The maximal discrete extension of $$SL_2(\mathcal {\scriptstyle {O}}_K)$$ S L 2 ( O K ) for an imaginary quadratic number field K

2019 ◽  
Vol 113 (1) ◽  
pp. 37-41
Author(s):  
Aloys Krieg ◽  
Joana Rodriguez ◽  
Annalena Wernz
Keyword(s):  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Imaginary Quadratic Number

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

2006 ◽  
Vol 02 (04) ◽  
pp. 569-590 ◽  
Author(s):  
BORKA JADRIJEVIĆ ◽  
VOLKER ZIEGLER
Keyword(s):  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Root Of Unity ◽  
Related Family ◽  
The Family ◽  
Thue Equation ◽  
Imaginary Quadratic Number ◽  
Pellian Equations

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

scholarly journals Picard Groups and Refined Discrete Logarithms

2005 ◽  
Vol 8 ◽  
pp. 1-16 ◽  
Author(s):  
W. Bley ◽  
M. Endres
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Finite Abelian Group ◽  
Discrete Logarithm ◽  
Quadratic Number ◽  
Prime Degree ◽  
Quadratic Number Field ◽  
Picard Groups ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

scholarly journals On a criterion for the class number of a quadratic number field to be one

1980 ◽  
Vol 79 ◽  
pp. 123-129 ◽  
Author(s):  
Masakazu Kutsuna
Keyword(s):  
Number Field ◽  
Class Number ◽  
Euclidean Algorithm ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Imaginary Quadratic Number ◽  
Necessary And Sufficient

G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:Theorem.It is necessary and sufficient for the class number of an imaginary quadratic number fieldD= 1 — 4m, m> 0,to be one that x2—x+m is prime for any integer x such that1 ≤x≤m— 2.

scholarly journals Linear Independence of -Logarithms over the Eisenstein Integers

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Peter Bundschuh ◽  
Keijo Väänänen
Keyword(s):  
Number Field ◽  
Linear Independence ◽  
Algebraic Number Field ◽  
Meromorphic Continuation ◽  
Quadratic Number ◽  
Arbitrary Integer ◽  
Quadratic Number Field ◽  
Linearly Independent ◽  
Imaginary Quadratic Number ◽  
Fixed Complex

For fixed complex with , the -logarithm is the meromorphic continuation of the series , into the whole complex plane. If is an algebraic number field, one may ask if are linearly independent over for satisfying . In 2004, Tachiya showed that this is true in the Subcase , , , and the present authors extended this result to arbitrary integer from an imaginary quadratic number field , and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if is the Eisenstein number field , an integer from , and a primitive third root of unity. Under these conditions, the linear independence holds also for , and both results are quantitative.

scholarly journals Graded rings of Hermitian modular forms with singularities

2022 ◽  
Author(s):  
Haowu Wang ◽  
Brandon Williams
Keyword(s):  
Number Field ◽  
Modular Forms ◽  
Polynomial Algebra ◽  
Projective Spaces ◽  
Graded Rings ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Hermitian Modular Forms ◽  
Imaginary Quadratic Number

AbstractWe study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $$\mathrm {SU}_{2,2}({\mathcal {O}}_K)$$ SU 2 , 2 ( O K ) where K is the imaginary-quadratic number field of discriminant $$-d$$ - d , $$d \in \{4, 7,8,11,15,19,20,24\}$$ d ∈ { 4 , 7 , 8 , 11 , 15 , 19 , 20 , 24 } we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

2016 ◽  
Vol 12 (06) ◽  
pp. 1625-1639
Author(s):  
Vladislav Petkov
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Quadratic Number ◽  
Converse Theorem ◽  
Quadratic Number Field ◽  
Converse Theorems ◽  
Double Dirichlet Series ◽  
Imaginary Quadratic Number ◽  
Metaplectic Cover ◽  
Integer Weight

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

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