Diophantine geometry

By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations.[3] The Hilbert–Hurwitz result from 1890 reducing the Diophantine geometry of curves of genus 0 to degrees 1 and 2 (conic sections) occurs in Chapter 17, as does Mordell's conjecture.In a hostile review of Lang's book, Mordell wrote: In recent times, powerful new geometric ideas and methods have been developed by means of which important new arithmetical theorems and related results have been found and proved and some of these are not easily proved otherwise.Despite a bad press initially, Lang's conception has been sufficiently widely accepted for a 2006 tribute to call the book "visionary".[6] Paul Vojta wrote: A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions, quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number.
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