Quadratic Number Rings Research Articles

IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

P-orderings: a metric viewpoint and the non-existence of simultaneous orderings

For a prime ideal ℘ and a subset S of a Dedekind ring R, a ℘- ordering of S is a sequence of elements of S with a certain minimizing property. These ℘-orderings were introduced in Bhargava (J. Reine Angew. Math., 490 (1997) 101) to generalize the usual factorial function and many classical results were thereby extended, including results about integer-valued polynomials. We consider ℘-orderings from the viewpoint of the ℘-adic metric on R. We find that the ℘-sequences of S depend only on the closure of S in R ̂ ℘ . When R is a Dedekind domain and R′ is the integral closure of R in a finite extension of the fraction field of R, we relate the ℘-sequences of R and R′. Lastly, we investigate orderings that are simultaneously ℘-orderings for all prime ideals ℘⊂R, and show that such simultaneous orderings do not exist for imaginary quadratic number rings.

On two-primary algebraic K-theory of quadratic number rings with focus on K₂

We give explicit formulas for the 2-rank of the algebraic K-groups of quadratic number rings. A 4-rank formula for K2 of quadratic number rings given in [1] provides further information about the actual group structure. The K2 claculations are based on 2- and 4-rank formulas for Picard groups of quadratic number fields. These formulas are derived in a completely elementary way from the classical 2-rank formula for the narrow ideal class group of a quadratic number field. We also lift the K2 results to higher algebraic K-groups.

A residue number system implementation of real orthogonal transforms

Previous work has focused on performing residue computations that are quantized within a dense ring of integers in the real domain. The aims of this paper are to provide an efficient algorithm for the approximation of real input signals, with arbitrarily small error, as elements of a quadratic number ring and to prove residual number system moduli restrictions for simplified multiplication within the ring. The new approximation scheme can be used for implementation of real-valued transforms and their multidimensional generalizations.

Shortest division chains in imaginaryquadratic number fields

Let O d be the set of algebraic integers in an imaginary quadratic number field Q[√ d ], d <0, where d is the discriminant of O d . Consider the Euclidean Algorithm (EA), applied to algebraic integers ξ, η∈ O d . It consists in computing a sequence of remainders ρ 0 =ξ, ρ 1 =η, ρ 2 ,…,ρ n+1 =0, where ρ i+1 =ρ i−1 −γ i ρ i for algebraic integers γ i ∈ O d , i=1,…, n . We show that except for d =−11 the number of divisions to be carried out is always minimized by choosing each γ i such that N(ρ i−1 −γ i ρ i ), the norm of ρ i−1 −γ i ρ i , is minimal. This result has been proven previously in special cases. It also applies to those imaginary quadratic number rings which are not Euclidean; in this case the division chains may be infinite. For d =−7, −8 the methods applied so far must be modified somewhat, and for d =−11 we provide a counterexample and a theorem which partially answers the question, how shortest division chains can be obtained.

Representing classes in the Brauer group of quadratic number rings as smash products

Representing classes in the Brauer group of quadratic number rings as smash products