Abelian-variety point-count asymptotic

Establish that for every abelian variety A over the rationals of dimension g, the product of normalized point counts ∏_{p≤x, p∉S_A} N_p/p^g is asymptotic to C(log x)^{rank A(Q)} as x tends to infinity, where S_A is the set of bad primes.

Background

The authors propose this as another direct generalization of the original Birch–Swinnerton-Dyer product asymptotic. Unlike the curve case, the trace-formula calculation for an abelian variety produces no additional Néron–Severi correction in the exponent.

References

We have that $$\prod_{\substack{p x p \not \in S_A} \frac{N_p}{pg} \sim C (\log x){rk(A)}$$ as $x \to \infty$ for some constant $C$ depending on $A$.

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Conjecture \ref{AVconj}, Section 1, subsection “The abelian variety case”