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Some vanishing results for the rational completed cohomology of Shimura varieties

This paper establishes that sufficiently regular infinitesimal weights in the locally analytic completed cohomology of a general Shimura variety appear exclusively in the middle degree, utilizing an almost Kodaira-type vanishing result in mixed characteristics developed by Bhatt.

Original authors: Kai-Wen Lan, Lue Pan

Published 2026-03-31

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CC BY 4.0

Original authors: Kai-Wen Lan, Lue Pan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a massive, complex orchestra playing in a giant, echoing cathedral. This orchestra represents Shimura Varieties, which are intricate geometric shapes that mathematicians use to study deep connections between numbers (arithmetic) and shapes (geometry).

The musicians are playing a specific piece of music called Completed Cohomology. This isn't just a single note; it's a vast, infinite library of sounds (data) that captures the "shape" of the cathedral at every possible level of detail.

The authors of this paper, Kai-Wen Lan and Lue Pan, are asking a very specific question: "If we listen to the music in the lower sections of the cathedral (the lower degrees), can we hear certain specific, very complex notes?"

Their answer is a resounding "No." They prove that if a note is "sufficiently regular" (a fancy way of saying it's complex enough and doesn't have a weird, repetitive pattern), it simply vanishes in the lower sections. It only appears in the very middle of the cathedral.

Here is how they figured this out, broken down into simple metaphors:

1. The Problem: Too Much Noise

The "completed cohomology" is like a recording of the orchestra that includes every possible echo and reverb. It's huge and messy. Mathematicians want to know which "frequencies" (mathematical weights) exist in the lower parts of the building. Intuitively, they suspected that the most complex, "regular" frequencies shouldn't exist there, but proving it was like trying to find a specific grain of sand in a hurricane.

2. The Secret Weapon: Bhatt's "Almost" Vanishing

The authors use a powerful new tool discovered by a mathematician named Bhargav Bhatt. Think of Bhatt's tool as a super-sensitive noise-canceling headphone.

In the world of math, there's a famous rule called the "Kodaira Vanishing Theorem." It's like saying, "If you play a song in a room with a specific shape, certain notes will naturally die out."
Bhatt created a version of this rule that works in "mixed characteristics" (a very tricky mathematical environment). It says that if you twist the music slightly (using a specific mathematical "line bundle"), the lower notes disappear entirely.

3. The Trick: The "Translation" Magic

The authors couldn't just use Bhatt's tool directly because the music they were studying (the Shimura variety) wasn't quite in the right shape. They needed to "untwist" the music to make Bhatt's rule work.

They used a technique called Translation Functors. Imagine you have a heavy, awkward box (the complex math object) that you can't lift.

Instead of lifting it directly, you attach it to a giant, empty box (a finite-dimensional representation).
You then "translate" the problem. You realize that the heavy box is actually just a small part of a much larger, simpler structure.
By looking at this larger structure, they could see that the "heavy box" (the complex notes they were looking for) was actually being crushed out of existence in the lower levels.

4. The "Sen" Operators: The Conductors' Whistles

A key part of their proof involves something called Geometric Sen Theory.

Imagine the orchestra has a conductor who blows a whistle (a differential operator) to tell the musicians to stop playing certain notes.
The authors showed that in the lower levels of the cathedral, these "whistles" are so loud and effective that they completely silence any "sufficiently regular" notes.
They proved that these whistles are essentially controlled by the geometry of a "flag variety" (a specific type of geometric shape), acting like a filter that only lets the "middle" notes pass through.

5. The Grand Conclusion

After combining the "noise-canceling headphones" (Bhatt's theorem) with the "translation trick" (moving the problem to a simpler setting) and the "conductors' whistles" (Sen theory), they arrived at their main result:

In the lower sections of the Shimura variety, the most "regular" and complex mathematical notes simply do not exist.

They only appear in the middle degree (the exact center of the building).

Why Does This Matter?

This is like discovering a fundamental law of acoustics for these mathematical cathedrals.

For Number Theory: It helps mathematicians understand which "patterns" of numbers can exist in certain contexts. If a pattern is too "regular," it can't hide in the lower levels.
For Future Research: It confirms a long-held belief that the "middle" of these shapes is the most interesting and rich place to look for deep mathematical secrets. It also provides a new, powerful method (using these "whistles" and "translation tricks") that other mathematicians can use to solve similar problems in different areas of math.

