An Observation on the Cyclicity of the Group of the $\mathbb{F}_p$-Rational Points of Abelian Surfaces

Abstract

Let $A$ be a principally polarized Abelian surface defined over $\mathbb{Q}$ with $\End(A)=\mathbb{Z}$ and $\widetilde{A}$ be the reduction at a good prime $p$. In this paper, we study the density of prime numbers $p$ for which $\widetilde{A}(\mathbb{F}_p)$ is a cyclic group and establish a conjecture which relates this density.