An additive function on a ring of integers in the imaginary quadratic field Q(√d) with class-number one

1995 ◽  
Vol 11 (1) ◽  
pp. 68-73
Author(s):  
Cai Tianxin
Keyword(s):  
Additive Function ◽  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

A certain eta-quotient and the class number of an imaginary quadratic field

2014 ◽  
Vol 10 (06) ◽  
pp. 1485-1499
Author(s):  
Takeshi Ogasawara
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Theta Series ◽  
Imaginary Quadratic Field

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Classification of Maximal Fuchsian Subsgroups of Some Bianchi Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 417-422 ◽  
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Quadratic Field ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

AbstractLet d = 1,2, or p, prime p ≡ 3 (mod 4). Let Od be the ring of integers of an imaginary quadratic field A complete classification of conjugacy classes of maximal non-elementary Fuchsian subgroups of PSL(2, Od) in PGL(2, Od) is given.

scholarly journals An explicit Baker-type lower bound of exponential values

2015 ◽  
Vol 145 (6) ◽  
pp. 1153-1182 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen ◽  
Kalle Leppälä ◽  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Linear Form ◽  
Quadratic Field ◽  
Rational Numbers ◽  
Rational Points ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1, . . . , eαm,m⩾ 2, whereα0= 0,α1, . . . ,αmarem+ 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies onmare improved.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals Generalized Ono invariant and Rabinovitch’s theorem for real quadratic fields

1988 ◽  
Vol 109 ◽  
pp. 117-124 ◽  
Author(s):  
Ryuji Sasaki
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Ring Of Integers ◽  
Real Quadratic

Let d be a square-free integer. Letand {1, ω} forms a Z-basis for the ring of integers of the quadratic field We denote by Δ and hd the discriminant and the class number of respectively.

scholarly journals Units in families of totally complex algebraic number fields

2004 ◽  
Vol 2004 (45) ◽  
pp. 2383-2400
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Algebraic Number ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Number Fields ◽  
Imaginary Quadratic Field ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Multidimensional Continued Fraction ◽  
Fundamental Units

Multidimensional continued fraction algorithms associated withGLn(ℤk), whereℤkis the ring of integers of an imaginary quadratic fieldK, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.

scholarly journals THE DIVISIBILITY OF THE CLASS NUMBER OF THE IMAGINARY QUADRATIC FIELD

2011 ◽  
Vol 54 (1) ◽  
pp. 149-154 ◽  
Author(s):  
ZHU MINHUI ◽  
WANG TINGTING
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Positive Integers

AbstractLet hK denote the class number of the imaginary quadratic field $K=\mathbf{Q}(\sqrt{2^{2m}-k^n})$, where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

scholarly journals Limiting equations for the class number of the ideals of an imaginary quadratic field

1965 ◽  
Vol 5 (2) ◽  
pp. 303-305
Author(s):  
O. Saparnijazov
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: О. Сапарниязов. Асимптотические равенства для числа классов идеалов мнимого квадратического поля O. Saparnijazovas. Menamojo kvadratinio skaičių kūno idealų klasių skaičiaus asimptotinės išraiškos

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