Prof. Dr. Elmar Große-Klönne

Profil

• Arithmetik unendlichdimensionaler Lie Algebren

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  Zeitraum: 01/2012 - 12/2015 Projektleitung: Prof. Dr. Elmar Große-Klönne

• CENTRAL--06 Automorphische Techniken in Arithmetischer Geometrie

  Quelle ↗

  Förderer: DAAD Zeitraum: 03/2015 - 12/2018 Projektleitung: Prof. Dr. Elmar Große-Klönne

• Forschungaufenthalt Yuval Flicker, Ohio State Univ.

  Quelle ↗

  Förderer: Alexander von Humboldt-Stiftung Zeitraum: 06/2011 - 08/2011 Projektleitung: Prof. Dr. Elmar Große-Klönne

• Galois moduli und modulare Heckealgebren

  Quelle ↗

  Förderer: DFG Sachbeihilfe Internationale Kooperation Zeitraum: 10/2024 - 09/2027 Projektleitung: Prof. Dr. Elmar Große-Klönne

• Garbentheoretische Methoden in der p-adischen und p-modularen Darstellungstheorie

  Quelle ↗

  Zeitraum: 01/2012 - 12/2015 Projektleitung: Prof. Dr. Elmar Große-Klönne

• Heisenberg-Professur: (I) Arithmetik Dwork´scher unit-root L-Funktionen; p-adische Modulformen (II) p-adische Kohomologie p-adischer Periodenbereiche (III) Koeffizientensysteme auf Gebäuden algebraischer Gruppen über lokalen Körpern (IV)...II

  Quelle ↗
  409-01-A · Algorithmik und Komplexität

  Förderer: DFG sonstige Programme Zeitraum: 06/2009 - 05/2011 Projektleitung: Prof. Dr. Elmar Große-Klönne

• IGRK 1800/1: Moduli and Automorphic Forms: Arithmetic and Geometric Aspects

  Quelle ↗
  409-01-A · Algorithmik und Komplexität

  Förderer: DFG Graduiertenkolleg Zeitraum: 07/2012 - 12/2016 Projektleitung: Prof. Dr. phil. Jürg Kramer

• Science & Startups Berlin

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• Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space

  2005

  Journal of Algebraic Geometry · 33 Zitationen · DOI

  We define Frobenius and monodromy operators on the de Rham cohomology of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , over a complete discrete valuation ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of mixed characteristic. For this we introduce log rigid cohomology and generalize the so-called Hyodo-Kato isomorphism to versions for non-proper <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel’d’s symmetric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by de Shalit (2005).

• On special representations of p-adic reductive groups

  2014

  Duke Mathematical Journal · 21 Zitationen · DOI

  Let F be a non-Archimedean locally compact field, and let G be a split connected reductive group over F. For a parabolic subgroup Q⊂G and a ring L, we consider the G-representation on the L-module C∞(G/Q,L)/∑Q'⊋QC∞(G/Q',L).(∗) Let I⊂G denote an Iwahori subgroup. We define a certain free finite rank-L module M (depending on Q; if Q is a Borel subgroup, then (∗) is the Steinberg representation and M is of rank 1) and construct an I-equivariant embedding of (∗) into C∞(I,M). This allows the computation of the I-invariants in (∗). We then prove that if L is a field with characteristic equal to the residue characteristic of F and if G is a classical group, then the G-representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for L=C); it had been conjectured by Vignéras. Herzig (for G=GLn(F)) and Abe (for general G) have given classification theorems for irreducible admissible modulo p representations of G in terms of supersingular representations. Some of their arguments rely on the present work.

• From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I

  2016

  Duke Mathematical Journal · 18 Zitationen · DOI

  Let o be the ring of integers in a finite extension K of Qp, and let k be its residue field. Let G be a split reductive group over Qp, and let T be a maximal split torus in G. Let H(G,I0) be the pro-p Iwahori–Hecke o-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) C(•) in the T-stable apartment, a period ϕ∈N(T) of C(•) of length r, and a homomorphism τ:Zp×→T compatible with ϕ, we construct a functor from the category Modfin(H(G,I0)) of finite-length H(G,I0)-modules to étale (φr,Γ)-modules over Fontaine’s ring OE. If G=GLd+1(Qp), then there are essentially two choices of (C(•),ϕ,τ) with r=1, both leading to a functor from Modfin(H(G,I0)) to étale (φ,Γ)-modules and hence to GalQp-representations. Both induce a bijection between the set of absolutely simple supersingular H(G,I0)⊗ok-modules of dimension d+1 and the set of irreducible representations of GalQp over k of dimension d+1. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of G over K. For d=1, we recover Colmez’s functor (when restricted to o-torsion GL2(Qp)-representations generated by their pro-p Iwahori invariants).

