1999/07/19 by Fumiharu Kato, Kato, Fumiharu
Computer Science · Mathematics · #11G15 #11Q25 #14G35 #14J29 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:11G15 #msc:11Q25 #msc:14G35 #msc:14J29
paper · pdf · doi:10.48550/arxiv.math/9907116
11 pages
arxiv created 1999/07/19 · openalex publication_date 1999/07/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of PEL-type, associated to a certain unitary group, whose connected components are isomorphic to Mumford's fake projective plane. We also give the canonical model, Shimura field and number of connected components of this Shimura surface. As a consequence of the construction we prove that the Mumford's fake projective plane is an arithmetic complex unit-ball quotient, and has the 7-th cyclotomic field as a field of definition. In Tohoku Math. J. 50 (1998), 537-555, Masanori Ishida and the author discussed two other possible fake projective planes. The author has also obtained the corresponding results as above for these fake projective planes and has written them in the above preprint, which is available in the web page: http://www.mpim-bonn.mpg.de/html/preprints/preprints.html