Weight–Monodromy Conjecture

Weight–Monodromy Conjecture

Definitions and mathematical statement

Put $V=H^i_{\mathrm{\acute et}}(X_{\bar K},\overline{\mathbb{Q}}_\ell)$ for a proper smooth variety over a nonarchimedean local field with residue cardinality $q$, and $\ell\ne p$. Inertia defines nilpotent monodromy $N:V\to V(-1)$. Its zero-centered increasing filtration $M$ is characterized by \[ N(M_j)\subseteq M_{j-2}(-1),\qquad N^a:\operatorname{gr}^M_aV\xrightarrow{\sim}\operatorname{gr}^M_{-a}V(-a). \] For any geometric Frobenius lift, every eigenvalue $\alpha$ on $\operatorname{gr}^M_jV$ should be algebraic and satisfy $|\iota(\alpha)|=q^{(i+j)/2}$ for every complex embedding $\iota$. This is purity of weight $i+j$. The original field and its $q$ remain fixed when defining monodromy on an open inertia subgroup; no semistability or projectivity assumption is added.

Formulation and concept references: [S1].

Short English statement

Do monodromy layers of local-field étale cohomology have exactly the Frobenius weights predicted by their degrees?

Research attempt

Prepared by GPT-6 Astra Ultra on 27 September 2026, with primary-source review, explicit example checks, and examination of the stated conditional reductions. This is an unresolved research attempt.

Scope and source check (27 September 2026).

The field $K$, residue cardinality $q$, and cohomological degree $i$ are those in the statement. No projectivity or semistable model is assumed for the target. Scholze [S1, Conjecture 1.13 and Theorem 1.14] states the general conjecture and proves a complete-intersection case. Wear [S3] establishes another complete-intersection case, in abelian varieties. The formality result [S2] assumes weight--monodromy; it cannot be used to establish that assumption. The Lefschetz-pencil paper [S4, Section 9] explicitly leaves a monodromy property for a pushforward unresolved. These are relevant results and reductions, not a proof in the full scope.

This attempt separates the formal nilpotent linear algebra from the geometric purity statement. It proves a reduction to primitive monodromy pieces, checks the Frobenius and twist signs, and explains precisely why passing to a finite extension preserves the claim. The remaining problem will be the weights of the primitive pieces, which are not determined by the Jordan form of monodromy.

The sign convention in one Jordan chain.

Choose a basis for the Tate line when writing matrices. Geometric Frobenius $F$ acts on $\mathbb Q_\ell(1)$ by $q^{-1}$, so it acts on the twist $V(-1)$ by $qF$. Equivariance of $N:V\to V(-1)$ therefore means \begin{equation}\label{wsref-0-77-6d-3a-72-65-6c-61-74-69-6f-6e} NF=qFN,\qquad FNF^{-1}=q^{-1}N. \end{equation} The untwisted underlying operator lowers Frobenius eigenvalues by a factor of $q$. Reversing this factor would reverse the predicted weights.

For a chain $e_0,\ldots,e_r$ with $Ne_a=e_{a-1}$ and $Ne_0=0$, the zero-centered monodromy grades are \[ \deg_M e_a=-r+2a. \] One compatible diagonal Frobenius is $Fe_a=\lambda q^a e_a$. The conjecture on this chain is exactly \[ |\iota(\lambda)|=q^{(i-r)/2}; \] then $\lambda q^a$ has weight $i-r+2a$, as required. For a length-two chain in degree one, the familiar eigenvalues $1,q$ belong to grades $-1,1$. This is also a direct test of the displayed isomorphism $N:\operatorname{gr}_1\to\operatorname{gr}_{-1}(-1)$: both sides have Frobenius eigenvalue $q$. A twist by $+1$ in that isomorphism would fail.

Nothing here assumes that all Frobenius operators are diagonalizable. Equation (wm:relation) maps a generalized $\alpha$-eigenspace to the generalized $\alpha/q$-eigenspace. Statements about eigenvalue weights must not be strengthened to semisimplicity without a separate argument.

A complete reduction to the primitive graded pieces.

For $r\ge0$, put \[ P_r=\ker\left(N^{r+1}:\operatorname{gr}^M_rV \longrightarrow\operatorname{gr}^M_{-r-2}V(-r-1)\right). \] These are Frobenius-stable spaces. Nilpotent linear algebra gives \begin{equation}\label{wsref-0-77-6d-3a-70-72-69-6d-69-74-69-76-65} \operatorname{gr}^M_j V\simeq \bigoplus_{\substack{r\ge |j|\\r\equiv j\pmod2}} P_r\left((r-j)/2\right). \end{equation} Here the map from a summand is the appropriate power of $N$, interpreted with its Tate twist. To verify the statement without assuming a Frobenius-stable Jordan basis, first choose an ordinary Jordan basis for $N$. A chain of length $r+1$ contributes once in grades $r,r-2,\ldots,-r$. In grade $r$, the kernel defining $P_r$ removes the contributions from every longer chain: on those contributions $N^{r+1}$ is injective. On a chain of length $r+1$ it vanishes. Thus the indicated maps give a direct sum decomposition in every grade. The maps and kernels themselves are intrinsic and Frobenius-equivariant, so the resulting decomposition is one of Frobenius representations.

If $P_r$ is pure of weight $i+r$, its twist in (wm:primitive) is pure of weight \[ i+r-2\frac{r-j}{2}=i+j. \] Conversely, if all the graded pieces have the conjectured weights, their subspaces $P_r$ do also. Thus the conjecture is equivalent to purity of these primitive spaces. This reduction does not construct their eigenvalues or prove they are algebraic; both requirements remain part of the purity assertion.

