Gaussian Number Field Research Articles

Iterated towers of number fields by a quadratic map defined over the Gaussian rationals

An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of $\mathbf{Z}_{2}$-extensions, we examine the parity of 2-ideal class numbers along the towers with some examples.

A Three Dimensional Modification of the Gaussian Number Field

Abstract For vectors in E 3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications. Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

A ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients

We determine the structure over $$\mathbb {Z}$$ of a ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients whose weights are multiples of 4 when the base field is the Gaussian number field $$\mathbb {Q}(\sqrt{-1})$$. Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds for such Hermitian modular forms of weight k with $$4\mid k$$, for $$p=2$$, 3. We remark that the bounds for $$p\ge 5$$ are already known.

Subconvexity for twisted 𝐿-functions on 𝐺𝐿₃ over the Gaussian number field

Let q ∈ Z [ i ] q \in \mathbb {Z} [i] be prime, and let χ \chi be the primitive quadratic Hecke character modulo q q . Let π \pi be a self-dual Hecke automorphic cusp form for S L 3 ( Z [ i ] ) \mathrm {SL}_3 (\mathbb {Z} [i] ) , and let f f be a Hecke cusp form for Γ 0 ( q ) ⊂ S L 2 ( Z [ i ] ) \Gamma _0 (q) \subset \mathrm {SL}_2 (\mathbb {Z} [i]) . Consider the twisted L L -functions L ( s , π ⊗ f ⊗ χ ) L (s, \pi \otimes f \otimes \chi ) and L ( s , π ⊗ χ ) L (s, \pi \otimes \chi ) on G L 3 × G L 2 \mathrm {GL}_3 \times \mathrm {GL}_2 and G L 3 \mathrm {GL}_3 . We prove the subconvexity bounds L ( 1 2 , π ⊗ f ⊗ χ ) ≪ ε , π , f N ( q ) 5 / 4 + ε , L ( 1 2 + i t , π ⊗ χ ) ≪ ε , π , t N ( q ) 5 / 8 + ε \begin{equation*} L \big (\tfrac 1 2, \pi \otimes f \otimes \chi \big ) \ll _{\, \varepsilon , \pi , f } \mathrm {N} (q)^{5/4 + \varepsilon }, \quad L \big (\tfrac 1 2 + it, \pi \otimes \chi \big ) \ll _{\, \varepsilon , \pi , t } \mathrm {N} (q)^{5/8 + \varepsilon } \end{equation*} for any ε > 0 \varepsilon > 0 .

Differential operators for Hermitian Jacobi forms and Hermitian modular forms

Kim (Arch Math (Basel) 79(3):208–215, 2002) constructs multilinear differential operators for Hermitian Jacobi forms and Hermitian modular forms. However, her work relies on incorrect actions of differential operators on spaces of Hermitian Jacobi forms and Hermitian modular forms. In particular, her results are incorrect if the underlying field is the Gaussian number field. We consider more general spaces of Hermitian Jacobi forms and Hermitian modular forms over \(\mathbb {Q}(i)\), which allow us to correct the corresponding results in Kim (2002).

A New Secure Authenticated Key Agreement Protocol in Gaussian Field

Key agreement protocols are a fundamental building block for ensuring authenticated and private communications between two parties over an insecure network. In this paper we propose an efficient and secure authenticated key agreement protocol based on DLP (Discrete Logarithm Problem) and Gaussian number field. The main purpose of this paper is to use the Gaussian integers; the set of all complex numbers ai b + with , ab Z ∈ in the Gaussian integers, to design new key agreement protocol that can help the system to be more secure. We show that our protocol meets the security attributes and strong against most of potential attacks. Also it provides most of desirable performance attributes.

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.

Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field

We prove the Kloosterman-Spectral sum formula for $\mathrm{PSL}_{2}(\mathbb{Z}[i])\backslash \mathrm{PSL}_{2}(\mathbb{C})$, and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta-function of the Gaussian number field.

A note on the mean value of the zeta and $L$-functions. X

The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$ of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$, and on a new version of the sum formula of Kuznetsov type for $\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$. Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$ gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting on the three dimensional hyperbolic space (the $K$-trivial situation). To solve this problem, it was necessary to include the $K$-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.

Asymptotic Normality of b-Additive Functions on Polynomial Sequences in the Gaussian Number Field

We consider the asymptotic behavior of b-additive functions f with respect to a base b of a canonical number system in the Gaussian number field. In particular, we get a normal limit law for f(P(z)) where P(z) is a polynomial with integer coefficients. Our methods are exponential sums over the Gaussian number field as well as certain results from the theory of uniform distribution.