Cusps of Picard modular surfaces

Abstract

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N -cusped surface. We also discuss a higher-rank analogue.

On the geometry of a Picard modular group

“…The converse direction says that the Picard group Γ = U(J, O 7 ) acts transitively on K-rational null vectors; this is can be reformulation as the statement that the quotient X/Γ has exactly one cusp, which is a consequence of the fact that K has class number 1 (see [14] or section 4.1 in [12]). The theoretical result in Proposition 3.1 is unfortunately difficult to make effective, as discussed in [8].…”

Degeneration of Hodge structures over Picard modular surfaces

“…By Lemma 3.11, this is a line bundle, and as X 2 D is contractible, the line bundle is trivial. The number N in the previous proposition is computed in several cases in [11]. Note also that the union of the N points (or the union of the N curves) is actually defined over the imaginary quadratic field E, but a priori each point is not.…”

A minimal volume arithmetic cusped complex hyperbolic orbifold

“…The volume of complex hyperbolic 2-orbifold H 2 C /G 2 was calculated firstly by John Parker [10] as π/27. Note that Stover [16] states that there are exactly two noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume π/27, whose fundamental groups are PU(2, 1; Z[ω]) and its sister G 2 . Observe, however, that [10] (1) Vertices:…”

“…Following [10] we will call it G 2 . Recently, Stover [16] has studied volumes of Picard modular surfaces. One of his main results is that there are exactly two of these with minimal volume, namely the Eisenstein-Picard modular surface and its sister, which corresponds to the group G 2 which we study here.…”