Titlebook: Lectures on the Mordell-Weil Theorem; Jean-Pierre Serre,Martin Brown Book 1989 Springer Fachmedien Wiesbaden 1989 Calc.Extension.Integral.

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,Hilbert’s Irreducibility Theorem,Hilbert’s irreducibility theorem has quite a large number of proofs, based on different principles; some give precise estimates for the number of integers that one seeks. There are several applications: the construction of elliptic curves over . having rank ≥ 9 and the construction of extensions of . with Galois groups. .., ..,...

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The Large Sieve,Let . be a thin set in Z., and .. the intersection of . with the ball of diameter . centred at the origin. When . = 1 we have seen in §9.7 (as a consequence of Siegel’s theorem) that |..| = ..) when . → ∞. To prove a similar result when . ≥ 2 one needs a different method, based on the large sieve inequality (cf. [Co]).

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978-3-528-08968-9Springer Fachmedien Wiesbaden 1989

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Lectures on the Mordell-Weil Theorem978-3-663-14060-3Series ISSN 0179-2156

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0179-2156 Overview: 978-3-528-08968-9978-3-663-14060-3Series ISSN 0179-2156

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Summary,where they introduced the, nowadays natural, viewpoint of algebraic geometry: if . and . are two birationally equivalent algebraic curves over . in .., their rational points correspond. Hence the importance of birational invariants, in particular the genus. They studied especially the case of genus zero: