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. 



Is there a p-adic Sato - Tate conjecture?
The answer is yes, and it is much simpler than the true Sato-Tate conjecture.
Let $\rho: G\_{\mathbb Q} \rightarrow GL\_2(\mathbb Z\_\ell)$ be the Galois representation attached to $E$ on its Tate module, and let $\Gamma$ be the image of $\rho$. That's a compact group, hence it has an Haar measure $\mu\_\Gamma$. Now there is a continuous map $tr : \Gamma \rightarrow \mathbb Z\_\ell$, the trace; let $\mu\_E$ be the measure on $\mathbb Z\_\ell$ image of $\mu\_\Gamma$ by $tr$.
Then $a\_E(p)$ is equidistributed for $\mu\_E$, in the sense that for every locally constant function $f$ on $\mathbb Z\_\ell$, $$\sum\_{p<x} f(a\_p(E)) / (x/\log x) \rightarrow \int f d\mu\_E$$ when $x \rightarrow \infty$. To prove that, note that $f$ factors through a quotient $\mathbb Z/\ell^n \mathbb Z$ of $\mathbb Z\_\ell$, and then the result is just Chebotarev's density theorem applied to the function $f \circ tr$ on the finite Galois group $\Gamma\_n$ image of $\Gamma$ in $GL\_2(\mathbb Z/\ell^n \mathbb Z)$.


Concetely, the measure $\mu\_E$ is easy to compute as soon as you know the image of $\rho$, and there are many theorem about that (cf. papers of Serre).


