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

2018/03/09 by Bloom, Samuel
#11G10 #11N36 #11N45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.03698

Abstract

Let E/\mathbb Q be an elliptic curve, and denote by N(p) the number of \mathbbF_p-points of the reduction modulo p of E. A conjecture of Koblitz, refined by Zywina, states that the number of primes p ≤ X at which N(p) is also prime is asymptotic to C_E ⋅ 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 \textrmGSp_2g \widehat\mathbb Z. Our main result, after David-Wu, finds a conditional lower bound on the number of primes at which # A_p ( \mathbbF_p) has few prime factors. We also present some experimental evidence in favor of a generalization of Koblitz's conjecture to this context.

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