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- W1001070037 abstract "As we saw already in Chapter I, there is a geometric analogue to the theorem that a curve over a number field has only a finite number of rational points. We consider a projective non-singular surface X, and a morphism onto a curve $$pi :X to Y$$ defined over an algebraically closed field of characteristic 0, so the generic fiber is a non-singular curve over the function field of Y. Rational points of this curve over finite extensions of k(Y) amount to sections of this fibering over finite coverings of Y. The height of these sections has a geometric definition, and we want to give bounds for those heights. There have been several methods in the function field case to obtain such bounds, which are of independent interest since they exhibit the diophantine geometry in a context independent of more refined arithmetic invariants found in the number field case. The purpose of this chapter is to describe some of these methods. The original proof of finiteness (without explicit bounds on heights, conjectured in [La 60a]) is due to Manin [Man 63] and the ideas of this proof will be given in §4.KeywordsLine SheafFunction FieldAbelian VarietyFinite ExtensionGeneric FiberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
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- W1001070037 date "1991-01-01" @default.
- W1001070037 modified "2023-10-14" @default.
- W1001070037 title "The Geometric Case of Mordell’s Conjecture" @default.
- W1001070037 doi "https://doi.org/10.1007/978-3-642-58227-1_6" @default.
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