scholarly journals INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS

2011 ◽  
Vol 07 (01) ◽  
pp. 1-8
Author(s):  
KIRTI JOSHI ◽  
CAMERON MCLEMAN
Keyword(s):  
Galois Representations ◽  
Number Fields ◽  
Modular Representations ◽  
Infinite Class ◽  
Fixed Field ◽  
Class Field ◽  
Class Field Towers ◽  
K 12 ◽  
Hilbert Class Field ◽  
Class Field Tower

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight k on SL2(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME.

On Hilbert class field tower for some quartic number fields

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Class Field ◽  
Hilbert Class Field ◽  
Class Field Tower

We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck,2 is isomorphic to ℤ/2ℤ×ℤ/2ℤ.

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

2018 ◽  
Vol 237 ◽  
pp. 166-187
Author(s):  
SOSUKE SASAKI
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Generalized Quaternion ◽  
Class Field Tower

Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

scholarly journals Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields

1993 ◽  
Vol 48 (3) ◽  
pp. 379-383 ◽  
Author(s):  
Elliot Benjamin
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Imaginary Quadratic Number ◽  
Hilbert Class Field ◽  
Class Field Tower ◽  
The Ideal

Letkbe an imaginary quadratic number field and letk1be the 2-Hilbert class field ofk. IfCk,2, the 2-Sylow subgroup of the ideal class group ofk, is elementary and |Ck,2|≥ 8, we show thatCk1,2is not cyclic. IfCk,2is isomorphic toZ/2Z×Z/4ZandCk1,2is elementary we show thatkhas finite 2-class field tower of length at most 2.

scholarly journals INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS

2008 ◽  
Vol 50 (1) ◽  
pp. 27-32 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Cyclotomic Field ◽  
Galois Groups ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers ◽  
Hilbert Class Field ◽  
Class Field Tower

AbstractWe use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field $\Q(\exp(2 \pi i/m))$ has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over $\Q(\exp(2 \pi i/m))$ for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of $\Q(\exp(2 \pi i/m))$.

scholarly journals Infinite class field towers of number fields of prime power discriminant

2020 ◽  
Vol 373 ◽  
pp. 107318
Author(s):  
Farshid Hajir ◽  
Christian Maire ◽  
Ravi Ramakrishna
Keyword(s):  
Prime Power ◽  
Number Fields ◽  
Infinite Class ◽  
Class Field ◽  
Class Field Towers

scholarly journals Infinite class field towers

2009 ◽  
Vol 344 (4) ◽  
pp. 923-928 ◽  
Author(s):  
Jing Long Hoelscher
Keyword(s):  
Infinite Class ◽  
Class Field ◽  
Class Field Towers

On Cyclic Fields of Odd Prime Degree p with Infinite Hilbert p-Class Field Towers

2002 ◽  
Vol 45 (1) ◽  
pp. 86-88 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Rational Numbers ◽  
Cyclic Extension ◽  
Prime Degree ◽  
Class Field ◽  
Positive Proportion ◽  
Class Field Towers ◽  
Class Field Tower

AbstractLet k be a cyclic extension of odd prime degree p of the field of rational numbers. If t denotes the number of primes that ramify in k, it is known that the Hilbert p-class field tower of k is infinite if t > 3 + 2 . For each t > 2 + , this paper shows that a positive proportion of such fields k have infinite Hilbert p-class field towers.

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