Supersingular primes and Bogomolov property

TL;DR

The paper extends Habegger’s Bogomolov-type results to fields generated by torsion points of elliptic curves. It leverages Plessis's general criterion, reducing the problem to finding a supersingular prime with specific properties; in the real-embedding setting, Elkies ensures infinitely many such primes, enabling the argument for both $L(E_{\mathrm{tor}})$ and $E(L(E_{\mathrm{tor}}))$ to satisfy the Bogomolov property, with CM cases handled via abelian-closure arguments. It also provides explicit height bounds in terms of a supersingular prime and treats the minimal field of definition $\mathbb{Q}(j_E)$ to unify the approach. The results yield a broad verification of the elliptic Bogomolov property for a large class of torsion-generated extensions and offer concrete lower bounds for heights. These findings connect Galois representations, supersingular primes, and height theory to establish discreteness phenomena for torsion-related fields.

Abstract

Let $E$ be an elliptic curve over a number field $K$ with at least one real embedding and $L$ be a finite extension of $K$. We generalize a result of Habegger to show that $L(E_{\text{tor}})$, the field generated by the torsion points of $E$ over $L$, has the Bogomolov property. Moreover, the Néron-Tate height on $E\big(L(E_{\text{tor}})\big)$ also satisfies a similar discreteness property. Our main tool is a general criterion of Plessis that reduces the problem to the existence of a supersingular prime for $E$ satisfying certain conditions.

Theorems & Definitions (14)

• Theorem 1.1
• Conjecture
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Proposition 2.2
• proof