Abstract: A basic question in Number theory, going back to Diophantos, is
to understand the rational solutions of systems of polynomial equations
with integer coefficients. This is recast usually in terms of rational
points on the algebraic variety V defined by the polynomial system. A very
intriguing principle is that the topology and geometry of the complex
points of V have a strong bearing on the set V(Q) of rational points. For
example, when V is a smooth projective curve, i.e., when V(C) is a compact
Riemann surface, the genus g of V has the following impact: When g=0 there
are many rational points or none, while for g=1, the cardinality of V(Q)
can be finite or infinite. A striking theorm of Faltings proved that V(Q)
is always finite when g is >1; in this case, X(C) is uniformized by the
hyperbolic plane. For algebraic surfaces V, a beautiful conjecture of Lang
asserts such a finiteness result when it is hyperbolic, in the sense that
there is no non-constant map from C to V(C). We will discuss this
conjecture and discuss what happens in the special case when V(C) is a
Picard modular surface X, which arises classically as a
(compactification of) a quotient of the unit ball in C^2 by an arithmetic
lattice Gamma in SU(2,1).