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Eugen Hellmann

Publications and source records attributed to Eugen Hellmann.

16 recordsLinked to original sources
arXiv2024

Patching and multiplicities of p-adic eigenforms

We prove the existence of non-classical $p$-adic automorphic eigenforms associated to a classical system of eigenvalues on definite unitary groups in $3$ variables. These eigenforms are associated to Galois representations which are crystalline but very critical at $p$. We use patching techniques related to the trianguline variety of local Galois representations and its local model. The new input is a comparison of the coherent sheaves appearing in the patching process with coherent sheaves on the Grothendieck--Springer version of the Steinberg variety given by a functor constructed by Bezrukavnikov.

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arXiv2021

On the derived category of the Iwahori-Hecke algebra

We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.

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arXiv2020

The universal family of semi-stable p-adic Galois representations

Let $K$ be a finite field extension of $Q_p$ and let $G_K$ be its absolute Galois group. We construct the universal family of filtered $(ϕ,N)$-modules, or (more generally) the universal family of $(ϕ,N)$-modules with a Hodge-Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semi-stable $G_K$-representations in $Q_p$-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over $Q_p$ in the sense of Huber. This has conjectural applications to the $p$-adic local Langlands program.

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arXiv2018

Density of automorphic points in deformation rings of polarized global Galois representations

Conjecturally, the Galois representations that are attached to essentially selfdual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvea-Mazur and Chenevier, and relies on the construction of non-classical $p$-adic automorphic forms, and the computation of the tangent space of the space of trianguline Galois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras.

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arXiv2017

A local model for the trianguline variety and applications

We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.

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arXiv2015

Smoothness and Classicality on eigenvarieties

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimension by making crucial use of the "patched eigenvariety".

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arXiv2015

Une interprétation modulaire de la variété trianguline

Using a patching module constructed in recent work of Caraiani, Emerton, Gee, Geraghty, Pa{š}k{ū}nas and Shin we construct some kind of analogue of an eigenvariety. We can show that this patched eigenvariety agrees with a union of irreducible components of a space of trianguline Galois representations. Building on this we discuss the relation with the modularity conjectures for the crystalline case, a conjecture of Breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of Bellaïche and Chenevier on the complete local ring at certain points of eigenvarieties.

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arXiv2014

Density of potentially crystalline representations of fixed weight

Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology.

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arXiv2012

Families of trianguline representations and finite slope spaces

We apply the theory of families of (phi,Gamma)-modules to trianguline families as defined by Chenevier. This yields a new definition of Kisin's finite slope subspace as well as higher dimensional analogues. Especially we show that these finite slope spaces contain eigenvarieties for unitary groups as closed subspaces. This implies that the representations arising from overconvergent p-adic automorphic forms on certain unitary groups are trianguline when restricted to the local Galois group.

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arXiv2012

Families of p-adic Galois representations and (phi,Gamma)-modules

We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.

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arXiv2011

On arithmetic families of filtered phi-modules and crystalline representations

We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.

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arXiv2011

On families of weakly admissible filtered phi-modules and the adjoint quotient of GL_d

We study the relation of the notion of weak admissibility in families of filtered phi-modules, as considered in a companion paper, with the adjoint quotient. We show that the weakly admissible subset is an open subvariety in the fibers over the adjoint quotient. Further we determine the image of the weakly admissible set in the adjoint quotient generalizing earlier work of Breuil and Schneider.

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arXiv2010

Connectedness of Kisin varieties for GL_2

We show that the Kisin varieties associated to simple $ϕ$-modules of rank $2$ are connected in the case of an arbitrary cocharacter. This proves that the connected components of the generic fiber of the flat deformation ring of an irreducible $2$-dimensional Galois representation of a local field are precisely the components where the multiplicities of the Hodge-Tate weights are fixed.

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arXiv2010

The image of the coefficient space in the universal deformation space

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parameterizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

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arXiv2008

On the structure of some moduli spaces of finite flat group schemes

We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of $mathbb{Q}_p$. We determine the connected components of this space and describe its irreducible components. These results prove a modified version of a conjecture of Kisin.

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