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Gabriel Dospinescu

Publications and source records attributed to Gabriel Dospinescu.

At least 19 recordsLinked to original sources
arXiv2026

On the de Rham flip-flopping in dual towers

We prove a version of de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces including those coming from dual basic local Shimura varieties. The main tool are comparison theorems expressing the two cohomologies as pro-\'etale cohomology of corresponding relative period sheaves that, by definition, satisfy pro-\'etale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of finite level coverings of the Drinfeld space of any dimension $d$ over $K$ are admissible as representations of $\mathbb{GL}_{d+1}(K)$.

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arXiv2024

A Jacquet-Langlands functor for $p$-adic locally analytic representations

We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level are independent of the actions of the $p$-adic Lie groups $G$ and $G_b$ of the two towers; this generalizes a result of Pan for the Lubin-Tate and Drinfeld spaces for $\mathrm{GL}_2$. We apply this theory to show that Scholze's $p$-adic Jacquet-Langlands functor commutes with the passage to locally analytic vectors, and is compatible with central characters of Lie algebras. We also prove that the compactly supported de Rham cohomology of the two towers are isomorphic as smooth representations of $G\times G_b$.

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arXiv2023

Factorization de la cohomologie étale p-adique de la tour de Drinfeld

For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower of coverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeld half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the $p$-adic geometric étale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finitness theorem for the arithmetic étale cohomology modulo $p$ which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all $F$; for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rm GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$.

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arXiv2023

Correspondance de Langlands locale $p$-adique et anneaux de Kisin

We use a ${\mathcal B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm {GL}}_2({\mathbf Q}_p )$ to give a construction of Kisin's rings and the attached universal Galois representations (in dimension 2 and for ${\mathbf Q}_p$) directly from the classical Langlands correspondence. This gives, in particular, a uniform proof of the geometric Breuil-Mézard conjecture in the supercuspidal case.

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arXiv2023

Gelfand-Kirillov dimension and the $p$-adic Jacquet-Langlands correspondence

We bound the Gelfand-Kirillov dimension of unitary Banach space representations of $p$-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and $p$-adic Banach space representations of the group of units of a quarternion algebra over $\mathbb Q_p$ appearing in the $p$-adic Jacquet-Langlands correspondence, deducing finiteness results in favourable cases.

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arXiv2022

Cohomologie des courbes analytiques $p$-adiques

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-étale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\ell$-adic pro-étale cohomology, for $\ell\neq p$.

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arXiv2021

p-adic etale cohomology of period domains

We compute the p-torsion and p-adic etale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. For the p-torsion case, we follow the method used by Orlik in his computations of the l-torsion etale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we don't use Huber's definition of etale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous etale cohomology with compact support.

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arXiv2020

Infinitesimal characters in arithmetic families

We associate infinitesimal characters to (twisted) families of $L$-parameters and $C$-parameters of $p$-adic reductive groups. We use the construction to study the action of the centre of the universal enveloping algebra on the locally analytic vectors in the Hecke eigenspaces in the completed cohomology.

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arXiv2019

Integral $p$-adic étale cohomology of Drinfeld symmetric spaces

We compute the integral $p$-adic étale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational $p$-adic étale cohomology from Colmez-Dospinescu-Nizioł. The main tools are: the computation of the integral de Rham cohomology from CDN and the integral $p$-adic comparison theorems of Bhatt-Morrow-Scholze and Česnavičius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used in CDN.

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arXiv2019

Cohomology of $p$-adic Stein spaces

We compute the $p$-adic étale and the pro-étale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-étale cohomology of $p$-adic Stein spaces.

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arXiv2017

Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique

We describe the de Rham complex of the étale coverings of Drinfeld's p-adic upper half-plane for GL_2(Q_p). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of the absolute Galois group of Q_p.

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arXiv2014

Irreducible components of deformation spaces: wild 2-adic exercises

We prove that the irreducible components of the space of framed deformations of the trivial 2-dimensional mod 2 representation of the absolute Galois group of Q_2 are in natural bijection with those of the trivial character, confirming a conjecture of Böckle. We deduce from this result that crystalline points are Zariski dense in that space: this provides the missing ingredient for the surjectivity of the p-adic local Langlands correspondence for GL_2(Q_p) in the case p=2 (the result was already known for p\geq 3).

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arXiv2013

The p-adic local Langlands correspondence for GL_2(Q_p)

The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of GL_2(Q_p) and absolutely irreducible 2-dimensional representations of G_{Q_p}. This had already been proved, by characteristic p methods, but only for p\geq 5.

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arXiv2013

Extensions de représentations de de Rham et vecteurs localement algébriques

Let $Π$ be an irreducible unitary completion of a locally algebraic ${\rm GL}_2(\qp)$-representation. We describe those first-order deformations of $Π$ which are themselves completions of a locally algebraic representation. This answers a question of Paskunas and has direct applications to the Breuil-Mézard conjecture.

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arXiv2013

Complétés universels de représentations de GL_2(Q_p)

Let Pi be a unitary representation of GL_2(Q_p), topologically of finite length. We describe the sub-representation Pi^{an} made of its locally analytic vectors, and its filtration by radius of analyticity, in terms of the phi-Gamma module attached to Pi via the p-adic local Langlands correspondence, and we deduce that the universal completion of Pi^{an} is Pi itself.

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