The Sato-Tate conjecture for hypersurfaces?

Question

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.

Answer 1

The conjecture is that associated to any cohomology group of an algebraic variety, or set of cohomology group on the algebraic variety, there should be a complex Lie group (with representations), which can be described in a number of ways:

The Zariski closure of the image of the Galois group inside $GL_n(\mathbb Q_l)$ is an algebraic group. Base change this group from $\mathbb Q_l$ to $\mathbb C$.

The Mumford-Tate group of the Hodge structure on the cohomology groups. This group fixes all algebraic cycle classes. If the algebraic cycle classes aren't defined over $\mathbb Q$, extend this group by the Galois group of the field they are defined over. Then base change to $\mathbb C$.

The image of the (conjectural) motivic Galois group in the representation associated to the motive in the (conjectural) Tannakian category of motives, base changed to $\mathbb C$.

These should all be the same group. This is "only" the Hodge conjecture and the Tate conjecture!

Then we need to make the weight adjustment on this group, which is parallel to dividing $Frob_p$ by $p^{w/2}$. To do this, consider all homomorphisms from this group to $\mathbb G_m$. These should correspond to characters of the Galois group / 1-dimensional Hodge structures with a finite Galois action / 1-dimensional motives. Take the kernel of all homomorphisms which correspond to powers of the cyclotomic character / Hodge classes with trivial Galois action / powers of the Tate motive. This gives a new, slightly smaller Lie group.

Take the maximal compact subgroup of this group. This is the Sato-Tate group. Take the Haar measure on it, and deduce a distribution for whatever data you care about (traces, eigenvalues, etc.). This is the conjectural distribution for the eigenvalues of Frobenius.

That is the Generalized Sato-Tate conjecture.

This follows, I believe, from holomorphicity of the L functions of the Galois representations corresponding to all the nontrivial representations of the Sato-Tate group, which should follow from the Langlands program.