• On the universal module of p -adic spherical Hecke algebras

  2014

  American Journal of Mathematics · 17 Zitationen · DOI

  Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $\chi:{\frak H}_V(\widetilde{G},\widetilde{K})\rightarrow B$ of the spherical Hecke algebra ${\frak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]} {\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the specialization $$ M_{\chi}(V)={\rm ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\frak H}_V(\widetilde{G},\widetilde{K}),\chi}B $$ of the universal ${\frak H}_V(\widetilde{G},\widetilde{K})$-module ${\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$ (including $\cal{O}_F$ and $\overline{\Bbb{F}}_p$), $V$, $B$ and $\chi$, arguing geometrically on the Bruhat Tits building we give a sufficient criterion for $M_{\chi}(V)$ to be $B$-free and to admit a $\widetilde{G}$-equivariant resolution by a Koszul complex built from finitely many copies of ${\rm ind}_{\widetilde{K}Z}^{\widetilde{G}}(V)$. This criterion is the exactness of certain fairly small and explicit ${\frak N}$-equivariant $R$-module complexes, where ${\frak N}$ is the group of $\cal{O}_F$-valued points of the unipotent radical of a Borel subgroup in $\widetilde{G}$. We verify it if $F={\Bbb Q}_p$ and if $V$ is an irreducible $\overline{\Bbb{F}}_p[\widetilde{K}]$-representation with highest weight in the (closed) bottom $p$-alcove, or a lift of it to $\cal{O}_F$. We use this to construct $p$-adic integral structures in certain locally algebraic representations of $\widetilde{G}$.

• Rigid analytic spaces with overconvergent structure sheaf

  2000

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 14 Zitationen · DOI

  We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincaré duality theorems and explain the relation with Berthelot's rigid cohomology.

• Integral structures in automorphic line bundles on the p-adic upper half plane

  2004

  Mathematische Annalen · 10 Zitationen · DOI

• Integral structures in the 𝑝-adic holomorphic discrete series

  2005

  Representation Theory of the American Mathematical Society · 8 Zitationen · DOI

  For a local non-Archimedean field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript d plus 1 Baseline left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathrm {GL}}_{d+1}(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant coherent sheaves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V Subscript script upper O Sub Subscript upper K"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathcal V}_{{\mathcal O}_K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the formal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript upper K"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathcal O}_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathfrak X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> underlying the symmetric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V Subscript script upper O Sub Subscript upper K"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathcal V}_{{\mathcal O}_K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript upper K"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathcal O}_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -lattices in (the sheaf version of) the holomorphic discrete series representations (in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -vector spaces) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript d plus 1 Baseline left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD">

• On Families of Pure Slope $L$-Functions

  2014

  arXiv (Cornell University) · 6 Zitationen · DOI

  Let $R$ be the ring of integers in a finite extension $K$ of $\mathbb{Q}_p$, let $k$ be its residue field and let $χ:π_1(X)\to R^{\times}=GL_{1}(R)$ be a "geometric" rank one representation of the arithmetic fundamental group of a smooth affine $k$-scheme $X$. We show that the locally $K$-analytic characters $κ:R^{\times}\to\mathbb{C}_p^{\times}$ are the $\mathbb{C}_p$-valued points of a $K$-rigid space ${\cal W}$ and that $$L(κ\circχ,T)=\prod_{\overline{x}\in X}\frac{1}{1-(κ\circχ)(Frob_{\overline{x}})T^{°(\overline{x})}},$$viewed as a two variable function in $T$ and $κ$, is meromorphic on $\mathbb{A}_{\mathbb{C}_p}^1\times{\cal W}$. On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) $σ$-modules, in the Grothendieck group of nuclear $σ$-modules.

• Integral structures in the $p$-adic holomorphic discrete series

  2014

  arXiv (Cornell University) · 5 Zitationen · DOI

  For a local non-Archimedean field $K$ we construct ${\rm GL}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are ${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete series representations (in $K$-vector spaces) of ${\rm GL}_{d+1}(K)$ as defined by P. Schneider \cite{schn}. We prove that the cohomology $H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t&gt;0$, for ${\mathcal V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other main topic of this paper: over a finite field $k$, the study of the cohomology of vector bundles on the natural normal crossings compactification $Y$ of the Deligne-Lusztig variety $Y^0$ for ${\rm GL}_{d+1}/k$ (so $Y^0$ is the open subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational hyperplanes).

• Compactifications of log morphisms

  2004

  Tohoku Mathematical Journal · 5 Zitationen · DOI

  We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.

