Profile
Profile

Prof. Dr. Michael Rapoport

E-mail: rapoport(at)math.uni-bonn.de
Phone: +49 228 73 7793
Homepage: http://www.math.uni-bonn.de/ag/alggeom/rapoport
Room: 1.031
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Former Research Area D (Leader)
Research Area DE
Former Research Area E
Former Research Area F
Date of birth: 02.Oct 1948
Mathscinet-Number: 144965

Academic Career

1976

These d'Etat, University of Paris-Sud, France

1976 - 1980

Assistant Professor, HU Berlin

1982 - 1986

Professor (C3), University of Heidelberg

1986 - 1989

Professor (C3), University of Bonn

1989 - 1996

Professor (C4), University of Wuppertal

1996 - 2003

Professor (C4), University of Cologne

2003 - 2012

Professor (C4), University of Bonn

Since 2012

Professor (W3), University of Bonn

Research Profile

My aim is to use algebraic geometry to establish higher reciprocity laws, which serve as a bridge between the field of arithmetic and the theory of automorphic forms. I am interested in the theory of Shimura varieties and their local variants, in particular Rapoport-Zink spaces. I am particularly fascinated by the possibility of constructing through them interesting Galois representations, of algebraic cycles on them and of deformations. My current research focusses on the following topics. I am interested in constructing arithmetic models of Shimura varieties through the correct formulation of a moduli problem whose solution gives such a model. For Shimura varieties attached to unitary groups this naturally leads to the problem of defining a crystalline discriminant of polarized p-divisible groups. So far, I have succeeded in this in the case of even height (joint work with S. Kudla). I am also interested in understanding the structure of natural stratifications of the reduction modulo p of integral models of Shimura varieties (the Newton stratification, the Ekedahl-Oort stratification, the Kottwitz-Rapoport stratification). In joint work with X. He, I have given an axiomatic framework for studying these questions. I am also interested in the Arithmetic Fundamental Lemma conjecture of W. Zhang. This conjecture predicts the intersection number of two arithmetic cycles of half-dimension on a Rapoport-Zink space. In joint work with U. Terstiege and W. Zhang I solved this conjecture in the minuscule case. Recently, in joint work with B. Smithling and W. Zhang, I extended the conjecture to ramified cases and solved it in a number of cases of small dimension.

In my future research I want to understand better the dependence of Rapoport-Zink spaces on the underlying group theory. My recent theorem with T. Zink on the Drinfeld moduli problem of p-divisible groups deals in a special case related to {\rm GL}_2 with the effect of changing the relevant cocharacter by a central cocharacter. In future work with S. Kudla and T. Zink, I want to treat the analogous problem for the group {\rm GU}_2 and apply this to the p-adic uniformization of certain Shimura curves. I want to understand the influence of exceptional isomorphisms between orthogonal groups and unitary groups on their associated Rapoport-Zink spaces (which exist, due to recent work of B. Howard/G. Pappas and of W. Kim). Another topic I want to explore is the relation between the conjectures of S. Kudla and myself on arithmetic divisors on Rapoport-Zink spaces and the conjectures of W. Zhang on arithmetic cycles of half-dimension. I want to understand the impact of Scholze's ideas and methods on the study of integral models of Shimura varieties. I also want to return to the theory of \Phi-modules which I started with G. Pappas and make progress after the recent contributions of M. Emerton/T. Gee.

Contribution to Research Areas

Former Research Area D
My contributions in this RA can be grouped into two main directions. First, there has been substantial progress in my joint project with S. Kudla in giving arithmetic interpretations of the Fourier coefficients of Eisenstein series. The case of the modular curve with no ramifications has been completely solved and is documented in the monograph [1] with S. Kudla and T. Yang. The basis for our results are the theory of Gross/Keating. In my seminar we elaborated an up-to-date account of this theory. Recently Kudla and I started to develop an analogous theory for the Shimura varieties associated to unitary groups [2].
Second, in the direction of Shimura varieties, I published a monograph [3] on the theory of period spaces with Dat and Orlik. It leads from the elementary theory of filtered vector spaces to the first published account of the determination of the Euler-Poincaré characteristic of p-adic period domains.
Former Research Area E
I developed further, in collaboration and close contact with G. Pappas and B. Smithling, the theory of local models of Shimura varieties. The paper [4] treats the case of ramified unitary groups, and states the highly relevant “coherence conjecture” which has been proved recently in spectacular work of X. Zhu. Smithling has proved the topological flatness conjecture of loc.cit.
Another topic developed in [5] is the theory of <br>mathcal G-bundles on algebraic curves. Some conjectures of loc. cit. have been proved by J. Heinloth.
Former Research Area F
I developed with G. Pappas [6] a theory of algebraic loop groups, generalizing the theory of Faltings to the non-constant, or twisted, case. In [7], Pappas and I gave a new framework for moduli spaces of Kisin modules it coefficient spaces, with possible applications to deformation problems of Galois representations.

