Take $E$ to be an elliptic curve over $\mathbb{Q}$, and consider the coefficients $a_E(p)$of the dirichlet series $L(E,s)$ so that $E(\mathbb{F}_p)=p+1-a_E(p)$.

The Sato-Tate conjecture (now proven) explains the distribution of $a_E(p)/\sqrt{p}$.

**Question:** For a prime $l>0$, is there an $l$-adic analogue of Sato-Tate? And if so, is it proven?

Precisely, is there a measure $\mu_E$ on $\mathbb{Z}_l$ so that the set $a_E(p)$ becomes equidistributed w.r.t $\mu_E$ as $p\rightarrow\infty$?

I searched for a while online and couldn't find anything.