Arithmetik über endlich erzeugten Körpern

• Applicant: Professor Dr. Moritz Kerz
• Subject Area: Mathematics
• Term: from 2009 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 155362679

• Final Report Year: 2015

Final Report Abstract

The Emmy Noether-group "Arithemtik über endlich erzeugten Körpern" studied the geometry and arithmetic of algebraic varieties over fields of arithmetic interest. Many of the questions are generalizations of classical arithmetic questions for curves over finite fields. For example we proved conjectures of K. Kato generalizing the Hasse principle for the Brauer group to a higher dimensional situation. We also studied various higher dimensional versions of class field theory. V. Drinfeld’s successful application of methods developed by Wiesend, which were improved in the work of our Emmy Noether-group, opened up new perspectives. Now methods used by our Emmy Noether-group could be related to Galois representations and the Langlands program. This led to an intense discussions with H. Esnault and a stimulating correspondence with P. Deligne and V. Drinfeld. One of the main problems suggested by Deligne in the course of this correspondence has been solved in joint work with S. Saito in a special case.

Publications

• Higher class field theory and the connected component, Manuscripta Math. 135 (2011), p. 63– 89
  Moritz Kerz
  
• Ideles in higher dimension, Math. Res. Letters 18, Number 4 (2011), p. 699–713
  Moritz Kerz
  
• A finiteness theorem for Galois representations of function fields over finite fields (after Deligne), Acta Mathematica Vietnamica 37, Number 4 (2012), p. 531–562
  Moritz Kerz, Hélène Esnault
  
• Cohomological Hasse principle and motivic cohomology for arithmetic schemes, Publications Mathématiques de l’IHES, Volume 115, Number 1 (2012), p. 123–183
  Moritz Kerz, Shuji Saito
  
• Delignes kompatible λ-adische Darstellungen, Mitteilungen der DMV 20(1), 25–27 (2012)
  Moritz Kerz
  
• Chow group of 0-cycles with modulus and higher dimensional class field theory (2013)
  Moritz Kerz, Shuji Saito
  (See online at https://doi.org/10.1215/00127094-3644902)
• Cohomological Hasse principle and resolution of quotient singularities, New York Journal of Mathematics (2013), 597–645
  Moritz Kerz, Shuji Saito
  
• Deformation of algebraic cycle classes in characteristic zero, Algebraic Geometry 1(3) (2014), 290–310
  Moritz Kerz, Spencer Bloch und Hélène Esnault
  (See online at https://doi.org/10.14231/AG-2014-015)
• Lefschetz theorem for abelian fundamental group with modulus, Algebra Number Theory 8(3) (2014), 689–701
  Moritz Kerz, Shuji Saito
  (See online at https://dx.doi.org/10.2140/ant.2014.8.689)
• p-adic deformation of algebraic cycle classes, Inventiones math. 195 (2014), 673–722
  Moritz Kerz, Spencer Bloch und Hélène Esnault
  (See online at https://doi.org/10.1007/s00222-013-0461-4)