Selected Publications

[1] Stephen S. Kudla, Michael Rapoport, Tonghai Yang
Modular forms and special cycles on Shimura curves
of Annals of Mathematics Studies : x+373
Publisher: Princeton University Press, Princeton, NJ
2006
ISBN: 978-0-691-12551-0; 0-691-12551-1
DOI: 10.1515/9781400837168
[2] Stephen Kudla, Michael Rapoport
Special cycles on unitary Shimura varieties I. Unramified local theory
Invent. Math. , 184: (3): 629--682
2011
DOI: 10.1007/s00222-010-0298-z
[3] Jean-François Dat, Sascha Orlik, Michael Rapoport
Period domains over finite and p-adic fields
of Cambridge Tracts in Mathematics : xxii+372
Publisher: Cambridge University Press, Cambridge
2010
ISBN: 978-0-521-19769-4
DOI: 10.1017/CBO9780511762482
[4] G. Pappas, M. Rapoport
Local models in the ramified case. III. Unitary groups
J. Inst. Math. Jussieu , 8: (3): 507--564
2009
DOI: 10.1017/S1474748009000139
[5] Georgios Pappas, Michael Rapoport
Some questions about G-bundles on curves
Algebraic and arithmetic structures of moduli spaces (Sapporo 2007)
of Adv. Stud. Pure Math. : 159--171
Publisher: Math. Soc. Japan, Tokyo
2010
[6] G. Pappas, M. Rapoport
Twisted loop groups and their affine flag varieties
With an appendix by T. Haines and Rapoport
Adv. Math. , 219: (1): 118--198
2008
DOI: 10.1016/j.aim.2008.04.006
[7] G. Pappas, M. Rapoport
Φ-modules and coefficient spaces
Mosc. Math. J. , 9: (3): 625--663, back matter
2009
[8] Michael Rapoport, Ulrich Terstiege, Wei Zhang
On the arithmetic fundamental lemma in the minuscule case
Compos. Math. , 149: (10): 1631--1666
2013
DOI: 10.1112/S0010437X13007239
[9] M. Rapoport, Th. Zink
Period spaces for p-divisible groups
of Annals of Mathematics Studies : xxii+324
Publisher: Princeton University Press, Princeton, NJ
1996
ISBN: 0-691-02782-X; 0-691-02781-1
DOI: 10.1515/9781400882601
[10] G. Laumon, M. Rapoport, U. Stuhler
{D}-elliptic sheaves and the Langlands correspondence
Invent. Math. , 113: (2): 217--338
1993
DOI: 10.1007/BF01244308
[13] Jr., Dan Burns, Michael Rapoport
On the Torelli problem for kählerian K-3 surfaces
Ann. Sci. École Norm. Sup. (4) , 8: (2): 235--273
1975

Publication List

Editorships

• Duke Math. J. (Associate Editor, 1995 - 2000)
• Ergebnisse der Mathematik, Springer Verlag (Editor, 1998 - 2003)
• International Mathematics Research Notices (Editor, since 2003)
• Algebra and Number Theory (Editor, since 2015)
• Epiga (Editor, since 2016)

Awards

1991

Akademiestipendium of the VW-foundation

1992

Leibniz Prize

2000

Prix Gay-Lussac/Humboldt of the French Ministry of Education

2003

Member of the Leopoldina (German National Academy of Sciences)

2011

Heinz Hopf Prize

2012

Teaching prize of the University of Bonn

2013

Staudt Prize

2013

Member of the Academia Europaea

Selected Invited Lectures

1993

Distinguished Ordway visitor in Mathematics, University of Minnesota, Minneapolis, MN, USA

1994

Invited speaker, International Congress of Mathematicians, Zürich, Switzerland

1995

Invited plenary speaker at Annual Conference of DMV, Ulm

2001

Distinguished Ordway visitor in Mathematics, University of Minnesota, Minneapolis, MN, USA

2011

Heinz Hopf Lectures, ETH Zürich, Switzerland

Habilitations

Torsten Wedhorn (2005)

Ulrich Görtz (2006)

Sascha Orlik (2007)

Eva Viehmann (2011)

Eugen Hellmann (2016)

Selected PhD students

Torsten Wedhorn (1998): “Ordinariness in Good Reductions of Schimura Varieties of PEL-Type”,
now Professor, TU Darmstadt

Sascha Orlik (1999): “Kohomologie von Periodenbereichen”,
now Professor, University of Wuppertal

Ulrich Görtz (2000): “On the flatness of certain Shimura varieties of PEL-Type”,
now Professor, University of Duisburg-Essen

Eva Viehmann (2005): “On affine Deligne-Lusztig varieties for GL_n”,
now Professor, TU Munich

Ulrich Terstiege (2009): “Intersections of Arithmetic Hirzebruch-Zagier Cycles”,
now Scientific Staff, RWTH Aachen

Eugen Hellmann (2011): “On arithmetic families of filtered f-modules and crystalline representations”,
now Professor, University of Münster

Peter Scholze (2012): “Perfectoid Spaces”,
now Professor (Hausdorff Chair), University of Bonn

Timo Richarz (2014): “On geometric Satake equivalences”,
now Scientific Staff, University of Duisburg-Essen

Daniel Kirch (2015): “Construction of a Rapoport-Zink space for split {\rm GU}(1, 1) in the ramified 2-adic case”,
now DFG-fellow, University of Paris VI, France

Andreas Mihatsch (2016): “Relative RZ-spaces and the Arithmetic Fundamental Lemma”

Supervised Theses

  • Master theses: 3
  • Diplom theses: 19
  • PhD theses: 14
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