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 $p\le X$ 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 ${\text{GSp}}_{2g}\hat{\mathbb{Z}}$. 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.

27 pages; comments welcome!

References in corpus (1)

• The Large Sieve and Galois Representations