  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
