Distribution of orders in number fields

Abstract

In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields.

Counting finite index subrings of $\mathbb Z^n$

“…The conditions for the columns of an n × n matrix to generate a multiplicatively closed sublattice of Z n define many equations in the matrix entries. For examples for n = 4 and 5, see [18,Lemmas 12 and 13]. It is possible that once n and e are large enough, varieties V t occur for which the functions b t (p) in Theorem 5.1 are not polynomials in p, and that these functions occur in formulas for f n (p e ).…”

“…The authors of [18] derive the asymptotic order of growth for N R 5 (X) up to a constant factor, despite not having a formula for ζ R Z 5 (s) analogous to those of Theorem 1.4. The main idea is to find the location and order of the rightmost pole of ζ R Z n (s) by computing f n (p e ) exactly for small e and giving estimates for larger e. A major motivation for the computations of the present paper is to try to prove stronger versions of Theorem 1.5.…”

“…In the statement of the theorem below, n = [K : Q] and r 2 is an explicitly computable positive integer that depends on the Galois group of the normal closure of K/Q; see [18] for details.…”

“…The authors of [18] suggest that among all unramified primes, those that split completely may control the asymptotic growth rate of N K (X). This suggests that the growth rate of the simpler function N R n (X) along with the Galois group of the normal closure of K may determine the growth rate of N K (X).…”

“…This suggests that the growth rate of the simpler function N R n (X) along with the Galois group of the normal closure of K may determine the growth rate of N K (X). For more information, see the discussion following [18,Theorem 4].…”