Evelyn nelson lecture 2012-2013
Let x_1,x_2,... be a sequence of n-tuples of roots of unity and suppose X is the vanishing locus of a polynomial in n variables and rational coefficients. For a prime number p, Tate and Voloch proved that if the p-adic distance between x_k and X tends to 0 then all but finitely many sequence members lie on X. Buium and Scanlon later generalized this result. The distribution of those x_k that lie on X is governed by the classical (and resolved) Manin-Mumford Conjecture. I will explain this conjecture and also the related (and open) Andre-Oort Conjecture towards which there has been much progress in the last years due to work of Klingler, Pila, Ullmo, Yafaev, Zannier and others. I will then present a variant of Tate and Voloch's discreteness result which was motivated by the Andre-Oort Conjecture.