scholarly journals On a conjecture of Lemmermeyer

2020 ◽  
Vol 16 (07) ◽  
pp. 1407-1424
Author(s):  
Siham Aouissi ◽  
Mohamed Talbi ◽  
Moulay Chrif Ismaili ◽  
Abdelmalek Azizi
Keyword(s):  
Class Groups ◽  
Root Of Unity ◽  
Cubic Fields

Let [Formula: see text] be a prime and denote by [Formula: see text] a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about [Formula: see text]-class groups of pure cubic fields [Formula: see text] and of their normal closures [Formula: see text]. The main goal of this paper is to reduce Lemmermeyer’s conjecture to a problem of unit theory by showing that the conjecture of Lemmermeyer follows from Conjecture 2.9.

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.

scholarly journals IRREDUCIBILITY OF SOME QUANTUM REPRESENTATIONS OF MAPPING CLASS GROUPS

2001 ◽  
Vol 10 (05) ◽  
pp. 763-767 ◽  
Author(s):  
JUSTIN ROBERTS
Keyword(s):  
Mapping Class Group ◽  
Prime Order ◽  
Class Group ◽  
Mapping Class Groups ◽  
Closed Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Root Of Unity

The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.

scholarly journals On 3-class groups of non-Galois cubic fields

1979 ◽  
Vol 35 (4) ◽  
pp. 395-402 ◽  
Author(s):  
Kiyoaki Iimura
Keyword(s):  
Class Groups ◽  
Cubic Fields

scholarly journals On 3-Class groups of certain pure cubic fields

2005 ◽  
Vol 72 (3) ◽  
pp. 471-476 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Class Groups ◽  
Additional Insight ◽  
Cubic Fields ◽  
Insight Into

Recently Calegari and Emerton made a conjecture about the 3-class groups of certain pure cubic fields and their normal closures. This paper proves their conjecture and provides additional insight into the structure of the 3-class groups of pure cubic fields and their normal closures.

scholarly journals Ranks of 3-class groups of non-Galois cubic fields

1976 ◽  
Vol 30 (4) ◽  
pp. 307-322 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Class Groups ◽  
Cubic Fields

scholarly journals The $2$-class groups of cubic fields and $2$-descents on elliptic curves

1992 ◽  
Vol 44 (4) ◽  
pp. 557-565 ◽  
Author(s):  
Mayumi Kawachi ◽  
Shin Nakano
Keyword(s):  
Elliptic Curves ◽  
Class Groups ◽  
Cubic Fields

scholarly journals On the Absolute Ideal Class Groups of Relatively Meta-Cyclic Number Fields of a Certain Type

1960 ◽  
Vol 17 ◽  
pp. 171-179 ◽  
Author(s):  
Taira Honda
Keyword(s):  
Number Field ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Ideal Class ◽  
Class Groups ◽  
Finite Degree ◽  
Ground Field ◽  
Root Of Unity ◽  
Ideal Class Groups ◽  
A Finite Group

Notations. The following notations will be used throughout this paper.˛: the identity of a finite group.Q: the rational number field.P: an algebraic number field of finite degree, fixed as the ground field.l: a prime number.ζl: a primitive l-th root of unity.

The 3‐class groups of non‐Galois cubic fields–II

Mathematika ◽  
1974 ◽  
Vol 21 (2) ◽  
pp. 168-188 ◽  
Author(s):  
T. Callahan
Keyword(s):  
Class Groups ◽  
Cubic Fields

Elliptic surfaces over ℙ1 and large class groups of number fields

2019 ◽  
Vol 15 (10) ◽  
pp. 2151-2162
Author(s):  
Jean Gillibert ◽  
Aaron Levin
Keyword(s):  
Elliptic Curve ◽  
Large Class ◽  
Class Group ◽  
Number Fields ◽  
Torsion Subgroup ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Elliptic Surfaces ◽  
Cubic Fields

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

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