Koblitz’s Conjecture for Abelian Varieties

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Consider a principally polarized abelian variety A of dimension d defined over a number field F. If \(\mathfrak p\) is a prime ideal in F such that A has good reduction at p, let \(N_{\mathfrak p}\) be the order of \(A\operatorname {mod}\mathfrak p\). We have formulae for the density p l of primes \(\mathfrak p\) such that \(N_{\mathfrak p}\) 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 \(C_A=\prod _\ell \frac {1-p_\ell }{1-1/\ell }\) is a positive constant. We conjecture that the number of primes \(\mathfrak p\) with norm up to n such that \(N_{\mathfrak p}\) is prime is given by the formula \(C_A\frac {n}{d\log (n)^2}\), generalizing a formula by N. Koblitz, conjectured in 1988 for elliptic curves. Numerical evidence that supports this conjectural formula is provided.