E. Hironaka — FSU

The most familiar form of Lang's conjecture states that a
complex projective variety *V* that contains no rational or
abelian subvarieties has only a finite number of the rational
points. The result for curves is the well-known Mordell
conjecture proved by Faltings ('83). Since then a new conjecture
(Caporaso, Harris, Mazur '97) has intrigued number theorists and
algebraic geometers:

Can we uniformly bound the number of rational solutions in terms of the genus?

Our talk concerns a lesser known variant of Lang's conjecture.
Assume *V* is a subset of the n-dimensional torus
*(S ^{1})^{n}* defined by trigonometric
equations (algebraic combinations of sine, cosine, etc), and
consider solutions which are rational multiples of π. If