An explicit bound for the least prime ideal in the Chebotarev density theorem

Jesse Thorner; Asif Zaman

doi:10.2140/ant.2017.11.1135

An explicit bound for the least prime ideal in the Chebotarev density theorem

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Abstract

We prove an explicit version of Weiss’ bound on the least norm of a prime ideal in the Chebotarev density theorem, which is a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. As an application, we prove the first explicit, nontrivial, and unconditional upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also consider applications to elliptic curves and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

Original language	English (US)
Pages (from-to)	1135-1197
Number of pages	63
Journal	Algebra and Number Theory
Volume	11
Issue number	5
DOIs	https://doi.org/10.2140/ant.2017.11.1135
State	Published - 2017
Externally published	Yes

Keywords

Binary quadratic forms
Chebotarev density theorem
Elliptic curves
Least prime ideal
Linnik’s theorem
Log-free zero density estimate
Modular forms

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.2140/ant.2017.11.1135

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title = "An explicit bound for the least prime ideal in the Chebotarev density theorem",

abstract = "We prove an explicit version of Weiss{\textquoteright} bound on the least norm of a prime ideal in the Chebotarev density theorem, which is a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. As an application, we prove the first explicit, nontrivial, and unconditional upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also consider applications to elliptic curves and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.",

keywords = "Binary quadratic forms, Chebotarev density theorem, Elliptic curves, Least prime ideal, Linnik{\textquoteright}s theorem, Log-free zero density estimate, Modular forms",

author = "Jesse Thorner and Asif Zaman",

note = "Funding Information: The authors thank John Friedlander, V. Kumar Murty, Robert Lemke Oliver, Ken Ono, David Zureick-Brown, and the anonymous referee for their comments and suggestions. Thorner conducted work on this paper while visiting Centre de Recherches Math{\'e}matiques (hosted by Chantal David, Andrew Granville, and Dimitris Koukoulopoulos) and Stanford University (hosted by Robert Lemke Oliver and Kannan Soundararajan); he is grateful to these departments and hosts for providing a vibrant work environment. Zaman was supported in part by an NSERC PGS-D scholarship. Publisher Copyright: {\textcopyright} 2017, Mathematical Sciences Publishers. All rights reserved.",

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doi = "10.2140/ant.2017.11.1135",

language = "English (US)",

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AU - Thorner, Jesse

AU - Zaman, Asif

N1 - Funding Information: The authors thank John Friedlander, V. Kumar Murty, Robert Lemke Oliver, Ken Ono, David Zureick-Brown, and the anonymous referee for their comments and suggestions. Thorner conducted work on this paper while visiting Centre de Recherches Mathématiques (hosted by Chantal David, Andrew Granville, and Dimitris Koukoulopoulos) and Stanford University (hosted by Robert Lemke Oliver and Kannan Soundararajan); he is grateful to these departments and hosts for providing a vibrant work environment. Zaman was supported in part by an NSERC PGS-D scholarship. Publisher Copyright: © 2017, Mathematical Sciences Publishers. All rights reserved.

PY - 2017

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N2 - We prove an explicit version of Weiss’ bound on the least norm of a prime ideal in the Chebotarev density theorem, which is a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. As an application, we prove the first explicit, nontrivial, and unconditional upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also consider applications to elliptic curves and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

AB - We prove an explicit version of Weiss’ bound on the least norm of a prime ideal in the Chebotarev density theorem, which is a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. As an application, we prove the first explicit, nontrivial, and unconditional upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also consider applications to elliptic curves and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

KW - Binary quadratic forms

KW - Chebotarev density theorem

KW - Elliptic curves

KW - Least prime ideal

KW - Linnik’s theorem

KW - Log-free zero density estimate

KW - Modular forms

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DO - 10.2140/ant.2017.11.1135

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JO - Algebra and Number Theory

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IS - 5

ER -

An explicit bound for the least prime ideal in the Chebotarev density theorem

Abstract

Keywords

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