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 (S1)n defined by trigonometric equations (algebraic combinations of sine, cosine, etc), and consider solutions which are rational multiples of π. If V contains no translates of subtori by finite order elements, then Lang's conjecture claims that V contains a finite number of rational points. A generalization of this version of Lang's conjecture was proved by Laurent ('82). In this talk, we will describe a powerful yet elementary result of Mann concerning Q-linear combinations of roots of unity, and show how Mann's result implies a uniform boundedness on numbers of rational solutions of V in terms of the degree of the defining polynomials of V.