION GROUP OVER NUMBER FIELDS OF PRIME DEGREE

2014 ◽  
Vol 57 (2) ◽  
pp. 465-473 ◽  
Author(s):  
FILIP NAJMAN
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
Torsion Group ◽  
Number Fields ◽  
Prime Degree

AbstractLet p be a prime and K a number field of degree p. We determine the finiteness of the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup or that it has a cyclic isogeny of prescribed degree.

p-Class groups of cyclic number fields of odd prime degree

2016 ◽  
Vol 10 ◽  
pp. 429-435
Author(s):  
Jose Valter Lopes Nunes ◽  
J. Carmelo Interlando ◽  
Trajano Pires da Nobrega Neto ◽  
Jose Othon Dantas Lopes
Keyword(s):  
Number Fields ◽  
Class Groups ◽  
Prime Degree

scholarly journals Local-global principles for Weil–Châtelet divisibility in positive characteristic

2017 ◽  
Vol 163 (2) ◽  
pp. 357-367 ◽  
Author(s):  
BRENDAN CREUTZ ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curve ◽  
Rational Point ◽  
Positive Characteristic ◽  
Number Fields ◽  
Global Field ◽  
Global Fields ◽  
Characteristic 2 ◽  
Global Principles ◽  
Global Principle

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

scholarly journals Finding Normal Integral Bases of Cyclic Number Fields of Prime Degree

2000 ◽  
Vol 30 (2) ◽  
pp. 129-136
Author(s):  
Vincenzo Acciaro ◽  
Claus Fieker
Keyword(s):  
Number Fields ◽  
Prime Degree ◽  
Integral Bases

ON THE DISTRIBUTION OF CYCLIC NUMBER FIELDS OF PRIME DEGREE

2012 ◽  
Vol 08 (06) ◽  
pp. 1463-1475
Author(s):  
SEOK HYEONG LEE ◽  
GYUJIN OH
Keyword(s):  
Riemann Hypothesis ◽  
Error Term ◽  
London Math ◽  
Number Fields ◽  
Positive Real ◽  
Galois Extensions ◽  
Prime Degree ◽  
Abelian Extensions ◽  
Generalized Riemann Hypothesis

Let NCp(X) denote the number of Cp Galois extensions of ℚ with absolute discriminant ≤ X. A well-known theorem of Wright [Density of discriminants of abelian extensions, Proc. London Math. Soc. 58 (1989) 17–50] implies that when p is prime, we have [Formula: see text] for some positive real c(p). In this paper, we improve this result by reducing the secondary error term to [Formula: see text]. Moreover, under Generalized Riemann Hypothesis, we obtain the following stronger result [Formula: see text] Here Rp(x) ∈ ℝ[x] is a polynomial of degree ⌊p(p-2)/3⌋-1. This confirms a speculation of Cohen, Diaz y Diaz and Olivier [Counting discriminants of number fields, J. Théor. Nombres Bordeaux 18 (2006) 573–593] in the case of C3 extensions.

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