scholarly journals Some real quadratic fields with infinite Hilbert 2-class field towers

2005 ◽  
Vol 31 (1) ◽  
pp. 175-181 ◽  
Author(s):  
Frank GERTHIII
Keyword(s):  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Class Field Towers ◽  
Real Quadratic

Power Residue Criteria for Quadratic Units and the Negative Pell Equation

2003 ◽  
Vol 46 (1) ◽  
pp. 39-53 ◽  
Author(s):  
Tommy Bülow
Keyword(s):  
Field Theory ◽  
Class Group ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Pell Equation ◽  
Class Field ◽  
Integer Power ◽  
Real Quadratic Fields ◽  
Solvability Criterion ◽  
Real Quadratic

AbstractLet d > 1 be a square-free integer. Power residue criteria for the fundamental unit εd of the real quadratic fields modulo a prime p (for certain d and p) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation x2 − dp2y2 = −1. The most important solvability criterion deals with all d for which has an elementary abelian 2-class group and p ≡ 5 (mod 8) or p ≡ 9 (mod 16).

scholarly journals Computing the Hilbert class field of real quadratic fields

1999 ◽  
Vol 69 (231) ◽  
pp. 1229-1245 ◽  
Author(s):  
Henri Cohen ◽  
Xavier-François Roblot
Keyword(s):  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Hilbert Class Field ◽  
Real Quadratic

scholarly journals Real Quadratic Fields with Abelian 2-Class Field Tower

1998 ◽  
Vol 73 (2) ◽  
pp. 182-194 ◽  
Author(s):  
Elliot Benjamin ◽  
Franz Lemmermeyer ◽  
C. Snyder
Keyword(s):  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Class Field Tower ◽  
Real Quadratic

LE 2-RANG DU GROUPE DE CLASSES DE CERTAINS CORPS BIQUADRATIQUES ET APPLICATIONS

2004 ◽  
Vol 15 (02) ◽  
pp. 169-182
Author(s):  
ABDELMALEK AZIZI ◽  
ALI MOUHIB
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Biquadratic Fields ◽  
Class Field Tower ◽  
Real Quadratic

Let K be a real biquadratic field and let k be a quadratic field with odd class number contained in K. The aim of this article is to determine the rank of the 2-class group of K and we give applications to the structure of the 2-class group of some biquadratic fields and to the 2-class field tower of some real quadratic fields. Résumé: Soient K un corps biquadratique réel et k un sous-corps quadratique de K dont le nombre de classes est impair. Dans ce papier on détermine le rang du 2-groupe de classes de K et on donne des applications à la structure du 2-groupe de classes de certains corps biquadratiques et aussi à la tour des 2-corps de classes de Hilbert de certains corps quadratiques réels.

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

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