Koblitz’s Conjecture for Abelian Varieties

Consider a principally polarized abelian variety A of dimension d defined over a number field F. If 𝔭 $$\mathfrak p$$ is a prime ideal in F such that A has good reduction at p, let N 𝔭 $$N_{\mathfrak p}$$ be the order of A mod 𝔭 $$A\operatorname {mod}\mathfrak p$$ . We have formulae for the density p ℓ of primes 𝔭 $$\mathfrak p$$ such that N 𝔭 $$N_{\mathfrak p}$$ is divisible by a fixed prime number ℓ in two cases: A is a CM abelian variety and the CM-field is contained in F, or A has trivial endomorphism ring and its dimension is 2, 6 or odd. In both cases, we can prove that C A = ∏ ℓ 1 − p ℓ 1 − 1 / ℓ $$C_A=\prod _\ell \frac {1-p_\ell }{1-1/\ell }$$ is a positive constant. We conjecture that the number of primes 𝔭 $$\mathfrak p$$ with norm up to n such that N 𝔭 $$N_{\mathfrak p}$$ is prime is given by the formula C A n d log ( n ) 2 $$C_A\frac {n}{d\log (n)^2}$$ , generalizing a formula by N. Koblitz, conjectured in 1988 for elliptic curves. Numerical evidence that supports this conjectural formula is provided.