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.

scholarly journals Hermitian Theta Series and Maaß Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
Annalena Wernz
Keyword(s):  
Number Field ◽  
Modular Forms ◽  
Unitary Group ◽  
Modular Group ◽  
Theta Series ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Special Unitary Group ◽  
Imaginary Quadratic Number

AbstractLet $$\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})$$ Γ n ( O K ) denote the Hermitian modular group of degree n over an imaginary quadratic number field $$\mathbb {K}$$ K and $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ its maximal discrete extension in the special unitary group $$SU(n,n;\mathbb {C})$$ S U ( n , n ; C ) . In this paper we study the action of $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ on Hermitian theta series and Maaß spaces. For $$n=2$$ n = 2 we will find theta lattices such that the corresponding theta series are modular forms with respect to $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ consolidates the different definitions of the spaces.

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.

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