Locally analytic completed cohomology

This paper computes the geometric Sen operator for arbitrary Shimura varieties using equivariant vector bundles and the Hodge-Tate period map, thereby establishing the rational vanishing of completed cohomology as predicted by the Calegari-Emerton conjectures.

The Gist

Imagine you are trying to understand the shape of a vast, invisible city called a Shimura Variety. This city is built using the rules of number theory (specifically prime numbers) and geometry. It's so complex that mathematicians have been trying to map its "cohomology"—a fancy word for counting the holes, loops, and tunnels in its structure—for decades.

The paper you provided, "Locally Analytic Completed Cohomology" by Juan Esteban Rodríguez Camargo, is like a new, high-tech drone that finally flies over this city and takes a perfect picture, revealing a surprising secret: Above a certain height, the city is completely empty.

Here is a breakdown of the paper's journey, using simple analogies.

1. The Problem: The Infinite City

Mathematicians study these Shimura varieties by looking at them at different "levels" of zoom.

• Low Zoom: You see a few buildings (finite levels).
• High Zoom: You zoom in infinitely, seeing every brick and dust particle (infinite level).

When you zoom in infinitely, the city becomes a "perfectoid space"—a strange, fluid object that behaves like a liquid rather than a solid building. The goal is to count the "holes" (cohomology) in this infinite city.

The Conjecture: Two mathematicians, Calegari and Emerton, guessed that if you look at the city from too high up (above the "middle degree"), there are no holes at all. It's just flat, empty space. But proving this for all types of these cities (not just the simple ones) was incredibly hard.

2. The New Tool: The "Geometric Sen Operator"

The author introduces a new tool called the Geometric Sen Operator. Think of this as a specialized wind sensor.

• The City's Wind: In this mathematical city, there is a "wind" (a mathematical force) blowing through the infinite levels. This wind is generated by the symmetries of the city.
• The Sensor: The Sen operator measures how this wind twists and turns the fabric of the city.
• The Discovery: The author calculated exactly how this wind behaves. He found that the wind is directly tied to the shape of a "Flag Variety" (a specific type of geometric map) and a "Period Map" (a bridge connecting the number-theoretic city to a geometric one).

The Analogy: Imagine the city is a giant, complex wind chime. The author figured out that the sound the chime makes (the cohomology) is determined entirely by how the wind hits a specific, simpler shape (the flag variety).

3. The "Locally Analytic" Lens

The paper focuses on "locally analytic" vectors.

• The Analogy: Imagine the city is made of a fuzzy, blurry material. "Locally analytic" means we are looking at the parts of the city that are smooth and sharp, ignoring the fuzzy bits.
• The author proves that if you look at the smooth parts of the infinite city, you can describe them using a simple "sheaf of functions" (a rulebook for how to write down numbers on the city).

4. The Big Result: The Vanishing Act

Using this new wind sensor and the rulebook, the author proves the Calegari-Emerton conjecture (at least for rational numbers, meaning we ignore tiny fractional errors).

The Result:
If you count the holes in the city above the middle height, the count is zero.

• Why? The "wind" (the Sen operator) pushes everything down. It acts like a gravity that forces all the complex structures to collapse into the lower levels. The "upper floors" of the city are so empty that they don't exist in the way we thought.
• The Metaphor: Imagine a skyscraper where the top 50 floors are made of smoke. If you try to put a solid object (a "hole") on the 60th floor, it falls right through because there's nothing there to hold it. The author proved mathematically that the "smoke" (the cohomology) vanishes completely above the middle floor.

5. Why This Matters

• Universal Truth: Previous proofs only worked for very simple, "nice" cities (like modular curves). This paper works for any Shimura variety, no matter how complicated.
• New Language: It translates a very hard problem in number theory into a problem about geometry and vector bundles (like mapping the wind on a flag). This opens the door for other mathematicians to use these geometric tools to solve other number theory puzzles.
• The "Arithmetic" Twist: The paper also defines an "Arithmetic Sen Operator," which is like a clock that ticks in sync with the prime numbers. It shows that the way the city is built is perfectly synchronized with the arithmetic of the universe.

Summary

Juan Esteban Rodríguez Camargo built a new mathematical telescope. He looked at the infinite, complex structures of Shimura varieties and realized that the "wind" blowing through them forces the upper levels to be empty. This confirms a major guess about the shape of these mathematical universes and provides a powerful new map for navigating the relationship between numbers and geometry.

