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].
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