By Julian Lowell Coolidge

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On the other hand, we have already observed in Chapter 1 that if g = 0 and deg D = 1 or 2 the corresponding curve, which is either Ga = A1 or Gm = A1 \ {0} has inﬁnitely many integral points (over a suitable ring of S-integers). Hence Siegel’s theorem provides a complete classiﬁcation of the algebraic curves admitting inﬁnitely many integral points. Let us analyze this classiﬁcation in view of the Chevalley-Weil theorem. Recall that given two (smooth, aﬃne) curves C1 , C2 admitting a dominant morphism π : C1 → C2 , if C1 (OS ) is inﬁnite, also C2 (OS ) will be inﬁnite.

Is in a sense a converse to the Chevalley-Weil Theorem discussed in the previous section. T. holds for certain coverings of rational varieties, which do ramify. T. is the non-surjectivity of the set-theoretic map between the sets of rational points. T. provided one admits coverings by possibly reducible varieties. 4. Let κ be a number ﬁeld, X be an algebraic variety deﬁned over κ of dimension d and π : X Ad a dominant rational map, also deﬁned over κ. Suppose that π admits no section θ : Ad X.

Xd , Y ] an irreducible polynomial of degree ≥ 1 in Y . Then for a Zariski-dense set of rational points (a1 , . . , ad ) ∈ κd the specialized polynomial F (a1 , . . , ad , Y ) ∈ κ[Y ] is irreducible. © Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2016 P. 3. Let V be an irreducible aﬃne algebraic variety of dimension d ≥ 1, π : V → Ad a dominant morphism, all deﬁned over a number ﬁeld κ; there exists a Zariski-dense subset of rational points (a1 , . . , ad ) ∈ Ad (κ) = κd such that each of their ﬁbre π −1 (a1 , .

### A History of Geometrical Methods by Julian Lowell Coolidge

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