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.
AMS Subject Classification: Primary 11M41; 11R29; secondary 11S40
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Acknowledgements
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.
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Kaplan, N., Marcinek, J. & Takloo-Bighash, R. Distribution of orders in number fields. Mathematical Sciences 2, 6 (2015). https://doi.org/10.1186/s40687-015-0027-8
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DOI: https://doi.org/10.1186/s40687-015-0027-8