Universal Quadratic Forms, Small Norms, and Traces in Families of Number Fields

Abstract

We obtain good estimates on the ranks of universal quadratic forms over Shanks’ family of the simplest cubic fields and several other families of totally real number fields. As the main tool, we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.

Number fields without universal quadratic forms of small rank exist in most degrees

“…Kala-Tinková [KT,Subsection 7•2] proved that there are (infinitely many) such fields L that contain n elements a 1 , a 2 , . .…”

“…Kala–Tinková [ KT , Subsection 7·2] proved that there are (infinitely many) such fields L that contain n elements and an element in the codifferent with for all i . By [ KT , Subsection 7·2 and Proof of Proposition 7·4], if an -lattice represents all the elements , then it has rank .…”

“…Further, the ranks can be arbitrarily large [ BK1 ], [ Ka ], ditto for multiquadratic fields of a given degree [ KS ]. Despite a number of other exciting results obtained in the last 25 years [ BK2, CL+, EK, KT, KY, Ki, KKP, KTZ, Ya ], ranks of universal forms over number fields, especially of higher degree, remain mysterious.…”

Cyclic polytope of the simplest cubic fields

“…The simplest cubic fields are the splitting fields of f a and were introduced by Shanks in [12] with the goal of finding number fields with a large class number. Since then, this family became a "testing ground" for many investigations of cubic number fields [7,9,15]. They have several favorable properties: the discriminant is a perfect square, expressed by ∆ = (a 2 + 3a + 9) 2 ; the field is monogenic for infinitely many values of a; the roots of f a are given by a simple explicit formula (see (2.1)) and converge to rational integers when a is large; and the regulator is explicitly known (and relatively small, thus giving a large class number).…”

“…3 ). The number |P ∩ Z 2 | can be used to bound the ranks of the universal quadratic forms defined over the number field [16], yet, the knowledge of |kP ∩ Z 2 | can be applied to sharpen those bounds (as recently was applied in [7]).…”