Hodge periods associated to algebraic varieties: what do they know? Do they know things?? Let's find out!

Jack Sempliner, Princeton University
Fine Hall 110

Faltings' stunning simultaneous resolution of the Shafarevich conjecture, the Tate conjecture and semisimplicity conjecture for abelian varieties over number fields, and the Mordell conjecture was and remains one of the landmark achievements of Diophantine geometry and early p-adic Hodge theory. In this talk, I'll discuss a beautiful recent work on reproving and generalizing the results of Faltings using a method which sidesteps the associated deep results on abelian varieties and semisimplicity in his original work. The new input is a clever method that, stated roughly, plays the p-adic Hodge theory of a p-adic model for the variety against the classical Hodge theory of a complex model for the variety to obtain expected results on the distribution of rational points on general type varieties.