Graded rings of Hermitian modular forms with singularities

We 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.

The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ , c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

Hyperbolic tessellations and generators of for imaginary quadratic fields

We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

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

Let $$\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})$$ 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.