In one sentence: The paper proves that the "upper floors" of these complex number-theoretic cities are empty, using a new geometric tool that measures how the city's internal symmetries twist and turn.

Technical Summary

1. Problem Statement and Context

The paper addresses the Calegari-Emerton conjectures regarding the vanishing of completed cohomology for Shimura varieties. Specifically, it focuses on the rational vanishing of these cohomology groups in degrees higher than the dimension of the variety.

• Completed Cohomology: Introduced by Emerton, the completed cohomology groups $\tilde{H}^i(K^p, \mathbb{Z}_p)$ are defined as the inverse limit of the direct limits of singular cohomology groups of Shimura varieties at finite levels, with coefficients in $\mathbb{Z}/p^s\mathbb{Z}$.
• The Conjecture: Calegari and Emerton conjectured that for a Shimura variety of dimension $d$, the completed cohomology groups vanish for $i > d$ (and similarly for compactly supported cohomology).
• Previous Progress:
  • Scholze proved the vanishing for the compactly supported case for Hodge-type Shimura varieties by utilizing the perfectoid nature of the infinite-level limit.
  • Hansen and Johansson extended this to pre-abelian type varieties.
  • He recently proved the integral version for general Shimura varieties using point-wise perfectoid criteria.
• The Gap: While the integral case was resolved, a proof for the rational case ($\mathbb{Q}_p$ coefficients) for arbitrary Shimura varieties (not necessarily of Hodge or pre-abelian type) that does not rely on the global perfectoidness of the infinite-level space was needed. The author aims to prove this using $p$-adic Hodge theory and geometric Sen theory.

2. Methodology and Tools

The author employs a sophisticated synthesis of modern arithmetic geometry and representation theory:

1. Log Adic Spaces and the Pro-Kummer-Étale Site: The work utilizes the theory of log adic spaces (from [DLLZ23b, DLLZ23a]) to handle the boundary of Shimura varieties via toroidal compactifications. This allows the use of the pro-Kummer-étale site, which is essential for studying the infinite-level Shimura variety $Sh_{K_p, \infty}$.
2. The Hodge-Tate Period Map ($\pi_{HT}$): A central tool is the Hodge-Tate period map, which maps the infinite-level Shimura variety to a flag variety $\mathbb{F}\ell$. This map is constructed using the Logarithmic Riemann-Hilbert correspondence, which associates de Rham local systems on the Shimura variety to filtered vector bundles with flat connections on the flag variety.
3. Geometric Sen Theory: The paper relies heavily on the geometric Sen operator theory developed in [RC26]. This theory provides a map (the geometric Sen operator) that relates the Lie algebra of the Galois group acting on the infinite-level cover to the sheaf of log differentials on the base space.
4. Solid Locally Analytic Representations: The author uses the framework of solid mathematics (condensed mathematics) and the theory of solid locally analytic representations (from [RJRC22, RJRC25]). This allows for a rigorous treatment of locally analytic vectors in the context of derived categories and infinite-dimensional representations.
5. Equivariant Vector Bundles: The computation involves constructing and analyzing $G$-equivariant vector bundles over flag varieties, specifically relating the adjoint representation of the group $G$ to the geometry of the period map.

3. Key Contributions and Results

The paper makes several groundbreaking contributions, culminating in the proof of the rational vanishing conjecture.

A. Computation of the Geometric Sen Operator

The primary technical achievement is the explicit computation of the geometric Sen operator for the tower of Shimura varieties.

• Theorem 1.1.4: The geometric Sen operator $\theta_{Sh}$ for the $K_p$-torsor $\pi_{K_p}: Sh_{K_p, \infty} \to Sh_{K_p}$ is shown to be naturally isomorphic to the pullback via the Hodge-Tate period map $\pi_{HT}$ of a specific surjective map of $G$-equivariant vector bundles over the flag variety $\mathbb{F}\ell_{\mathbb{C}_p}$:
  $$\mathfrak{g}_0^\vee \to \mathfrak{n}_0^\vee$$
  Here, $\mathfrak{g}_0$ is the vector bundle associated with the adjoint representation, and $\mathfrak{n}_0$ corresponds to the unipotent radical of the parabolic subgroup fixing a point in the flag variety.
• Significance: This identifies the geometric Sen operator with the Kodaira-Spencer map (twisted by a Hodge-Tate twist). This computation is crucial because it translates the analytic action of the Galois group into algebraic geometry on the flag variety.

