Number Theory Seminar: Tom Scanlon

Abstract

The usual Mordell-Lang conjecture, a theorem of Faltings, asserts that if A is an abelian variety over C, \Gamma < A(C) is a finitely generated subgroup, and X \subseteq A is a closed subvariety, then X(C) \cap \Gamma is a finite union of cosets of subgroups of \Gamma.  If one were to ask instead that A be defined over a field K of positive characteristic, then such a conclusion cannot hold in general as if A were defined over a finite field, F: A \to A were the associate Frobenius morphism, X \subseteq A were defined over the same finite field, and P \in Y(K) \cap \Gamma, then { Fn(P): n \in N } \subseteqY(K) \cap \Gamma.  Other anomalous intersections may arise as sums of such orbits. Some years ago, in joint work with Moosa, I showed that these are essentially the only counterexamples to a na&iuml;ve translation of the Mordell-Lang conjecture to semiabelian varieties defined over a finite field.   Our proof which was long but elementary yields bounds which explicitly depend on the characteristic.  In these lectures, I shall explain how to deduce characteristic independent bounds from a differential algebraic argument.