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.

On quaternion algebras over the composite of quadratic number fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

Redei reciprocity, governing fields and negative Pell

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation \[{x^2} - d{y^2} = - 1\] .

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}).