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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.

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up vote 12 down vote accepted

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).

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Thanks! I thought briefly about using Chebatorev but somehow confused myself into thinking it didn't apply. This clarifies things. – jacob Apr 9 '14 at 5:12

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