Cohomologie de de Rham du revêtement modéré de l'espace de Drinfeld

Technical Summary: De Rham Cohomology of the First Cover of the Drinfeld Space

Problem Statement
The paper investigates the De Rham cohomology of the first cover, denoted $M^1_{Dr}$, within the Drinfeld tower of étale covers over the Drinfeld symmetric space $H^d_K$. While the $\ell$-adic cohomology ($\ell \neq p$) of this tower is well-established as providing a geometric realization of the local Langlands and Jacquet-Langlands correspondences (via works by Harris, Taylor, Boyer, etc.), the $p$-adic case (specifically De Rham cohomology) presents significant challenges. The primary difficulty lies in the absence of a known semi-stable (or generalized semi-stable) model for the first cover $M^1_{Dr}$, unlike the base space $M^0_{Dr}$ which admits a Deligne model. Without such a model, standard techniques involving the comparison of generic and special fibers (such as the study of vanishing cycles) are not directly applicable in the De Rham setting.

Methodology
The author employs a purely local approach, avoiding global geometric arguments where possible. The methodology relies on three main pillars:

1. Generalization of Excision Results: The paper extends a result by Grosse-Klönne regarding the excision of cohomology in rigid analytic spaces. Specifically, Theorem B establishes that for a formal scheme $X$ of generalized semi-stable reduction and a tame cyclic cover $\pi: T \to X_\eta$, the De Rham cohomology of the tube over an intersection of irreducible components of the special fiber is isomorphic to the cohomology of the tube over the intersection minus the "boundary" components. This allows the reduction of the global cohomology calculation to local computations on specific open subsets.

2. Explicit Geometric Description: Utilizing previous work by the author, the first cover $\Sigma_1$ (the analytic space associated with $M^1_{Dr}$) is described explicitly as a Kummer cover (a cyclic cover of order prime to $p$) of the Drinfeld space. Over specific open sets corresponding to vertices of the Bruhat-Tits building, this cover is shown to be isomorphic to a cyclic cover of a monomial torus.

3. Comparison with Deligne-Lusztig Varieties: The paper identifies that the special fiber of the cover over a vertex of the Bruhat-Tits building is isomorphic to a Deligne-Lusztig variety $DL^d_F$. The author computes the De Rham cohomology of these cyclic covers of monomial tori and relates them to the rigid cohomology of the Deligne-Lusztig varieties.

Key Contributions and Results

• Theorem 5.1 (Excision for Cyclic Covers): The paper proves a generalized excision theorem for the De Rham cohomology of tame cyclic covers over formal schemes of generalized semi-stable reduction. This result is crucial for reducing the cohomology of the entire cover to the cohomology of specific "tubes" over the irreducible components of the special fiber.
• Theorem 5.8 (Cohomology of Cyclic Covers of Monomial Tori): The author provides an explicit description of the De Rham cohomology of a tame cyclic cover of a monomial torus. The cohomology is shown to decompose into isotypic components corresponding to the characters of the Galois group, with a basis given by logarithmic differentials.
• Theorem 6.1 (Local Calculation): For a primitive character $\theta$ of the Galois group $F_{q^{d+1}}^*$, the De Rham cohomology with compact support of the first cover, restricted to the $\theta$-isotypic component, is concentrated in degree $d$. Specifically:
  $$H^i_{dR,c}(\Sigma_1)[\theta] \cong \begin{cases} \bigoplus_{s \in BT_0} H^d_{rig,c}(DL^d_F / \breve{K})[\theta] & \text{if } i = d \\ 0 & \text{otherwise} \end{cases}$$
  Here, $BT_0$ denotes the set of vertices of the semi-simple Bruhat-Tits building.
• Theorem 6.9 (Realization of Jacquet-Langlands): The paper establishes that the supercuspidal part of the De Rham cohomology of the first cover realizes the local Jacquet-Langlands correspondence. Specifically, for a primitive character $\theta$, the space of $D^*$-equivariant maps from the representation $\rho(\theta)$ to the cohomology $H^i_{dR,c}(M^1_{Dr})$ is isomorphic to the representation $JL(\rho(\theta))$ of $GL_{d+1}(K)$ if $i=d$, and zero otherwise.

Significance
The paper claims to provide a purely local proof that the supercuspidal part of the De Rham cohomology of the first Drinfeld cover realizes the local Jacquet-Langlands correspondence. This serves as a De Rham analogue to the classical $\ell$-adic results established by Wang and others.

The significance lies in:

1. Bridging Cohomology Theories: It demonstrates that the "supercuspidal" information encoded in the $\ell$-adic cohomology is also present in the De Rham cohomology, provided one ignores the action of the Weil group (which is invisible in the De Rham setting).
2. Overcoming Model Limitations: By utilizing the generalized excision theorem (Theorem B) and the explicit description of the cover as a Kummer extension, the author bypasses the need for a global semi-stable model of $M^1_{Dr}$, a major obstruction in previous attempts.
3. Methodological Extension: The techniques developed, particularly the excision result for cyclic covers, are noted as potentially useful for studying the first cover of the Lubin-Tate space, where similar geometric complexities arise.

The author notes that while the result is proven for $n=1$ (the first cover) and arbitrary dimension $d$, the situation for higher levels ($n > 1$) is significantly more complex, and a purely local approach may not suffice for those cases. The work relies heavily on the geometric description of the first cover obtained in the author's prior publications.