Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

2023/09/27 by Anup B. Dixit, Dixit, Anup B., Sushant Kala +1 · 1 citation
Mathematics · #11G50 #11M41 #11R04 #11R06 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2309.15872

openalex publication_date 2023/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on GRH. We also introduce and study a condition on prime ideals with small norms for arbitrary infinite extensions, in the spirit of a prime splitting condition for infinite Galois extensions introduced by E. Bombieri and U. Zannier.

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• A p-adic criterion for Lehmer's conjecture

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