Arizona Winter School 1998
Buium's Course Abstract

Differential algebras and diophantine geometry

Following classical work of Ritt and Kolchin, in differential algebra it is possible to develop an analogue of algebraic geometry in which algebraic equations are replaced by algebraic differential equations. This “new geometry” (which can be called “differential algebraic geometry”) can then be applied to diophantine questions over function fields, in particular to Lang's conjecture over function fields [1,2,3] and to what one may call the “abc theorem for abelian varieties over functions fields” [4]. On the other hand one can develop an arithmetic analogue of “differential algebraic geometry” which becomes relevant in arithmetic questions like the Manin-Mumford conjecture [5], the “arithmetic analogue” of Manin's theorem of the kernel [6], and modular forms [7].