16
$\begingroup$

In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.

Suppose that $X$ is a smooth proper variety over a local field (which does not necessarily have good reduction). Write $\rho_\ell$ for the Galois representation given by $H^n_{et}(X, \mathbb{Z}_\ell)$, for some prime $\ell$ not equal to the residue characteristic. Suppose the residue field is finite. Then Serre and Tate conjectured:

(1) The restriction of $\operatorname{Tr} \rho_\ell$ to the inertia subgroup is locally constant, takes values in $\mathbb{Z}$, and is independent of $\ell$.

(2) The characteristic polynomial of $\rho_\ell(\pi)$ has rational coefficients independent of $\ell$, and that its roots have absolute value $q^{-k/2}$ for some $0 \leq k \leq 2n$. (For any choice of $\pi$ of Frobenius element in the Galois group; here, $q$ is the cardinality of the residue field.)

Are these conjectures now proven, or are they still open? And if they are proven, what is a reference for this?

$\endgroup$
4
  • 1
    $\begingroup$ If you assume in (2) that $X$ has good reduction, then I think this is the content of the Weil conjectures (as proven by Deligne in Weil II). If you don't, then probably you want to consider the action of $\pi$ on the inertial-invariants of $H^n(X)$, and then, isn't this the content of Deligne's monodromy purity conjecture? $\endgroup$
    – unknown
    Jun 5, 2011 at 1:18
  • $\begingroup$ Is there a typo in the range $0\le k\le 2n?$ $\endgroup$
    – shenghao
    Jun 5, 2011 at 11:06
  • $\begingroup$ No, I don't think so. For example, say $X = E$ is an elliptic curve and $n = 1$. Then if $E$ has good (or potentially good) reduction, this is true for $k = 1$ (by the Weil conjectures); when $E$ has multiplicative (or potentially multiplicative) reduction, then we can represent $E$ by a Tate curve, so this is true with $k = 0$ for one of the eigenvalues and $k = 2$ for the other eigenvalue. $\endgroup$ Jun 5, 2011 at 19:34
  • 3
    $\begingroup$ The article you mention is by Serre and Tate. $\endgroup$ Jun 7, 2011 at 7:26

1 Answer 1

2
$\begingroup$

The article of Katz "review of $\ell$-adic cohomology" in Motives (PSPM 55, 1990) contains a good survey of the partial results in the direction of those conjectures that were known then. There hasn't been many progresses since.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.