scholarly journals Torsion of rational elliptic curves over different types of cubic fields

2020 ◽  
Vol 16 (06) ◽  
pp. 1307-1323
Author(s):  
Daeyeol Jeon ◽  
Andreas Schweizer
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group ◽  
Number Fields ◽  
Cubic Field ◽  
Different Types ◽  
Cubic Fields ◽  
Cubic Number Fields ◽  
Totally Real

Let [Formula: see text] be an elliptic curve defined over [Formula: see text], and let [Formula: see text] be the torsion group [Formula: see text] for some cubic field [Formula: see text] which does not occur over [Formula: see text]. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) [Formula: see text] can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves [Formula: see text] together with cubic fields [Formula: see text] so that [Formula: see text].

Torsion Groups of Elliptic Curves with Integral j-Invariant over General Cubic Number Fields

1997 ◽  
Vol 07 (03) ◽  
pp. 353-413 ◽  
Author(s):  
Attila Pethö ◽  
Thomas Weis ◽  
Horst G. Zimmer
Keyword(s):  
Elliptic Curves ◽  
Torsion Group ◽  
Number Fields ◽  
Elimination Theory ◽  
Group Problem ◽  
New Methods ◽  
Cubic Fields ◽  
Cubic Number Fields ◽  
Torsion Groups ◽  
Cyclic Cubic Fields

In [15] and [16] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely any) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.

scholarly journals EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS

2012 ◽  
Vol 08 (05) ◽  
pp. 1231-1246 ◽  
Author(s):  
FILIP NAJMAN
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Torsion Group ◽  
Modular Curve ◽  
Quartic Field ◽  
Cubic Fields

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = ℤ/mℤ⊕ℤ/nℤ, where m|n, be a torsion group such that the modular curve X1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

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 On the torsion group of elliptic curves induced by D(4)-triples

2014 ◽  
Vol 22 (2) ◽  
pp. 79-90 ◽  
Author(s):  
Andrej Dujella ◽  
Miljen Mikić
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group

AbstractA D(4)-m-tuple is a set of m integers such that the product of any two of them increased by 4 is a perfect square. A problem of extendibility of D(4)-m-tuples is closely connected with the properties of elliptic curves associated with them. In this paper we prove that the torsion group of an elliptic curve associated with a D(4)-triple can be either ℤ/2ℤ × ℤ/2ℤ or ℤ/2ℤ × ℤ/6ℤ, except for the D(4)-triple {−1, 3, 4} when the torsion group is ℤ/2ℤ × ℤ/4ℤ.

scholarly journals The Selmer groups and the ambiguous ideal class groups of cubic fields

1996 ◽  
Vol 54 (2) ◽  
pp. 267-274
Author(s):  
Yen-Mei J. Chen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Selmer Groups ◽  
Ideal Class ◽  
Selmer Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Cubic Fields ◽  
Rational Isogeny

In this paper, we study a family of elliptic curves with CM by which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields. We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.

A note on Cusick's theorem on units in totally real cubic fields

1984 ◽  
Vol 95 (1) ◽  
pp. 1-2 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Present Note ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

Let ε = ε1, with conjugates ε2, ε3, be a unit in a totally real cubic field, and let . Let ε be a unit for which T (ε) is least and let η be a unit, not a power of ε, for which T(η) is least. It was shown by Cusick[l] that ε,η form a pair of fundamental units under certain conditions. The purpose of the present note is to show that these conditions are unnecessary and that ε, η form a pair of fundamental units in all cases.

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