In short: The authors built a mathematical filter that proves the most complex, orderly sounds in these geometric structures are strictly forbidden from the lower floors. They can only exist in the middle, making the "middle" the exclusive VIP lounge for these special mathematical notes.

1. Problem Statement

The paper addresses the structure of the $p$ -adically completed cohomology of Shimura varieties, specifically focusing on the locally analytic vectors within these cohomology groups.

Context: Emerton introduced completed cohomology $\tilde{H}^i$ as a $p$ -adic analogue of spaces of automorphic forms. It is conjectured (Calegari–Emerton) that for a Shimura variety of dimension $d$ , the cohomology vanishes for degrees $i > d$ , and the "most interesting" behavior occurs at the middle degree $i=d$ .
The Gap: While vanishing results for $i > d$ are known (Scholze, Calegari–Emerton, etc.), and vanishing for $i < d$ is known for specific cases or under strong regularity assumptions (Lan, Li–Schwermer), a general vanishing theorem for the rational completed cohomology $\tilde{H}^{i, \text{la}}_{\mathbb{C}}$ in degrees $i < d$ based on the infinitesimal character of the $G(\mathbb{Q}_p)$ -action was missing.
Goal: To prove that for sufficiently "regular" infinitesimal characters, the locally analytic completed cohomology $\tilde{H}^{i, \text{la}}_{\mathbb{C}}$ vanishes for all $i < d$ .

2. Methodology

The authors employ a sophisticated blend of $p$ -adic geometry, representation theory, and algebraic geometry. The proof strategy can be broken down into four main pillars:

A. Almost Kodaira Vanishing (Bhatt's Theorem)

The foundation of the proof is a mixed-characteristic vanishing theorem by Bhargav Bhatt.

The authors utilize Bhatt's result which states that for a flat projective scheme $X$ over $\mathcal{O}_C$ (the ring of integers of a $p$ -adic field) and an ample line bundle $L$ , the cohomology of certain sheaves twisted by $L^{-1}$ vanishes in degrees below the dimension.
They adapt this to the pro-étale and pro-Kummer étale sites of Shimura varieties, dealing with the infinite-level towers and the specific sheaves of locally analytic functions.

B. Geometric Sen Theory and Hodge–Tate Period Morphisms

The authors leverage the work of Juan Esteban Rodríguez Camargo and Scholze regarding the geometry of Shimura varieties at infinite level.

Hodge–Tate Period Morphism ( $\pi_{HT}$ ): They use the morphism from the infinite-level Shimura variety (or its toroidal compactification) to a partial flag variety $\mathbb{F}\ell$ .
Locally Analytic Sheaves: They define a sheaf $\mathcal{O}^{\text{la}}_S$ of locally analytic functions on the infinite-level tower.
Geometric Sen Operators: A crucial insight is that the action of the Lie algebra $\mathfrak{g}$ on $\mathcal{O}^{\text{la}}_S$ is governed by the geometry of the flag variety. Specifically, the subalgebra $\mathfrak{n}_0$ (associated with the unipotent radical of the parabolic) acts trivially on $\mathcal{O}^{\text{la}}_S$ . This allows the action of the center of the universal enveloping algebra $Z(U(\mathfrak{g}))$ to factor through the center of the Levi subalgebra $Z(U(\mathfrak{m}))$ .

C. Translation Functors (Beilinson–Bernstein Localization)

To "untwist" the cohomology and apply the vanishing theorem, the authors use translation functors from the representation theory of Lie algebras.

They consider a finite-dimensional representation $V_0$ of $G$ with a specific extremal weight $\lambda_0$ .
By tensoring with $V_0$ and using the exact sequences of automorphic vector bundles, they relate the cohomology of the "twisted" sheaf (which vanishes by the Kodaira-type theorem) to the cohomology of the untwisted sheaf.
This process generates an ideal $I$ in the center $Z(U(\mathfrak{g}))$ that annihilates the cohomology in low degrees.

D. Regularity Conditions

The authors define a new notion of regularity called $\mu_h$ -sufficient regularity.

Unlike classical regularity (which requires the weight to be regular with respect to the Weyl group), this condition is tailored to the specific geometry of the Shimura datum and the Hodge cocharacter $\mu_h$ .
It accounts for the distinction between "shorter" and "longer" roots in types B and C, and handles cases where the group has compact factors.