The identity also gives exact dimension checks. A chain of length $r+1$ contributes one dimension to $P_r$, so \[ \dim V=\sum_{r\ge0}(r+1)\dim P_r,\qquad \dim\operatorname{gr}_j^M V=\dim\operatorname{gr}_{-j}^M V. \] These numerical symmetries are consequences of monodromy alone. They are necessary but do not prove a statement about complex absolute values.

Changing a Frobenius lift does not change the purity test.

An open inertia subgroup acts as $\exp(t_\ell(\sigma)N)$. Since $N$ strictly lowers the monodromy filtration, this subgroup acts trivially on its associated graded. Therefore the residual inertia image $H$ on the total graded space is finite. A Frobenius operator normalizes $H$.

Suppose $F'=hF$ is another geometric lift, with $h\in H$. Choose $a>0$ so that conjugation by $F^a$ is trivial on $H$; this is possible because $\operatorname{Aut}(H)$ is finite. Then $(hF)^a=h_aF^a$ for some $h_a\in H$, and $h_a$ commutes with $F^a$. If $b$ is an exponent of the finite group $H$, it follows that \[ (F')^{ab}=F^{ab} \] on the associated graded. Thus the multisets of eigenvalues of the two operators have the same $ab$-th powers. Algebraicity and every specified absolute-value weight for one multiset imply them for the other. Indeed an element whose positive integral power is algebraic is algebraic, and taking absolute values recovers the unique positive $ab$-th root. This proves the lift independence needed here without pretending that inertia was already trivial on $V$.

Finite extensions and the original residue cardinality.

Let $K'/K$ have residue degree $f$, so $q'=q^f$. Restriction rescales the underlying monodromy operator by a nonzero scalar if a different tame generator is used; kernels and images of its powers, and hence the zero-centered filtration, do not change. A geometric Frobenius for $K'$ acts on the graded space as $hF^f$ for a finite inertia element $h$. The preceding argument, applied to $F^f$, compares their positive powers.

Consequently purity over $K'$ of weight $i+j$ is equivalent to purity over $K$ of the same weight. On eigenvalues the numerical calculation is \[ |\iota(\alpha^f)|=(q^f)^{(i+j)/2} \quad\Longleftrightarrow\quad |\iota(\alpha)|=q^{(i+j)/2}. \] Algebraicity descends in the same way, and every embedding of the field generated by $\alpha$ restricts to an embedding of the field generated by $\alpha^f$. The additional finite-inertia powers cause no change. This justifies an extension used in a geometric argument; it does not allow changing $q$ while keeping the same Frobenius eigenvalue.

A genuine solved case and an algebraic countertest.

If $X$ has a proper smooth model over the valuation ring, smooth proper base change identifies $V$ with the cohomology of the special fiber. Inertia is trivial, $N=0$, and only $\operatorname{gr}_0^M V$ survives. The finite-field purity theorem gives precisely weight $i$. These established geometric inputs, recalled in [S1], prove the good-reduction case. Merely knowing $N=0$ for an abstract representation would not give finite-field geometric purity.

For a concrete test, take two length-two chains in formal degree $i=1$, and set \[ F=\operatorname{diag}(q^s,q^{s+1},q^{-s},q^{1-s}), \qquad Ne_1=e_0,\quad Nf_1=f_0, \] in the basis $(e_0,e_1,f_0,f_1)$, with $s=1$, for instance. Equation (wm:relation) holds, both monodromy grades have dimension two, and $\det F=q^2$, the determinant size expected for four dimensions of weight one on average. Yet grade $-1$ has eigenvalues $q^s,q^{-s}$, which do not both have weight zero. The top grade fails in the corresponding way. This is a formal Frobenius--monodromy pair, not a claimed cohomology representation or a counterexample to the conjecture. It shows exactly why nilpotence, the chain isomorphisms, and even the total determinant do not imply purity.

Verification and the geometric gap.

The local checks construct chains of several lengths and verify $FNF^{-1}=q^{-1}N$, the grade dimensions, the primitive count, and the four-dimensional determinant countertest with rational matrices. They also check the exponents after a residue-degree extension. These calculations audit signs and linear algebra, not geometric realizability.

The first missing statement is that the actual geometric $P_r$ are pure of weight $i+r$. In a semistable spectral-sequence approach, purity of individual special-fiber strata is not enough: one must control the induced monodromy maps after taking cohomology of the differentials. The unresolved pushforward condition in [S4] is a precise manifestation of that issue. Possible continuation would prove the needed pairing nondegeneracy in a specified geometric family, produce a comparison preserving these primitive pieces, or establish the monodromy isomorphisms on the relevant spectral-sequence page. The linear algebra here supplies no such theorem.

UNRESOLVED. The attempt proves formal reductions and normalization checks, and explains an established good-reduction subcase. It neither adds a hypothesis to the catalogue target nor proves its remaining mixed-characteristic cases. No novelty or formal proof certification is asserted.

Sources

• S1 Peter Scholze, Perfectoid spaces, arXiv1111.4914v1,banner21November2011/internal27November2024,51pages. https://arxiv.org/pdf/1111.4914v1. Formulation source.
• S2 Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture. https://arxiv.org/abs/2607.14517. Further reference; not a proof endorsement.
• S3 Perfectoid covers of abelian varieties and the weight-monodromy conjecture. https://arxiv.org/abs/2303.05610. Further reference; not a proof endorsement.
• S4 Semistable Lefschetz pencils. https://page.mi.fu-berlin.de/esnault/preprints/helene/151_mw.pdf. Further reference; not a proof endorsement.