Abstract
AbstractIn 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.AMS Subject Classification:Primary 11M41; 11R29; secondary 11S40
Highlights
Let K/Q be an extension of degree n with ring of integers OK
An order O is a subring of OK with identity that is a Z-module of rank n
Combining everything done so far, one concludes that the function f (s) in the statement of Theorem 3 has a pole at s = 1 of order r a( f )b( f )
Summary
Let K/Q be an extension of degree n with ring of integers OK. An order O is a subring of OK with identity that is a Z-module of rank n. We use the expression (3) to write the number of unitary subrings of a given index in terms of volumes of certain p-adic sets. Multivariable cone integral For a finite extension F of Qp, we let be OF its ring of integers, p the maximal ideal, |.|F its normalized absolute value, and vF the corresponding discrete valuation.
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