• On the crystalline cohomology of Deligne–Lusztig varieties

  2006

  Finite Fields and Their Applications · 4 Zitationen · DOI

• On families of pure slope $L$-functions

  2003

  Documenta Mathematica · 3 Zitationen · DOI

  Let R be the ring of integers in a finite extension K of \mathbb{Q}_p , let k be its residue field and let \chi:\pi_1(X)\to R^{\times}=GL_{1}(R) be a “geometric” rank one representation of the arithmetic fundamental group of a smooth affine k -scheme X . We show that the locally K -analytic characters \kappa:R^{\times}\to\mathbb{C}_p^{\times} are the \mathbb{C}_p -valued points of a K -rigid space {\cal W} and that L(\kappa\circ\chi,T)=\prod_{\overline{x}\in X}\frac{1}{1-(\kappa \circ\chi)(Frob_{\overline{x}})T^{\deg(\overline{x})}}, viewed as a two variable function in T and \kappa , is meromorphic on \mathbb{A}_{\mathbb{C}_p}^1\times{\cal W} . On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) \sigma -modules, in the Grothendieck group of nuclear \sigma -modules.

• Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space

  2014

  arXiv (Cornell University) · 2 Zitationen · DOI

  We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper $Y$, for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of $Y$. We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's symmetric space $X$ and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of $X$ given by de Shalit.

• From pro-$p$ Iwahori–Hecke Modules to $(\varphi,\Gamma)$-Modules, II

  2016

  International Mathematics Research Notices · 1 Zitationen · DOI

  Let |${\mathfrak o}$| be the ring of integers in a finite extension field of |${\mathbb Q}_p$|⁠, let |$k$| be its residue field. Let |$G$| be a split reductive group over |${\mathbb Q}_p$|⁠, let |${\mathcal H}(G,I_0)$| be its pro-|$p$| Iwahori–Hecke |${\mathfrak o}$|-algebra. In [2] we introduced a general principle how to assign to a certain additionally chosen datum |$(C^{(\bullet)},\phi,\tau)$| an exact functor |$M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$| from finite length |${\mathcal H}(G,I_0)$|-modules to |$(\varphi^r,\Gamma)$|-modules. In this paper we concretely work out such data |$(C^{(\bullet)},\phi,\tau)$| for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular |${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$|-modules with the set of |$(\varphi^r,\Gamma)$|-modules satisfying a certain symmetry condition.

• On the universal module of $p$-adic spherical Hecke algebras

  2014

  arXiv (Cornell University) · 1 Zitationen · DOI

  Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $χ:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the specialization $$M_χ(V)={\rm ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\mathfrak H}_V(\widetilde{G},\widetilde{K}),χ}B$$ of the universal ${\mathfrak H}_V(\widetilde{G},\widetilde{K})$-module ${\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$ (including ${\mathcal O}_F$ and $\overline{\mathbb F}_p$), $V$, $B$ and $χ$, arguing geometrically on the Bruhat Tits building we give a sufficient criterion for $M_χ(V)$ to be $B$-free and to admit a $\widetilde{G}$-equivariant resolution by a Koszul complex built from finitely many copies of ${\rm ind}_{\widetilde{K}Z}^{\widetilde{G}}(V)$. This criterion is the exactness of certain fairly small and explicit ${\mathfrak N}$-equivariant $R$-module complexes, where ${\mathfrak N}$ is the group of ${\mathcal O}_F$-valued points of the unipotent radical of a Borel subgroup in $\widetilde{G}$. We verify it if $F={\mathbb Q}_p$ and if $V$ is an irreducible $\overline{\mathbb F}_p[\widetilde{K}]$-representation with highest weight in the (closed) bottom $p$-alcove, or a lift of it to ${\mathcal O}_F$. We use this to construct $p$-adic integral structures in certain locally algebraic representations of $\widetilde{G}$.

• Locally unitary principal series representations of ${\rm GL}_{d+1}(F)$

  2014

  arXiv (Cornell University) · 1 Zitationen · DOI

  For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.

• p-torsion coefficient systems for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="normal">SL</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="double-struck">Q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="normal">GL</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="double-struck">Q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>

  2011

  Journal of Algebra · 1 Zitationen · DOI

• Families of two dimensional modular $(φ,Γ)$-modules

  2024

  arXiv (Cornell University) · DOI

  Let $F/{\mathbb Q}_p$ be a finite unramified extension, let $k$ be a finite extension of the residue field of $F$. We provide explicit constructions of integral structures for all rank two étale Lubin-Tate $(φ,{\mathcal O}_F^{\times})$-modules over $k$. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of $(φ,{\mathcal O}_F^{\times})$-modules. These results reveal new combinatorial structures of the moduli stack of $(φ,{\mathcal O}_F^{\times})$-modules, and allow us, in particular, to rederive the fact that the Serre weights assigned to a two dimensional ${\rm Gal}(\overline{F}/F)$-representation over $k$ can be read off from the geometry of the stack.

