Note On Elliptic Primitive Points

Abstract

Let $E$ be an elliptic curve of rank $\text{rk}(E)\ge 1$, and let $P\in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p\le x$ for which $⟨P⟩=E({\mathbb{F}}_p)$ is expected to be $\pi (x,E,P)=\delta (E,P)x/\log x+o(x/\log x)$, where $\delta (E,P)\ge 0$ is a constant. This note proves the lower bound $\pi (x,E,P)\gg x/\log x$.

Sixteen Pages. Keywords: Elliptic Prime; Primitive Point; Lange-Trotter Conjecture. arXiv admin note: text overlap with arXiv:1702.06814

References in corpus (2)

• A refinement of Koblitz's conjecture
• Character sums for elliptic curve densities