3. Key Contributions and Results

Main Theorem (Theorem 1.1.2 / Theorem 6.1.3)

Let $(G, X)$ be a Shimura datum of dimension $d$ . Let $\tilde{H}^{i, \text{la}}_{\mathbb{C}}$ be the subspace of locally analytic vectors in the $p$ -adically completed cohomology with coefficients in $\mathbb{C}$ .

Result: There exists an integer $n > 0$ and an ideal $I \subset Z(U(\mathfrak{g}))$ (defining the locus of non-sufficiently regular infinitesimal characters) such that $I^n$ annihilates $\tilde{H}^{i, \text{la}}_{\mathbb{C}}$ for all $i < d$ .
Corollary: If an infinitesimal character $[\lambda]$ is $\mu_h$ -sufficiently regular, then the isotypic component $\tilde{H}^{i, \text{la}}_{\mathbb{C}, [\lambda]}$ is zero for all $i < d$ .

Technical Innovations

Generalization of Bhatt's Theorem: The authors extend Bhatt's mixed-characteristic Kodaira vanishing to the setting of pro-Kummer étale cohomology with coefficients in locally analytic functions, removing the need for the line bundle to extend to an integral model in the strictest sense.
Refined Regularity: They introduce $\mu_h$ -sufficient regularity, which is weaker than the regularity conditions required in previous works (like Lan [Lan16]) for certain cases (e.g., Shimura curves vs. Hilbert modular varieties) and more precise for types B and C.
Handling Non-Simply-Connected Groups: The proof carefully handles the case where the derived group $G^{\text{der}}$ is not simply connected by reducing the problem to the simply-connected cover and using induction on the level structure.

4. Significance and Impact

Support for the "Middle Degree" Conjecture: The result provides strong evidence for the belief that the completed cohomology of Shimura varieties is concentrated in the middle degree $d$ when restricted to sufficiently regular infinitesimal characters.
Rational vs. Integral: Unlike many recent vanishing results (e.g., by Caraiani–Scholze, Koskivirta) that rely on torsion coefficients and mod $p$ representations, this paper works entirely with rational coefficients ( $\mathbb{Q}_p$ and $\mathbb{C}$ ). This makes the result applicable to the study of $p$ -adic automorphic forms and their infinitesimal characters directly.
New Tools for $p$ -adic Langlands: The interplay between the geometric Sen operators, the Hodge–Tate period map, and translation functors provides a robust framework for analyzing the local structure of completed cohomology.
Reproving and Extending Previous Results: The method reproves the vanishing of cohomology for automorphic local systems (Lan [Lan16]) and extends it to "big" (infinite-dimensional) local systems that are not necessarily finite-dimensional.
Future Directions: The authors note that their method suggests the existence of ideals in the full enveloping algebra $U(\mathfrak{g})$ (not just the center) that annihilate low-degree cohomology, a direction explored in a follow-up paper by Lue Pan for Hilbert modular varieties.

Summary of the Proof Logic

Setup: Identify $\tilde{H}^{i, \text{la}}_{\mathbb{C}}$ with the cohomology of a sheaf $\mathcal{O}^{\text{la}}_S$ on the infinite-level Shimura variety.
Reduction to Flag Variety: Use the Hodge–Tate period morphism to push forward $\mathcal{O}^{\text{la}}_S$ to the flag variety $\mathbb{F}\ell$ . The action of $\mathfrak{g}$ factors through the Levi $\mathfrak{m}$ .
Vanishing on Flag Variety: Apply Bhatt's almost Kodaira vanishing to show that the cohomology of the twisted sheaf (by an ample line bundle) vanishes in degrees $< d$ .
Translation: Use the translation functor (tensoring with a finite-dimensional representation $V_0$ ) to relate the twisted cohomology back to the untwisted cohomology.
Annihilation: The failure of the translation to be an isomorphism is controlled by the ideal $I$ of non-regular characters. This implies that $I^n$ kills the low-degree cohomology.
Descent: Extend the result from the simply-connected cover to the general Shimura datum via level-shrinking and induction arguments.

This paper represents a significant advancement in the $p$ -adic Langlands program, establishing a rigorous vanishing theorem for rational completed cohomology based on infinitesimal characters.