Treffer: DIOPHANTINE GEOMETRY ON CURVES OVER FUNCTION FIELDS

In these notes we give a reasonably self contained proof of three of the main theorems of the diophantine geometry of curves over function fields of characteristic zero. Let F be a function field of dimension one over the field of the complex numbers C i. e. a field of transcendence degree one over C. Let X F be a smooth projective curve over F. We prove that:-if the genus of X F is zero then it is isomorphic, over F , to the projective line P 1 .-If the genus of X F is one and X F is not isomorphic (over the algebraic closure of F) to a curve defined over C, then the set of F-rational points of X F has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil).-If the genus of X F is strictly bigger then one and X F is not isomorphic (over the algebraic closure of F) to a curve defined over C, then the set of F-rational points of X F is finite (former Mordell conjecture). The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [11] or [8]).