Weight–monodromy conjecture

Let $K$ be a non-archimedean local field with ring of integers $\mathcal O_K$ and finite residue field $\mathbb F_q$ of characteristic $p$. Fix a separable closure $\overline K$, a prime $\ell\ne p$, and a geometric Frobenius $\Phi\in\operatorname{Gal}(\overline K/K)$, meaning any lift of the inverse of $x\mapsto x^q$. Let $I\subseteq\operatorname{Gal}(\overline K/K)$ be the inertia subgroup and $t_\ell\colon I\to\mathbb Z_\ell(1)$ the $\ell$-adic tame character.

Let $X$ be a proper smooth variety over $K$ and fix $w\ge0$. Put

$$V=H^w_{\mathrm{\acute et}}(X_{\overline K},\mathbb Q_\ell).$$

By Grothendieck's $\ell$-adic monodromy theorem, there is a unique nilpotent map $N\colon V\to V(-1)$ such that every $\sigma$ in some open subgroup of $I$ acts by $\exp\bigl(t_\ell(\sigma)N\bigr)$. The monodromy filtration $M_\bullet V$ is the unique increasing filtration with

$$N(M_kV)\subseteq M_{k-2}V(-1),\qquad N^k\colon \operatorname{Gr}^M_kV\xrightarrow{\ \sim\ }\operatorname{Gr}^M_{-k}V(-k)\quad(k\ge0).$$

It is stable under $\Phi$.

Conjecture (Deligne). For every such $K$, $\ell$, $X$ and $w$, and every integer $k$, each eigenvalue $\alpha$ of $\Phi$ on $\operatorname{Gr}^M_kV$ is an algebraic number such that

$$|\iota(\alpha)|=q^{(w+k)/2}\quad\text{for every field embedding }\iota\colon\overline{\mathbb Q}_\ell\to\mathbb C .$$

In words, the monodromy filtration, shifted by $w$, is the weight filtration of Frobenius.

If $X$ has a proper strictly semistable model over $\mathcal O_K$, the conjecture is equivalent to an assertion about the Rapoport–Zink weight spectral sequence $E_1^{-r,w+r}\Rightarrow V$ [4]. It asks that $N^r\colon E_2^{-r,w+r}\to E_2^{r,w-r}(-r)$ be an isomorphism for every $r\ge0$. When $K$ has characteristic $p$ the conjecture is a theorem. The open problem is the case of characteristic $0$, that is, finite extensions of $\mathbb Q_p$.

Background

The conjecture is the local counterpart of the Weil conjectures. When $X$ has good reduction, inertia acts trivially, $N=0$, and the statement is Deligne's theorem that $V$ is pure of weight $w$ [1, 2]. In general the monodromy operator records how $X$ degenerates. The conjecture predicts that the weights of Frobenius are determined by this purely combinatorial datum. Deligne formulated it by analogy with the limiting mixed Hodge structure of a degeneration over a disc. There the corresponding statement is a theorem of Schmid and Steenbrink.

The conjecture is central to the arithmetic of Shimura varieties and to local–global compatibility in the Langlands program. It implies that the local factors of Hasse–Weil zeta functions at primes of bad reduction are determined by the Weil–Deligne representation in a precise way. Illusie's survey [3] gives the standard account of the local monodromy theorem and the conjecture.