Galois covers of ℙ1 and number fields with large class groups

2021 ◽  
pp. 1-28
Author(s):  
Jean Gillibert ◽  
Pierre Gillibert
Keyword(s):  
Large Class ◽  
Class Group ◽  
Number Fields ◽  
Finite Subgroup ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Galois Extensions ◽  
Galois Covers ◽  
New Formula

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number 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].

scholarly journals A Note on Ideal Class Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 239-247 ◽  
Author(s):  
Kenkichi Iwasawa
Keyword(s):  
Class Group ◽  
Cyclotomic Field ◽  
Theoretical Method ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ideal Class Groups ◽  
The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

The Kernel of and the 4-Rank of K2(O)

1992 ◽  
Vol 35 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Ruth I. Berger
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ring Of Integers

AbstractAn upper bound is given for the order of the kernel of the map on Sideal class groups that is induced by For some special types of number fields F the connection between the size of the above kernel for and the units and norms in are examined. Let K2(O) denote the Milnor K-group of the ring of integers of a number field. In some cases a formula by Conner, Hurrelbrink and Kolster is extended to show how closely the 4-rank of is related to the 4-rank of the S-ideal class group of

scholarly journals Tournaments and Ideal Class Groups

1995 ◽  
Vol 38 (3) ◽  
pp. 330-333
Author(s):  
Robert J. Kingan
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Quadratic Number Fields ◽  
The Ideal

AbstractResults are given for a class of square {0,1}-matrices which provide information about the 4-rank of the ideal class group of certain quadratic number fields.

scholarly journals Exponents of the class groups of imaginary Abelian number fields

1987 ◽  
Vol 35 (2) ◽  
pp. 231-246 ◽  
Author(s):  
A. G. Earnest
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Abelian Field ◽  
The Ideal

It is a classical result, deriving from the Gaussian theory of genera of integral binary quadratic forms, that there exist only finitely many imaginary quadratic fields for which the ideal class group is a group of exponent two. This finiteness has been shown to extend to all those totally imaginary quadratic extensions of any fixed totally real algebraic number field. In this paper we put forward the conjecture that there exist only finitely many imaginary abelian algebraic number fields which have ideal class groups of exponent two, and we examine the extent to which existing methods can be brought to bear on this conjecture. One consequence of the validity of the conjecture would be a proof of the existence of finite abelian groups which do not occur as the ideal class group of any imaginary abelian field.

Circulant Graphs and 4-Ranks of Ideal Class Groups

1994 ◽  
Vol 46 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Jurgen Hurrelbrink
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Number Fields ◽  
Regular Graphs ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Ideal Class Groups ◽  
Yield Information ◽  
The Ideal

AbstractThis is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

scholarly journals On nilpotent factors of congruent ideal class groups of Galois extensions

1976 ◽  
Vol 62 ◽  
pp. 13-28 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Finite Degree ◽  
Ring Of Integers ◽  
Ideal Class Groups ◽  
The Ideal

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

scholarly journals The 2-Ideal Class Groups of ℚ(ζl)

2001 ◽  
Vol 162 ◽  
pp. 1-18 ◽  
Author(s):  
Pietro Cornacchia
Keyword(s):  
Class Group ◽  
Ideal Class ◽  
Galois Module ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
The Ideal

For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

scholarly journals On the Galois module structure of ideal class groups

2001 ◽  
Vol 164 ◽  
pp. 133-146 ◽  
Author(s):  
Toru Komatsu ◽  
Shin Nakano
Keyword(s):  
Class Group ◽  
Class Numbers ◽  
Module Structure ◽  
Ideal Class ◽  
Galois Module ◽  
Ideal Class Group ◽  
Class Groups ◽  
Prime Degree ◽  
Ideal Class Groups ◽  
The Ideal

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

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