Koblitz's conjecture for abelian varieties

Let A/QA/\mathbb{Q} be an abelian variety satisfying the hypothesis (TrivA)(\operatorname{Triv}_A), and suppose that CA0C_A\neq 0. Koblitz's conjecture for abelian varieties.

#{px#Ap(Fp) is prime}Ax(logx)2.\#\{p\leq x\Bigm| \#A_p(\mathbb{F}_p)\text{ is prime}\}\asymp_A\frac{x}{(\log x)^2}.

In particular, if AA is generic, then

#{px#Ap(Fp) is prime}CAxg(logx)2.\#\{p\leq x\Bigm| \#A_p(\mathbb{F}_p)\text{ is prime}\}\sim C_A\frac{x}{g(\log x)^2}.

This extends Koblitz's prediction from elliptic curves to higher-dimensional abelian varieties; the supplied text gives no resolution status.

Sources & referencesView supporting material

Primary source

Samuel Bloom, “Almost prime values of the order of abelian varieties over finite fields”, arXiv:1803.03698 (2018).

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