arXiv subjects

Adrian Zenteno

Publications and source records attributed to Adrian Zenteno.

9 recordsLinked to original sources

arXiv2025

Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to $\ell$, and splits into two cases according to $G$ is of type $A_{\ell}$ or not. Furthermore, we discuss explicit formulas for computing the dimensions of such representations. The motivation for this work arises, in part, from a desire to obtain new results on two classical problems concerning Galois representations: the large image conjecture for automorphic Galois representations and the inverse Galois problem. We conclude the paper by giving some remarks on potential implications in these addresses.

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arXiv2023

Galois representations with large image in the global Langlands correspondence

The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $\pi$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ \rho_{\pi,\ell} \}_\ell$ of $n$-dimensional $\ell$-adic Galois representations of $\text{Gal}(\overline{F}/F)$. In general, it is expected that the image of the residual Galois representation $\overline{\rho}_{\pi,\ell}$ of $\rho_{\pi,\ell}$ should be as large as possible for almost all primes $\ell$, unless there is an automorphic reason for the image to be small. In this paper, we study the images of certain compatible systems of Galois representations $\{\rho_{\pi,\ell} \}_\ell$ associated to regular algebraic, polarizable, cuspidal automorphic representations $\pi$ of $\text{GL}_n(\mathbb{A}_F)$ by using only standard techniques and currently available tools (e.g., Fontaine-Laffaille theory, Serre's modularity conjecture, classification of the maximal subgroups of Lie type groups, and known results about irreducibility of automorphic Galois representations and Langlands functoriality). In particular, when $F$ is a totally real field and $n$ is an odd prime number $\leq 293$, we prove that (under certain automorphic conditions) the images of the residual representations $\overline{\rho}_{\pi,\ell}$ are as large as possible for infinitely many primes $\ell$. In fact, we prove the large image conjecture (i.e., large image for almost all primes $\ell$) when $F=\mathbb{Q}$ and $n=5$.

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arXiv2023

Bianchi and Hilbert-Blumenthal quaternionic orbifolds

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by Díaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal quaternionic orbifolds in terms of fundamental units of real quadratic fields. These are 7-dimensional solvmanifolds which are virtual ${\mathbb T}^6$ bundles over the circle with monodromy a linear Anosov diffeomorphism of the 6-torus.

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arXiv2021

On the images of certain $G_2$-valued automorphic Galois representations

In this paper we study the images of certain families $\{ρ_{π,\ell} \}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations $π$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Taïbi.

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arXiv2020

Automorphic Galois representations and the inverse Galois problem for certain groups of type $D_{m}$

Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}Ω^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of $\mbox{GL}_{2m}(\mathbb{A}_\mathbb{Q})$..

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arXiv2019

Lübeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups

In this paper we prove that for each integer of the form $n=4\varpi$ (where $\varpi$ is a prime between $17$ and $73$) at least one of the following groups: $PΩ^+_n(\mathbb{F}_{\ell^s})$, $PSO^+_n(\mathbb{F}_{\ell^s})$, $PO_n^+(\mathbb{F}_{\ell^s})$ or $PGO^+_n(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for almost all primes $\ell$ and infinitely many integers $s > 0$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic of F. Lübeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $GL_n(\mathbb{A}_\mathbb{Q})$.

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arXiv2018

On the images of the Galois representations attached to generic automorphic representations of GSp(4)

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.

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arXiv2018

On the images of the Galois representations attached to certain RAESDC automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$

In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that preserve some kind of geometric structure, so they are commonly called subgroups of geometric type. In this paper we will prove the existence of infinitely many compatible systems $\{ ρ_\ell \}_\ell$ of $n$-dimensional Galois representations associated to regular algebraic, essentially self-dual, cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$ ($n$ even) such that, for almost all primes $\ell$, the image of $\overlineρ_{\ell}$ (the semi-simplification of the reduction of $ρ_\ell$) cannot be contained in a maximal subgroup of geometric type of an $n$-dimensional symplectic or orthogonal group. Then, we apply this result to some 12-dimensional representations to give heuristic evidence towards the inverse Galois problem for even-dimensional orthogonal groups.

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arXiv2017

Constructing Hilbert modular forms whithout exceptional primes

In this paper we construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre's modularity conjecture, to totally real fields.

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