Almost prime values of the order of abelian varieties over finite fields

Abstract

Let $E/\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \leq X$ at which $N(p)$ is also prime is asymptotic to $C_E \cdot X / \log(X)^2$, where $C_E$ is an arithmetically-defined non-negative constant. Following Miri-Murty (2001) and others, Y.R. Liu (2006) and David-Wu (2012) study the number of prime factors of $N(p)$. We generalize their arguments to abelian varieties $A / \mathbb Q$ whose adelic Galois representation has open image in $\textrm{GSp}_{2g} \widehat{\mathbb Z}$. Our main result, after David-Wu, finds a conditional lower bound on the number of primes at which $\# A_p ( \mathbb{F}_p)$ has few prime factors. We also present some experimental evidence in favor of a generalization of Koblitz's conjecture to this context.