B. Description of Locally Analytic Completed Cohomology

The paper establishes a bridge between completed cohomology and sheaf cohomology of locally analytic functions.

• Theorem 1.1.8: There is an isomorphism between the locally analytic vectors of the completed cohomology (after base change to $\mathbb{C}_p$) and the sheaf cohomology of a sheaf of locally analytic functions, denoted $\mathcal{O}_{Sh}^{la}$, on the underlying topological space of the infinite-level Shimura variety:
  $$(\tilde{H}^i(K^p, \mathbb{Z}_p) \widehat{\otimes}_{\mathbb{Z}_p} \mathbb{C}_p)^{la} \cong H^i_{sheaf}(|Sh_{K_p, \infty}|, \mathcal{O}_{Sh}^{la})$$
• Vanishing of the Unipotent Action: A critical intermediate result (Theorem 1.1.9) shows that the action of the sub-Lie algebroid $\mathfrak{n}_0$ (associated with the unipotent radical) on the sheaf $\mathcal{O}_{Sh}^{la}$ vanishes. This is a direct consequence of the computation of the geometric Sen operator.

C. Arithmetic Sen Operator and Decompletion

The author defines and computes an arithmetic Sen operator for the completed cohomology.

• Theorem 1.1.10: The locally analytic completed cohomology admits an arithmetic Sen operator given by $-\theta_\mu$, where $\theta_\mu$ is the element in the Lie algebra corresponding to the Hodge cocharacter $\mu$.
• This result allows for a "decompletion" of the cohomology, expressing it in terms of representations of the Galois group over a smaller field ($L_\infty$), which is essential for analyzing the structure of the cohomology groups.

D. Proof of Rational Vanishing

• Theorem 1.1.3 (Corollary 6.2.12): The paper proves the rational version of the Calegari-Emerton conjecture for arbitrary Shimura varieties.
  $$\tilde{H}^i(K^p, \mathbb{Z}_p)[1/p] = \tilde{H}^i_c(K^p, \mathbb{Z}_p)[1/p] = 0 \quad \text{for } i > d$$
• Mechanism of Proof:
  1. The locally analytic vectors of the completed cohomology are identified with the sheaf cohomology of $\mathcal{O}_{Sh}^{la}$.
  2. The vanishing of the action of $\mathfrak{n}_0$ (derived from the geometric Sen operator computation) implies that the cohomology is controlled by the geometry of the flag variety.
  3. The underlying topological space of the infinite-level Shimura variety has cohomological dimension $d$ (a result by Scholze).
  4. Consequently, the sheaf cohomology vanishes for $i > d$.
  5. Since the locally analytic vectors are dense in the admissible representations (by [ST03]), the vanishing extends to the rational completed cohomology.

4. Significance

• Generality: Unlike previous proofs that relied on the variety being of Hodge type or pre-abelian type (which ensures the infinite-level limit is perfectoid), this proof works for arbitrary Shimura varieties. It relies purely on the $p$-adic Hodge theory of the varieties and the properties of the period map.
• Methodological Innovation: The paper successfully integrates geometric Sen theory with the Hodge-Tate period map and solid representation theory. This provides a new toolkit for studying the local-global compatibility in the Langlands program, specifically for the local components of automorphic representations.
• Structural Insight: By computing the geometric Sen operator explicitly in terms of the flag variety geometry, the author reveals a deep structural link between the Galois action on cohomology and the Kodaira-Spencer map. This clarifies the role of the unipotent radical in the vanishing of cohomology.
• Foundation for Future Work: The explicit description of locally analytic completed cohomology as sheaf cohomology on the infinite-level space provides a robust framework for further investigations into the local Langlands correspondence for $p$-adic groups and the study of overconvergent modular forms.

In summary, Rodríguez Camargo provides a definitive proof of the rational vanishing of completed cohomology for all Shimura varieties by leveraging the geometric Sen operator to translate the problem into the cohomological dimension of the infinite-level space, bypassing the need for global perfectoidness assumptions.