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

Thorner, Jesse; Zaman, Asif

doi:10.2140/ant.2017.11.1135

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

Jesse Thorner and Asif Zaman

Vol. 11 (2017), No. 5, 1135–1197

DOI: 10.2140/ant.2017.11.1135

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.

Keywords

Chebotarev density theorem, least prime ideal, Linnik's theorem, binary quadratic forms, elliptic curves, modular forms, log-free zero density estimate

Mathematical Subject Classification 2010

Primary: 11R44

Secondary: 11M41, 14H52

Milestones

Received: 12 May 2016

Revised: 25 October 2016

Accepted: 10 March 2017

Published: 12 July 2017

Authors

Jesse Thorner
Department of Mathematics Stanford University Building 380 Sloan Mathematical Center Stanford, CA 94305 United States

Asif Zaman
Department of Mathematics University of Toronto Room 6290 40 St. George St. Toronto, ON M5S 2E4 Canada

Vol. 11, No. 5, 2017