Arithmetic of a fake projective plane and related elliptic surfaces
نویسنده
چکیده
The purpose of the present paper is to explain the fake projective plane constructed by J. H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J. H. Keum as well as the one constructed by M. N. Ishida in terms of p-adic uniformization.
منابع مشابه
Fake Projective Planes
1.1. A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. Such a surface is known to be projective algebraic and it is the quotient of the (open) unit ball B in C (B is the symmetric space of PU(2, 1)) by a torsion-free cocompact discrete subgroup of PU(2, 1) whose Euler-Poincaré cha...
متن کاملArithmetic Structure of Cmsz Fake Projective Planes
In [CMSZ2], Cartwright, Mantero, Steger, and Zappa discovered a unitary group in three variables with respect to the quadratic extension Q( √ −15)/Q whose integral model over the integer ring with the prime 2 inverted gives rise to a diadic discrete group acting transitively on vertices of Bruhat-Tits building over Q2. Inside the integral model are three subgroups to which the restricted action...
متن کاملProperty (FA) and Lattices in su(2, 1)
In this paper we consider Property (FA) for lattices in SU(2, 1). First, we prove that SU(2, 1;O3) has Property (FA). We then prove that the arithmetic lattices in SU(2, 1) of second type arising from congruence subgroups studied by Rapoport–Zink and Rogawski cannot split as a nontrivial free product with amalgamation; one such example is Mumford’s fake projective plane. In fact, we prove that ...
متن کاملProjective Surfaces with Many Nodes
We prove that a smooth projective complex surface X, not necessarily minimal, contains h(X) − 1 disjoint (−2)-curves if and only if X is isomorphic to a relatively minimal ruled rational surface F2 or P or a fake projective plane. We also describe smooth projective complex surfaces X with h(X) − 2 disjoint (−2)-curves.
متن کاملOn Abelian Surfaces with Potential Quaternionic Multiplication
An abelian surface A over a field K has potential quaternionic multiplication if the ring End K̄ (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is...
متن کامل