arXiv subjects

Damien Junger

Publications and source records attributed to Damien Junger.

8 recordsLinked to original sources

arXiv2026

$\mathbb{G}_m$-cohomology of $p$-adic Stein spaces

We compute the \'etale $\mathbb{G}_m$-cohomology of some $p$-adic rigid analytic Stein spaces. The computation is done by considering the filtration induced by the subgroup of principal units $U=1+ \mathfrak{m} \mathcal{O}^+$ of $\mathbb{G}_m$. We then determine the $U$-cohomology via methods from $p$-adic Hodge theory (passage to the pro-\'etale site, comparison theorems with $p$-adic cohomologies), while the $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.

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arXiv2023

Pour une définition commune des courbes elliptiques et modules de Drinfeld

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role similar to that of elliptic curves. This work grew out with the will of finding a common definition for these objects, depending only on the ring of coefficients, and thus elevating this analogy to a common theory. To that end, we introduce a class of algebraic $A$-modules for a finitely generated Dedekind ring $A$, called "modules élémentaires", which naturally generalize Drinfeld modules, forms of the multiplicative group, and elliptic curves over a field (when $A$ has the corresponding form). The objective of this text is the classification of these "modules élémentaires".

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arXiv2023

Cohomologie mod $p$ des fibrés en droites équivariants sur le demi-plan de Drinfeld

We give a classification of all equivariant line of bundles on the semi-stable model $\hat{\mathbb{H}}$ of the Drinfeld upper half plane $\mathbb{H}$ on $\mathbb{Q}_p$ for a certain subgroup $[G]_2$ of ${\rm GL}_2(\mathbb{Q}_p)$ of index $2$. Then we study the cohomology groups of these line bundles that we restrict on the special fiber $\bar{\mathbb{H}}$ and this process provides a whole family of mod $p$ representations. In particular, we exhibit a class of $[G]_2$-equivariant line bundles on $\bar{\mathbb{H}}$ (the so-called positives of weight $-1$) and show that they are in one-to-one correspondence with the irreducible supersingular representations of $[G]_2$ (a notion we define) by taking the contragredient of the global sections.

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arXiv2022

Cohomologie de de Rham du revêtement modéré de la tour de Lubin-Tate

In this article, we study the De Rham cohomology of the first cover in the Lubin-Tate tower. In particular, we get a purely local proof that the supercuspidal part realizes the local Jacquet-Langlands correspondence for ${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig varieties. The representations obtained are analogous to the ones appearing in the $\ell$-adic cohomology if we forget the action of the Weil group. The proof relies on the generalization of an excision result of Grosse-Klönne and on the existence of a semi-stable model constructed by Yoshida for which we give a more explicit description.

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arXiv2022

Un autre calcul des fonctions inversibles sur l'espace symétrique de Drinfeld

In this article, we give an explicit description of the invertible functions on the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. We identify them with some distribution spaces over the profinite set of $K$-rationnal points of the projective space. The strategy consists of constructing a map from these distributions to the invertible functions following the methods of Schneider-Stuhler, Iovita-Spiess, de Shalit. We show that it is compatible with the isomorphisms they constructed to compute étale and de Rham cohomology in degree $1$ and that this property forces our desired map to be an isomorphism. In particular, we get a $\mathbb{Z}$-structure on these cohomology groups.

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arXiv2022

Équations pour le premier revêtement de l'espace symétrique de Drinfeld

The goal of this work is to study some aspects of the geometry of the first cover $Σ^1$ in the Drinfeld tower over $\mathbb{H}^d_K$ the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. It is a cyclic étale cover of order prime to $p$ and even of Kummer type from the vanishing of the Picard group of $\mathbb{H}^d_K$ shown in a previous work of the author. It is then completely described by a certain class of invertible functions on $\mathbb{H}^d_K$ via the Kummer exact sequence and the main result of this article gives an explicit description of this class thus providing "equations" for $Σ^1$. This statement extends and uses crucially the local description over a vertex obtained by Wang (and originally by Teitelbaum in dimension 1). One of the main consequence of our global equation is the description of invertible functions of $Σ^1$ in terms of the invertible functions of $\mathbb{H}^d_K$.

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arXiv2022

Cohomologie analytique des arrangements d'hyperplans

In this article, we study the cohomology of some analytic sheaves on the complementary in the projective space of a suitable infinite collection of hyperplane like the Drinfel'd symetric space. In particular, the sheaf of invertible functions on these rigid spaces has no cohomology in degree greater or equal to $1$. This proves the vanishing of the Picard goup and the methods used give a convenient description of the global invertible functions.

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arXiv2022

Cohomologie de de Rham du rev\^etement mod\'er\'e de l'espace de Drinfeld

In this article, we study the De Rham cohomology of the first cover in the Drinfel'd tower. In particular, we get a purely local proof that the supercuspidal part realizes the local Jacquet-Langlands correspondence for ${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig varieties. The representations obtained are analogous to the ones appearing in the $\ell$-adic cohomology if we forget the action of the Weil group. The proof relies on the generalization of an excision result of Grosse-Kl\"onne and on the explicit description of the first cover as a cyclic cover obtained by the author on a previous work.

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