# Serre and Tate's conjectures on étale cohomology

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?

• 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? Jun 5, 2011 at 1:18
• Is there a typo in the range $0\le k\le 2n?$ Jun 5, 2011 at 11:06
• 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. Jun 5, 2011 at 19:34
• The article you mention is by Serre and Tate. Jun 7, 2011 at 7:26

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.