2025/02/05 by Plessis, Arnaud, Sahoo, Satyabrat
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2502.03039
Let v be a finite place of a number field K and write Knr,v for the maximal field extension of K in which v is unramified. The purpose of this paper is split up into two parts. The first one generalizes a theorem of Pottmeyer: If E is an elliptic curve defined over K with split multiplicative reduction at v, then the Néron-Tate height of a non-torsion point P∈ E(K) is bounded from below by C / ev(P)2 ev(P)+1, where C>0 is an absolute constant and ev(P) is the maximum of all ramification indices ew(K(P) \vert K) with w\vert v. Among other things, we refine this result by showing that given a simple abelian variety A defined over K that is degenerate at v, the Néron-Tate height of a non-torsion point P∈ A(K) is at least C / lcmw\vert v \ew(K(P)\vert K)\2, where C>0 is an absolute constant. We then give applications towards Lehmer's conjecture. Next, we provide the first examples of polynomials ϕ∈ K[X] of degree at least 2 so that the canonical height hϕ of any point in \bbP1(Knr,v) is either 0 or bounded from below by an absolute positive constant.