Published June 10, 2015
| Published + Submitted
Journal Article
Open
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.
Additional Information
© 2015 Kaplan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Received: 14 August 2014; Accepted: 15 April 2015; Published: 10 June 2015. This work owes a great deal of intellectual debt to Ricky Liu's paper [19]. This paper has its genesis in the Princeton senior thesis [18]. The first author wishes to acknowledge support from a National Science Foundation Graduate Research Fellowship. The second author wishes to acknowledge support from the National Science Foundation (Award number DMS-0701753), a grant from the National Security Agency (Award number 111011), and a Collaboration Grant from the Simons Foundation (Award number 245977). In the course of the preparation of this work, we benefited from conversations with Nir Avni, Manjul Bhargava, Alice Medvedev, Alireza Salehi-Golsefidi, and Christopher Voll. We wish to thank Bhama Srinivasan for useful conversations, and especially for her crucial observation Lemma 1, and Eun Hye Lee for numerical computations on UIC's Argo cluster to provide support for Equation (1). Thanks are also due to the referee for reading the paper very carefully, and for pointing out errors and inconsistencies that have led to the improvement of the paper. The authors declare that they have no competing interests.Attached Files
Published - art_3A10.1186_2Fs40687-015-0027-8.pdf
Submitted - 1408.1374
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Additional details
- Eprint ID
- 78280
- Resolver ID
- CaltechAUTHORS:20170616-110336086
- NSF Graduate Research Fellowship
- DMS-0701753
- NSF
- 111011
- National Security Agency
- 245977
- Simons Foundation
- Created
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2017-06-16Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field