Lang–Trotter conjecture for generic abelian varieties

Let $A$ be a generic abelian $g$-fold over $\mathbb{Q}$, and let $\pi_{A,T}(x)$ count primes $p\leq x$ of good reduction for which $a_p(A)=T$. Write $H_A$ for the adelic Galois image, $m_A$ for its conductor, and $m_{A,T}=m_A\prod_{\ell\mid m_A}\ell^{v_\ell(T)}$ for $T\ne0$. Lang–Trotter conjecture for generic abelian varieties. Assuming the preceding equidistribution conjecture, for every $T\ne0$ one has

$$\pi_{A,T}(x)\sim C(A,T)\frac{\sqrt{x}}{\log x},$$

where

$$C(A,T)=2\Phi_{\operatorname{USp}(2g)}(0)\frac{m_{A,T}|H_A(m_{A,T},T)|}{|H_A(m_{A,T})|}\prod_{\ell\nmid m_A}\frac{\ell^{v_\ell(T)+1}|\operatorname{GSp}_{2g}(\ell^{v_\ell(T)+1},T)|}{|\operatorname{GSp}_{2g}(\ell^{v_\ell(T)+1})|}.$$

Here $C(A,T)\geq0$. This is the higher-dimensional Lang–Trotter prediction for generic abelian varieties and remains open in general.