• On $ψ$-lattices in modular $(φ,Γ)$-modules

  2024

  arXiv (Cornell University) · DOI

  Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a finite field extension of the residue field of $F$. Generalizing the $ψ$-lattices which Colmez constructed in étale $(φ,Γ)$-modules over $k[[t]][t^{-1}]$, we define, study and exemplify $ψ$-lattices in étale $(φ,Γ)$-modules over $k[[t_1,\ldots,t_d]][\prod_it_i^{-1}]$ for arbitrary $d\in{\mathbb N}$.

• On $$\psi $$-Lattices in Modular $$(\varphi ,\Gamma )$$-Modules

  2022

  Infosys science foundation series · DOI

• From pro-$p$ Iwahori-Hecke modules to $(\varphi,Γ)$-modules I

  2015

  arXiv (Cornell University)

• Locally algebraic automorphisms of the $PGL_2(F)$-tree and $o$-torsion representations

  2015

  Bulletin de la Société mathématique de France · DOI

• Locally algebraic automorphisms of the ${\rm PGL}_2(F)$-tree and ${\mathfrak o}$-torsion representations

  2015

  arXiv (Cornell University) · DOI

  For a local field $F$ and an Artinian local coefficient ring $Λ$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(Λ)$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $Λ$ to ${\mathfrak C}^{(e)}(Λ)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(Λ)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $Λ$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${\mathfrak C}^{(e)}(Λ)$ with trivial central character is equivalent to a category of representations of a certain subgroup of ${\rm Aut}(X)$ consisting of "locally algebraic automorphisms of level $e$". For $e=1$ there is a functor from this category to that of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is a bijection between irreducible objects. Finally, we present a parallel of Colmez' functor $V\mapsto {\bf D}(V)$: to objects in ${\mathfrak C}^{(e)}(Λ)$ (for any $F$) we assign certain étale $(φ,Γ)$-modules over an Iwasawa algebra ${\mathfrak o}[[\widehat{N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawa algebra ${\mathfrak o}[[{N}_{0}]]$. This assignment preserves finite generation.

• Sheaves of bounded $p$-adic logarithmic differential forms

  2014

  arXiv (Cornell University) · DOI

  Let $K$ be a local field, $X$ the Drinfel'd symmetric space $X$ of dimension $d$ over $K$ and ${\mathfrak X}$ the natural formal ${\mathcal O}_K$-scheme underlying $X$; thus $G={\rm GL}\sb {d+1}(K)$ acts on $X$ and ${\mathfrak X}$. Given a $K$-rational $G$-representation $M$ we construct a $G$-equivariant subsheaf ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$ of ${\mathcal O}_K$-lattices in the constant sheaf $M$ on ${\mathfrak X}$. We study the cohomology of sheaves of logarithmic differential forms on $X$ (or ${\mathfrak X}$) with coefficients in ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$. In the second part we give general criteria for two conjectures of P. Schneider on $p$-adic Hodge decompositions of the cohomology of $p$-adic local systems on projective varieties uniformized by $X$. Applying the results of the first part we prove the conjectures in certain cases.

• Integral structures in automorphic line bundles on the $p$-adic upper half plane

  2014

  arXiv (Cornell University) · DOI

  Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in ${\mathcal O}_X(k)\otimes_K\widehat{K}$, as a coherent sheaf on the formal model $\widehat{\mathfrak{X}}$ underlying $X\otimes_K\widehat{K}$. Here $\widehat{K}/K$ is ramified of degree $2$. This generalizes a construction of Teitelbaum from the case of even weight $k$ to arbitrary integer weight $k$. We compute $H^*(\widetilde{\mathfrak{X}},{\mathcal O}_{\widehat{\mathfrak X}}(k))$ and obtain applications to the de Rham cohomology $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ with coefficients in the $k$-th symmetric power of the standard representation of ${\rm SL}_2(K)$ (where $k\ge0$) of projective curves $Γ\backslash X$ uniformized by $X$: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$, we re-prove the Hodge decomposition of $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ and show that the monodromy operator on $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ respects integral de Rham structures and is induced by a "universal"{} monodromy operator defined on $\widehat{\mathfrak{X}}$, i.e. before passing to the $Γ$-quotient.

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Name

Prof. Dr. Elmar Große-Klönne

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Algebra und Zahlentheorie

Telefon

+49 30 2093-45402

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26.4.2026, 01:05:23