The split-reduction asymptotic conjecture for abelian surfaces

Let $K$ be a number field and let $A/K$ be a principally-polarizable abelian surface whose absolute endomorphism ring satisfies

$$\operatorname{End}_{\overline K} A\cong{\mathbb Z}.$$

Let $\pi_{\operatorname{split}}(A/K,z)$ count prime ideals of norm at most $z$ at which $A$ has split reduction. The split-reduction asymptotic conjecture. There is a constant $C_A>0$ such that

$$\pi_{\operatorname{split}}(A/K,z)\sim C_A\frac{\sqrt z}{\log z}\qquad\text{as }z\to\infty.$$

This gives a precise form of the paper’s expected square-root-over-logarithm growth and is compared with the Lang–Trotter conjecture. The claim remains open in the supplied source.