scholarly journals Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree

2021 ◽  
pp. 1
Author(s):  
Tomislav Gužvić
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Prime Degree

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.

scholarly journals Determinants of subquotients of Galois representations associated with abelian varieties

2013 ◽  
Vol 13 (3) ◽  
pp. 517-559 ◽  
Author(s):  
Eric Larson ◽  
Dmitry Vaintrob
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Number Fields ◽  
Torsion Points ◽  
Prime Degree ◽  
Rho A

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

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.

scholarly journals Darmon points on elliptic curves over number fields of arbitrary signature

2015 ◽  
Vol 111 (2) ◽  
pp. 484-518 ◽  
Author(s):  
Xavier Guitart ◽  
Marc Masdeu ◽  
Mehmet Haluk Şengün
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Arbitrary Signature

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].

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