arXiv subjects

Gebhard Böckle

Publications and source records attributed to Gebhard Böckle.

16 recordsLinked to original sources

arXiv2026

The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields

Let $K$ be a function field in positive characteristic, $\infty$ be a fixed place of $K$ and $K_\infty$ be the completion of $K$ at $\infty$. By the work of Serre, it is well known that, for a suitable arithmetic subgroup $Γ\subset GL_2(K)$, the $Γ$-unstable region of the Bruhat-Tits tree for $GL_2(K_\infty)$ is naturally homotopy equivalent to the spherical Tits building for $GL_2(K)$. Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for $GL_r(K_\infty)$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $Γ$-unstable region, for $Γ\subset GL_r(K)$ a principal congruence subgroup.

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arXiv2026

On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

Let $\{ρ_λ:G_K\rightarrow GL_n(\overline E_λ)\}$ be a semisimple E-rational compatible system of a number field K. In a first step, building upon the theory of pseudocharacters [Ro96],[Ch14], we attach to each $ρ_λ$ an algebraic monodromy group $G_λ$ defined over $E_λ$ and also prove that the compatible system can be descended to a strongly E'-rational compatible system $\{ρ_{λ'}: G_K\rightarrow GL_n(E'_{λ'})\}$ for some finite extension E'/E. Secondly, we demonstrate that the maximal potentially abelian quotient of $G_λ$ is independent of $λ$ in a strong sense. Finally, as an application, we generalize a result of Patrikis--Snowden--Wiles on residual irreducibility of compatible systems.

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arXiv2024

Weak abelian direct summands and irreducibility of Galois representations

Let $ρ_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $ρ_\ell$ and completely characterize them, for example, if the algebraic monodromy of $ρ_\ell$ is connected. If $ρ_\ell$ is in addition $E$-rational for some number field $E$, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when $K$ is totally real and $ρ_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $π$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $ρ_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $ρ_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes.

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arXiv2023

On local Galois deformation rings

We show that framed deformation rings of mod $p$ representations of the absolute Galois group of a $p$-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application we prove density results of loci with prescribed $p$-adic Hodge theoretic properties.

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arXiv2023

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $Π$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $Π$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations.

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arXiv2022

Zariski density of crystalline points

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a $p$-adic field. We also show that these points are dense in the subspace parameterizing deformations with determinant equal to a fixed crystalline character. Our proof is purely local and works for all $p$-adic fields and all residual Galois representations.

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arXiv2021

Deformation rings and images of Galois representations

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ . (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity map on $\mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $\mathcal{G}(R)$ given by the reduction map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $\mathcal{G}$, and we study in the case at hand in great detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$ are necessary for the above results to hold.

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arXiv2019

The generic monodromy of Drinfeld modular varieties in special characteristic

By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.

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arXiv2019

$\hat{G}$-local systems on smooth projective curves are potentially automorphic

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\hat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.

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arXiv2019

On the semisimplicity of reductions and adelic openness for $E$-rational compatible systems over global function fields

Let $X$ be a normal geometrically connected variety over a finite field $κ$ of characteristic~$p$. Let $E$ be a number field. Using automorphic methods over global function fields, we derive properties of the geometric monodromy groups of arbitrary connected $E$-rational semisimple compatible systems $(ρ_λ)$ of $n$-dimensional representations of the arithmetic fundamental group $π_1(X)$, where $λ$ ranges over the finite places of $E$ not above $p$: Let $Λ_λ$ be any $π_1(X)$-stable lattice in $E_λ^n$ under $ρ_λ$. Then for almost all $λ$, the schematic closure of the geometric monodromy $ρ_λ(π_1(X_{\overlineκ}))$ in $\mathrm{Aut}_{\mathcal{O}_λ}(Λ_λ)$ is a semisimple $\mathcal{O}_λ$-group scheme, and its special fiber agrees with the Nori envelope of the geometric monodromy of the mod-$λ$ reduction of $ρ_λ$. A comparable result under different hypotheses was recently proved by Cadoret, Hui and Tamagawa by other methods. We also provide natural criteria for the image of $π_1(X_{\overlineκ})$ under $\prod_λρ_λ$ to have adelic open image in an appropriate sense.

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arXiv2017

A variational open image theorem in positive characteristic

In this note we prove a variational open adelic image theorem for the Galois action on the cohomology of smooth proper $S$-schemes where $S$ is a smooth variety over a finitely generated field of positive characteristic. A central tool is a recent result of Cadoret, Hui and Tamagawa.

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arXiv2016

Number of irreducible mod l rank 2 sheaves on curves over finite fields

Let X be a smooth projective curve of genus g over a finite field F_q of characteristic p. Consider primes l different from p. We formulate some questions related to a well known counting formula of Drinfeld. Drinfeld counts rank 2, irreducible l-adic sheaves on the base change X_n of X to F_{q^n} as n varies. We would like to count rank 2, irreducible mod l sheaves on X_n as n varies. Drinfeld's l-adic count gives an upper bound for the mod l count. We conjecture that Drinfeld's count is the correct asymptotic for the count of rank 2, irreducible mod l sheaves on X_n as n varies with (n,\ell)=1.

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arXiv2013

Cartier Crystals

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we will explain how Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory of $τ$-crystals as developed by Pink and the second author.

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arXiv2012

On computing quaternion quotient graphs for function fields

Let $Λ$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units $Λ^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group $Λ^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $Λ^*\backslash\mathcal{T}$ is {\em almost} Ramanujan.

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arXiv2009

Cartier Modules: finiteness results

On a locally Noetherian scheme X over a field of positive characteristic p we study the category of coherent O_X-modules M equipped with a p^{-e}-linear map, i.e. an additive map C: O_X \to O_X satisfying rC(m)=C(r^{p^e}m) for all m in M, r in O_X. The notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main reasult in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F-finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of Hartshorne-Speiser, Lyubeznik, Sharp, Enescu-Hochster, and Hochster about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F-finite scheme X Lyubeznkik's F-finite modules have finite length.

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arXiv2005

Uniformizable families of $t$-motives

Abelian $t$-modules and the dual notion of $t$-motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question, and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obtain a generalization of a uniformizability criterion of Anderson and, among other things, we establish that the locus of uniformizability is Berkovich open.

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