Congruence Between Bernoulli Numbers and Class Numbers of Imaginary Quadratic Fields

2014 ◽  
pp. 95-102
Author(s):  
Tomoyoshi Ibukiyama ◽  
Masanobu Kaneko
Keyword(s):  
Bernoulli Numbers ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields

scholarly journals Class numbers of imaginary quadratic fields

2003 ◽  
Vol 73 (246) ◽  
pp. 907-939 ◽  
Author(s):  
Mark Watkins
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields

scholarly journals On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

2021 ◽  
Vol 197 (1) ◽  
pp. 105-110
Author(s):  
Jaitra Chattopadhyay ◽  
Subramani Muthukrishnan
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields

scholarly journals Divisibility criteria for class numbers of imaginary quadratic fields

2006 ◽  
Vol 125 (3) ◽  
pp. 285-289 ◽  
Author(s):  
Paul Jenkins ◽  
Ken Ono
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields

scholarly journals REMARKS ON THE DIVISIBILITY OF THE CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

2011 ◽  
Vol 53 (2) ◽  
pp. 379-389 ◽  
Author(s):  
AKIKO ITO
Keyword(s):  
Prime Number ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe consider the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$, where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - 3^n})$ when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-1})$ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 7 mod 8.

scholarly journals The fundamental unit and bounds for class numbers of real quadratic fields

1991 ◽  
Vol 124 ◽  
pp. 181-197 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

Coefficients of half-integral weight modular forms and indivisibility of class numbers of imaginary quadratic fields

2016 ◽  
Vol 12 (02) ◽  
pp. 561-565
Author(s):  
Haobo Dai
Keyword(s):  
Modular Forms ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

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