On dependence of rational points on elliptic curves

TL;DR

This paper investigates the dependence of rational points on elliptic curves over $\\mathbb{Q}$, providing explicit bounds on primes related to subgroup membership under the assumption of the General Riemann Hypothesis, especially for non-CM, semistable, or non-exceptional cases.

Contribution

It offers explicit prime bounds for detecting rational point membership in subgroups of elliptic curves assuming GRH, extending previous results without such bounds.

Findings

Provides explicit prime bounds under GRH for elliptic curves without CM.

Applies to semistable curves and those with no exceptional primes.

Enhances understanding of rational points and subgroup detection on elliptic curves.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $\Gamma$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\in\Gamma$ if and only if $P\in \Gamma$ mod $p$ for finitely many primes $p$. In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when $E$ does not have complex multiplication and either $E$ is a semistable curve or $E$ has no exceptional prime.