# Prof. Dr. Urs Hartl, Mathematisches Institut

Member of CRC 1442 Geometry: Deformations and RigidityInvestigator in Mathematics Münster

Investigator in Mathematics Münster

Private Homepage | https://www.uni-muenster.de/Arithm/hartl/index.html |

Research Interests | Arithmetic algebraic geometry Arithmetic of function fields Algebraic Number Theory Structur theory of Shimura varieties and their function field analogs p-adic Hodge theory Model theory of valued fields |

Project membershipMathematics Münster | A: Arithmetic and GroupsA1: Arithmetic, geometry and representations |

Current Publications | • E. Arasteh Rad, U. Hartl Category of C-Motives over Finite Fields. Journal of Number Theory Vol. 232, 2022, pp 283-316 online• Hartl, Urs; Singh, Rajineesh Kumar A short review on local shtukas and divisible local Anderson modules. Perfectoid SpacesInfosys Science Foundation Series in Mathematical Sciences, 2022 online• Arasteh Rad Esmail, Hartl Urs Uniformizing the Moduli Stacks of Global G-Shtukas. International Mathematics Research Notices Vol. 21, 2021, pp 16121-16192 online• U. Hartl, E. Viehmann The generic fiber of moduli spaces of bounded local G-shtukas. Journal of the Institute of Mathematics of Jussieu Vol. 2021, 2021, pp 1-80 online• U. Hartl, R. Singh Periods of Drinfeld modules and local shtukas with complex multiplication - Errata. Journal of the Institute of Mathematics of Jussieu Vol. 2021, 2021, pp 1-5 online• Hartl Urs, Kim Wansu Local Shtukas, Hodge-Pink Structures and Galois Representations. t-motives: Hodge structures, transcendence and other motivic aspects, 2020, pp 183-260 online• Hartl Urs, Hellmann Eugen The universal familly of semi-stable p-adic Galois representations. Algebra and Number Theory Vol. 14, 2020, pp 1055-1121 online• Hartl Urs, Singh Rajneesh Kumar Periods of Drinfeld modules and local shtukas with complex multiplication
. Journal of the Institute of Mathematics of Jussieu Vol. 19, 2020, pp 175-208 online• Hartl Urs, Juschka Ann-Kristin Pink's Theory of Hodge Structures and the Hodge Conjecture over Function Fields. in t-motives: Hodge structures, transcendence and other motivic aspects, 2020, pp 31-182 online |

Current Projects | • CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations p-adic Galois representations in finite Zp-modules are equivalent to (phi,Gamma)-modules for Qp. In this project, we develop the theory of (phi,Gamma)-modules further in the direction of finite extensions of Qp and their function field analogues. We will also use (phi,Gamma)-modules to construct moduli spaces of p-adic Galois representations. We aim to decompose special fibres on these moduli spaces into cycles in a way that mirrors multiplicity formulas in representation theory. online• CRC 1442: Geometry: Deformation and Rigidity - A03: Special cycles on moduli spaces of G-shtukas Moduli stacks of global G-shtukas are the function field analogue of Shimura varieties. Their geometry was investigated by the PI and his collaborators. The long term goal of the project is to investigate arithmetic properties of these moduli stacks, special cycles on them and the intersection numbers of the latter. In particular, we want to develop a function field analogue of the Kudla programme for Shimura varieties and prove arithmetic fundamental lemmas for function fields. online• EXC 2044 - A1: Arithmetic, geometry and representations The Langlands programme relates representations of (the adele valued points of) reductive groups G over Q - so-called automorphic representations - with certain representations of the absolute Galois group of Q. This programme includes the study of these objects over general global fields (finite extension of Q or Fp (t)) and local fields as well. In its local form the classical programme onlyconsidered l-adic Galois representations of p-adic fields for unequal primes l neq p. In order to allow for a p-adic variation of the objects, it is absolutely crucial to extend it to the case l = p. In the global situation, the automorphic representations in question can often be realised in (or studied via) the cohomology of a tower of Shimura varieties (or related moduli spaces) attached to the group G. We will focus on the following directions within this programme: The p-adic and mod p Langlands programme asks for an extension of such a correspondence involving certain continuous representations with p-adic respectively mod p coefficients. Broadening the perspective to p-adic automorphic forms should, for example, enable us to capture all Galois representations, not just those having a particular Hodge theoretic behaviour at primes dividing p. This extended programme requires the introduction of derived categories. We will study differential graded Hecke algebras and their derived categories on the reductive group side. On the Galois side, we hope to use derived versions of the moduli spaces of p-adic Galois representations introduced by Emerton and Gee. The geometric Langlands programme is a categorification of the Langlands programme. We plan to unify the different approaches using motivic methods. In another direction, we study the geometry and arithmetic of moduli stacks of global G-shtukas over function fields. Their cohomology has been the crucial tool to establish large parts of the local and global Langlands programme over function fields. Variants of G-shtukas are also used to construct and investigate families of p-adic Galois representations.Cohomology theories are a universal tool pervading large parts of algebraic and arithmetic geometry. We will develop and study cohomology theories, especially in mixed characteristic, that generalise and unify étale cohomology, crystalline cohomology and de Rham cohomology as well as Hochschild cohomology in the non-commutative setting. Developing (topological) cyclic homology in new contexts is an important aim. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations. online |

E-Mail | urs dot hartl at uni-muenster dot de |

Phone | +49 251 83-33702 |

FAX | +49 251 83-33786 |

Room | 317 |

Secretary | Sekretariat Harenbrock/Reckermann Frau Ina Reckermann Telefon +49 251 83-33700 Fax +49 251 83-33786 Zimmer 316 |

Address | Prof. Dr. Urs Hartl Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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