The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the eigenvalues of a random element of an associated compact group. There are conjectural generalizations for curves of higher genus and abelian varieties of higher dimension in terms of more complicated compact groups.

What's conjectured about the corresponding question for hypersurfaces?

To be more precise, consider a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ defined over $\mathbb{Z}$. Its only interesting $\ell$-adic cohomology should be in the middle degree, and its rank is known. What's conjectured about the distribution of the eigenvalues of Frobenius at $p$ acting on it as $p$ varies?

**Edit:** If, as ulrich suggests, the expectation here is that it resembles the case of curves / abelian varieties, what about the case of more general varieties? In general it seems like the cup product restricts Frobenius eigenvalues in some a priori complicated way.

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