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

Anup B. Dixit; Sushant Kala

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arXiv · 2309.15872

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

Anup B. Dixit·Sushant Kala

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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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Anup B. Dixit, Sushant Kala. 2023-09-27. Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function. https://arxiv.org/abs/2309.15872

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