Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In

Serre presents nine conjectures *C*$_1$-*C*$_9$ concerning Galois representations on $l$-adic cohomology of nonsingular projective algebraic varieties defined over a local or global field. What is the status of those conjectures today? The first two of them are part of the Weil conjectures but how about the rest? What (partial) progress has been made towards their resolution? Let me recall the conjectures below (skipping the first two).

Let $K_v$ be a local field of residue characteristic $p$, let $Y$ be a nonsingular projective variety over $K_v$, and let $m\ge 0$. If $G_v$ is the absolute Galois group of $K_v$ the functoriality of $l$-adic cohomology ($l \neq p$) gives us a representation $\rho_l\colon G_v \rightarrow H^m(\overline{Y}, \mathbb{Q}_l) =: V$ (here $\overline{Y}$ is the base change of $Y$ to the separable closure of $K_v$). One can measure how ramified $\rho_l$ is by introducing $\epsilon = \dim V - \dim V^{I_v}$ ($I_v$ is the inertia in $G_v$) and $\delta$ which is a little bit more involved to define (it is the inner product of $\mathrm{Tr }\ \rho_l|_{I_v}$ with the "Swan character" of $\rho_l$, see 2.1 in [S]). The conductor exponent of $\rho_l$ is then $f = \epsilon + \delta$.

*C*$_3$: The integers $\epsilon$, $\delta$ and $f$ are independent of $l$.

Status: ???

*C*$_4$: $\mathrm{Tr }\ \rho_l|_{I_v}$ takes values in $\mathbb{Z}$ and is independent of $l$.

Status: ???

Consider the geometric Frobenius $\pi \in G_v/I_v$. Then $\rho_l(\pi)$ acts on $V^{I_v}$ and we get a polynomial $P_{\rho_l}(T) = \det(1 - \rho_l(\pi) T)$.

*C*$_5$: $P_{\rho_l}$ has coefficients in $\mathbb{Z}$ and is independent of $l$.

Status: ???

Assuming the latter conjecture split $P_{\rho_l}(T) = \prod (1 - \lambda_\alpha T)$ and let $Nv$ denote the cardinality of the residue field of $K_v$.

*C*$_6$: For each $\alpha$ there is an integer $m(\alpha)$ between $0$ and $m$ such that $|\alpha| = (Nv)^{m(\alpha)/2}$.

Status: ???

*C*$_7$: If $\epsilon = 0$ (i.e., if $\rho_l$ is unramified) then all $m(\alpha)$ are equal to $m$.

Status: ???

*C*$_8$: Let $g$ be an element of $G_v$ whose image in $G_v/I_v$ is an integral power of Frobenius. Then the characteristic polynomial of $\rho_l(g)$ has coefficients in $\mathbb{Q}$ and is independent of $l$.

Status: ???

The last conjecture *C*$_9$ concerns the zeta function $\zeta(s)$ of a nonsingular projective variety $X$ defined over a global field $K$ (and fixed $m \ge 0$ as above), as well as the completed version $\xi(s)$ of $\zeta(s)$. It's a bit of a trek to define $\zeta(s)$ and $\xi(s)$ so I'll refer to [S] $\S$3, $\S$4 for that. Both $\zeta(s)$ and $\xi(s)$ are holomorphic functions on some right half-plane.

*C*$_9$: $\zeta(s)$ and $\xi(s)$ admit meromorphic continuations to the complex plane. In addition, $\xi(s)$ satisfies the functional equation $\xi(s) = w\xi(m + 1 - s)$ with $w = \pm 1$.

Status: Open. Afterall, a special case of this is meromorphic continuation of $L$ functions of elliptic curves over number fields.

As the answers come in feel free to edit the status fields above adding more information (and I will try to do that myself).

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It is my understanding that the "independent of l" conjectures are still quite wide open. Weizhe Zheng Sur l'indépendance de en cohomologie l-adique sur les corps locaux Annales scientifiques de l'ENS 42, fascicule 2 (2009), 291-334 is probably among the most advanced results there are. –  Olivier Feb 17 '12 at 13:13