Arithmetic 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 is simply transitive. Moreover, a slight inspection shows that two of them act freely also on simplicies of the other dimensions. Such a situation evokes that Mumford [Mu79] had also obtained a discrete group with the same properties (but in a different unitary group) to construct an algebraic surface with Pg = q = 0, c 2 1 = 3c2 = 9, and with the ample canonical class, so-called, fake projective planes, which is among the most interesting classes of algebraic surfaces. Like that Mumford’s group produces such a surface through diadic uniformization, these two groups give rise to two fake projective planes. It has been shown in [IK98] that these three fake projective planes are not isomorphic to among others. In [Ka99], in the mean time, it was proved that Mumford’s fake projective plane is a unitary Shimura variety. A clue to this result is that Mumford’s discrete group is, as it is clear from the construction, a congruence subgroup in the integral model of the unitary group, as well as that it is arithmetic. However, it is not clear whether the groups in [CMSZ2] are characterized by congruence conditions or not, because the definition of the groups therein only gives us generators in matrices, although they are certainly arithmetic. This obscurity should be resolved, for it is the key point to settle whether the other two fake projective planes also have such a nice arithmetic structure as Mumford’s one has. This point is exactly what we will discuss in this paper. In the next section we will construct congruence subgroups in the unitary group characterized by conditions in (modulo 3)-reduction, and will prove that they coincide up to scalars with the groups defined in [CMSZ2]. Properly speaking, we give another way of construction of these groups, and hence the main stream of our argument is independent from that of [CMSZ2], although we have been much motivated by it. From this we proceed to construct the Shimura varieties in the following section, which mimics the argument in [Ka99]. Our main theorem (Theorem 3.6) states that, for each of the two groups, there exists a Shimura variety with the reflex field Q( √ −15)