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

TL;DR

This paper investigates the distribution of primes for which the number of points on abelian varieties over finite fields has few prime factors, extending conjectures and results known for elliptic curves to higher-dimensional cases.

Contribution

It generalizes previous work on elliptic curves to abelian varieties with open Galois image, providing lower bounds and experimental evidence for a Koblitz-type conjecture.

Findings

Conditional lower bounds on primes with few prime factors in point counts

Extension of Koblitz's conjecture to abelian varieties

Experimental support for the generalized conjecture

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.