Projects in Mathematics Muenster

Research area A: Arithmetic and Groups
Unit A1: Arithmetic, geometry and representations

Further 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

Research Interests

Research Interests

$\bullet$ Arithmetic algebraic geometry.
$\bullet$ Algebraic number theory.
$\bullet$ Arithmetic of function fields.
$\bullet$ Structure theory of Shimura varieties and their function field analogues.
$\bullet$ $p$-adic Hodge theory.
$\bullet$ $p$-adic Langlands programme.
$\bullet$ Model theory.

Selected Publications

Selected Publications of Urs Hartl

$\bullet$ U. Hartl and R. K. Singh. Periods of Drinfeld modules and local shtukas with complex multiplication. J. Inst. Math. Jussieu, 1–34, 2018.

$\bullet$ E. Arasteh Rad and U. Hartl. Local $\mathbb P$-shtukas and their relation to global $\mathfrak G$-shtukas. Münster J. Math., 7(2):623–670, 2014.

$\bullet$ U. Hartl. On a conjecture of Rapoport and Zink. Invent. Math., 193(3):627–696, 2013.

$\bullet$ U. Hartl and E. Viehmann. Foliations in deformation spaces of local $G$-shtukas. Adv. Math., 229(1):54–78, 2012.

$\bullet$ U. Hartl and E. Viehmann. The Newton stratification on deformations of local $G$-shtukas. J. Reine Angew. Math., 656:87–129, 2011.

$\bullet$ U. Hartl. Period spaces for Hodge structures in equal characteristic. Ann. of Math. (2), 173(3):1241–1358, 2011.

$\bullet$ U. Hartl. A dictionary between Fontaine-theory and its analogue in equal characteristic. J. Number Theory, 129(7):1734–1757, 2009.

$\bullet$ U. Hartl. Uniformizing the stacks of abelian sheaves. In Number fields and function fields–two parallel worlds, volume 239 of Progr. Math., pages 167–222. Birkhäuser, Boston, 2005.

$\bullet$ U. Hartl and R. Pink. Vector bundles with a Frobenius structure on the punctured unit disc. Compos. Math., 140(3):689–716, 2004.

$\bullet$ U. Hartl and W. Lütkebohmert. On rigid-analytic Picard varieties. J. Reine Angew. Math., 528:101–148, 2000.

Current Publications

$\bullet $ Urs Hartl and Eva Viehmann. The generic fibre of moduli spaces of bounded local $G$-shtukas. Journal of the Institute of Mathematics of Jussieu, pages 1–80, July 2021. doi:10.1017/S1474748021000293.

$\bullet $ Urs Hartl and Rajneesh Kumar Singh. Product formulas for periods of CM abelian varieties and the function field analog. Journal of Number Theory, 218:62–151, January 2021. doi:10.1016/j.jnt.2020.06.017.

$\bullet $ Urs Hartl and Chia-Fu Yu. Arithmetic Satake compactifications and algebraic Drinfeld modular forms. arXiv e-prints, September 2020. arXiv:2009.13934.

$\bullet $ Esmail Arasteh Rad and Urs Hartl. Category of C-motives over finite fields. Journal of Number Theory, July 2020. doi:10.1016/j.jnt.2020.06.015.

$\bullet $ Urs Hartl and Eugen Hellmann. The universal family of semistable $p$-adic Galois representations. Algebra Number Theory, 14(5):1055–1121, July 2020. doi:10.2140/ant.2020.14.1055.

$\bullet $ U. Hartl and W. Kim. Local Shtukas, Hodge-Pink Structures and Galois Representations, pages 157–218. European Mathematical Society (EMS), Zürich, May 2020. doi:10.4171/198.

$\bullet $ U. Hartl and A.-K. Juschka. Pink's theory of Hodge structures and the Hodge conjecture over function fields, pages 35–155. European Mathematical Society (EMS), Zürich, May 2020. doi:10.4171/198.

$\bullet $ Urs Hartl and Rajneesh Kumar Singh. Periods of Drinfeld modules and local shtukas with complex multiplication. J. Inst. Math. Jussieu, 19(1):175–208, January 2020. doi:10.1017/s1474748017000494.

$\bullet $ Urs Hartl. Isogenies of abelian Anderson A-modules and A-motives. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 19(4):1429–1470, December 2019. doi:10.2422/2036-2145.201612_003.

$\bullet $ Esmail Arasteh Rad and Urs Hartl. Uniformizing the moduli stacks of global G-shtukas. International Mathematics Research Notices, November 2019. doi:10.1093/imrn/rnz223.

$\bullet $ Urs Hartl and Rajneesh Kumar Singh. Local shtukas and divisible local Anderson modules. Canadian Journal of Mathematics, 71(5):1163–1207, March 2019. doi:10.4153/cjm-2018-016-2.

$\bullet $ Urs Hartl and Ambrus Pál. Crystalline Chebotarëv density theorems. arXiv e-prints, November 2018. arXiv:1811.07084.

$\bullet $ Urs Hartl and Eva Viehmann. Foliations in deformation spaces of local $G$-shtukas. Adv. Math., 229(1):54–78, January 2012. doi:10.1016/j.aim.2011.08.011.

$\bullet $ Urs Hartl and Eva Viehmann. The Newton stratification on deformations of local $G$-shtukas. J. Reine Angew. Math., 2011(656):87–129, July 2011. doi:10.1515/crelle.2011.044.