Title: Mordell-Lang and Integral Points on Abelian Varieties in Characteristic p

Abstract: I will discuss prior work on the generalized Mordell-Lang conjecture and on the finiteness of integral points on abelian varieties in characteristic p. The number field analogues of both of these questions are completely answered (thanks to work of several people, especially Faltings), and serve as generalizations of Mordell's Conjecture on finiteness of rational points on curves of genus > 2 and Siegel's Theorem on finiteness of integral points on elliptic curves, respectively. Significant progress has been made on the characteristic p analogues of these questions (again, by several people), but much remains unknown. I will survey this work, and then perhaps speculate about further progress. In particular, I will perhaps try to explain the connection between these two problems which seem to me to be intimately connected in characteristic p (unlike in characteristic 0, as far as I can tell).

Abstract: I will discuss prior work on the generalized Mordell-Lang conjecture and on the finiteness of integral points on abelian varieties in characteristic p. The number field analogues of both of these questions are completely answered (thanks to work of several people, especially Faltings), and serve as generalizations of Mordell's Conjecture on finiteness of rational points on curves of genus > 2 and Siegel's Theorem on finiteness of integral points on elliptic curves, respectively. Significant progress has been made on the characteristic p analogues of these questions (again, by several people), but much remains unknown. I will survey this work, and then perhaps speculate about further progress. In particular, I will perhaps try to explain the connection between these two problems which seem to me to be intimately connected in characteristic p (unlike in characteristic 0, as far as I can tell).

## Date:

Mon, 18/11/2019 - 14:30 to 15:30

## Location:

Ross 70