On some families of imaginary quadratic fields

Visualizing imaginary quadratic fields

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...

Some remarks for codes and lattices over imaginary quadratic fields

Let ` > 0 be a square-free integer, ` ≡ 3 mod 4, K = Q( √ −`), and OK the ring of integers of K. Codes C over rings R := OK/pOK determine lattices Λ`(C) over K. The theta series θΛ`(C) of such lattice can be written in terms of the complete weight enumerator of C. For any `′ > ` the first `+1 4 terms of their corresponding theta functions are the same with those of Λ`(C). In [6] it was conjectu...

Visualising the arithmetic of quadratic imaginary fields

We study the orbit of R under the Bianchi group PSL2(OK), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK , is a geometric realisation, as an intricate circle packing, of the arithmetic of K. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of √ −∆ and describe the curvatures of t...

Class numbers of imaginary quadratic fields

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N . The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N . Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work ...

Zp-Extensions of Imaginary Quadratic Fields