Part of the Aspects of Mathematics / Aspekte der Mathematik book series (ASMA, volume E 15)
The aim of this course is the study of rational and integral points on algebraic varieties, especially on curves or abelian varieties. Before the end of the last century only special cases had been considered. The first general results are found, around 1890, in the work of Hurwitz and Hilbert [HH] where they introduced the, nowadays natural, viewpoint of algebraic geometry: if X and X′ are two birationally equivalent algebraic curves over Q in P 2, their rational points correspond. Hence the importance of birational invariants, in particular the genus. They studied especially the case of genus zero:
Theorem. A curve of genus zero is isomorphic to a conic. If it has a rational point over Q, then it is isomorphic to P 1, and thus has an infinite number of rational points over Q.
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© Springer Fachmedien Wiesbaden 1989