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.

  • $\begingroup$ There is no significant difference between higher dimensional abelian varieties and hypersurfaces: replace $H^1$ of the abelian variety with the middle dimensional cohomology of the hypersurface. $\endgroup$
    – naf
    Dec 15 '13 at 5:46
  • $\begingroup$ @ulrich: that's certainly plausible. Do you know a reference where a conjecture of this form is asserted? I was only able to find conjectures about abelian varieties and curves. What's expected for more general varieties? $\endgroup$ Dec 15 '13 at 20:04
  • $\begingroup$ As you probably know, even for elliptic curves the form of the conjecture depends on whether you have CM or not. I guess in general there will be some sort of Mumford-Tate group (that's where things like the cup product will come in) and the conjecture should say that Frobenii are equidistributed within the real points of that group. $\endgroup$
    – eric
    Dec 15 '13 at 21:25
  • $\begingroup$ @eric: yes, it's called the Sato-Tate group in the literature. But again I've only seen discussion of curves and abelian varieties. $\endgroup$ Dec 15 '13 at 22:08
  • 1
    $\begingroup$ See, for example, the last section of the article by Serre in "Motives", PSMP 55, for a formulation for arbitrary motives. $\endgroup$
    – naf
    Dec 16 '13 at 4:26

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.

  • 1
    $\begingroup$ 1. I got this from reading the same reference as Ulrich, but I too do not remember explicitly where it is in that reference. 2. The cup product shows up in that the representations of all these groups should preserve the ring structure by their respective definitions. $\endgroup$
    – Will Sawin
    Dec 17 '13 at 21:37

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