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Ramla Abdellatif

Publications and source records attributed to Ramla Abdellatif.

9 recordsLinked to original sources
arXiv2026

Singular ideals over arbitrary fields for the cyclic-headed snakes

We study the Steinberg algebras with coefficients in an arbitrary field K for the cyclic-headed snake groupoids, which are basic examples of non-Hausdorff groupoids. We are particularly interested in elements of this algebra that are no longer continuous, known as singular functions. These functions form an ideal, which may contain proper subsets that are themselves ideals of the Steinberg algebra. We provide three conditions under which such 'proper subset' ideals exist: first, when the number of heads of the snake divides the characteristic of the base field; second, when the base field is of non-prime characteristic; and third, when certain cyclotomic polynomials split over the base field. We also show the existence of many further subset ideals not covered by these conditions. We fully explore the cases of the two- and three-headed snakes. In the three-headed snake, we prove that the ideal of singular functions properly contains non-zero ideals of the Steinberg algebra if, and only if, the base field K is a splitting field of x^2 + x + 1, the third cyclotomic polynomial. Consequently, there are always proper subset ideals when K has characteristic a prime not congruent to -1 mod 3.

math.RA↗
arXiv2026

Unlikely intersection in higher-dimensional formal groups

In this article, we identify a class of higher-dimensional formal groups over the ring of $p$-adic integers that are uniquely determined by their $p$-power torsion points. More precisely, we prove that if two simple finite-height formal groups share infinitely many torsion points, then they are equal. This extends a rigidity theorem of Berger \cite{LB1} from the one-dimensional setting to a higher-dimensional family of simple formal groups.

math.NT↗
arXiv2024

Constructing $2$-dimensional Lubin-Tate formal groups over $\mathbb{Z}_{p}$ (I)

In this paper, we construct a class of $2$-dimensional formal groups over $\mathbb{Z}_p$ that provide a higher-dimensional analogue of the usual $1$-dimensional Lubin-Tate formal groups, then we initiate the study of the extensions generated by their $p^{n}$-torsion points. For instance, we prove that the coordinates of the $p^{\infty}$-torsion points of such a formal group generate an abelian extension over a certain unramified extension of $\mathbb{Q}_{p}$, and we study some ramification properties of these abelian extensions. In particular, we prove that the extension generated by the coordinates of the $p$-torsion points is in general totally ramified.

math.NT↗
arXiv2023

Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

Let G be a split Kac-Moody group over a non-archimedean local field. We define a completion of the Iwahori-Hecke algebra of G. We determine its center and prove that it is isomorphic to the spherical Hecke algebra of G using the Satake isomorphism. This is thus similar to the situation of reductive groups. Our main tool is the masure I associated to this setting, which is the analogue of the Bruhat-Tits building for reductive groups. Then, for each special and spherical facet F , we associate a Hecke algebra. In the Kac-Moody setting, this construction was known only for the spherical subgroup and for the Iwahori subgroup.

math.RT↗
arXiv2022

From Fontaine-Mazur conjecture to analytic pro-p groups -- A survey

Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic geometry. Several formulations were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it. Boston's seminal paper in 1992 gave a range of purely group-theoretic methods rather than representation-theoretic ones to prove some special cases of this conjecture. Such methods have been later successfully carried on by Maire and his co-authors, and brings different informations on the objects involved in the conjecture. This survey article aims to review what is known in this direction and to present some interesting related questions the authors work on.

math.NT↗
arXiv2020

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

This paper surveys what is known about (conjectural) $p$-adic and $p$-modular semisimple Langlands correspondences in the non-supercuspidal setting for the unramified quasi-split unitary group $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$. It focuses in particular on the potential of deformation theory to relate these correspondences.

math.NT↗
arXiv2018

Irreducible $p$-modular representations of unramified $U(2,1)$

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $π$ of $G$ over $\overline{\mathbf{F}}_p$, with an underlying irreducible smooth representation $σ$ of a maximal compact open subgroup $K$, we prove that $π$ admits eigenvectors for an appropriate Hecke operator $T_σ$, and we classify those $π$ with non-zero eigenvalues for $T_σ$ by a tree argument; as a corollary, we show $π$ is supersingular if and only if it is supercuspidal.

math.RT↗
arXiv2010

Une étude des représentations modulo $p$ de SL(2,F)

Following what Barthel-Livné and Breuil made for GL(2,F), we study mod $p$ representations of SL(2,F) for F a complete non-archimedean local field of residual characteristic p and with finite residue field. In particular, we link these representations to the mod p representations of GL(2,F) and, when F = Q_p, we give an explicit description of the so-called supersingular representations, that do appear by packets of size at most 2.

math.RT↗