Integral points on algebraic varieties. An introduction to Diophantine geometry (Q2796414)

This remarkable small booklet gives a very readable introduction to some of the most modern results in Diophantine geometry, illustrated by a large number of enlightening examples. Many sketches of proofs are given, explaining in simple terms some of the main ideas of the subject; the choice of the author is to keep the technicalities at a level as low as possible, avoiding full proofs when the details are cumbersome, but providing hints and references. Before studying the original papers, reading these lecture notes will be a tremendous help.NEWLINENEWLINEThe first section introduces the topics and provides the necessary background from algebraic geometry for understanding the statements of the basic conjectures, essentially those due to Lang and Vojta. Partial results in this direction, mainly the ones by Vojta and Faltings, are described together with a number of illustrations by means of concrete explicit examples. Two versions of the Chevalley-Weil Theorem are proved.NEWLINENEWLINEThe second chapter is an introduction to Diophantine approximation, featuring the theorems of Thue, Siegel, Roth and Ridout in one dimension, the subspace theorem of Schmidt and its extension by Schlickewei in higher dimension. The emphasis of these lecture notes is on the subspace theorem and its applications; this is a very powerful tool but the results it produces are not effective.NEWLINENEWLINEThe next chapter deals with the finiteness theorems of Thue and Siegel on Diophantine equations. The interdependence between \(S\)-unit equations in two variables \(u+v=1\) and the hyperelliptic equation \(y^2=f(x)\) is proved. A subspace theorem approach to the Theorem of Siegel, Mahler and Lang on \(S\)-integral points on curves is outlined, as usual with concrete examples.NEWLINENEWLINEAn important application of this theory is related with Hilbert irreducibility Theorem and the notion of universal Hilbert sequences. Without proof, the author also connects this topic with algebraic groups.NEWLINENEWLINEThe last chapter, dealing with integral points on surfaces, is arguably the most original part of the book. It gives many recent results, not available in already published books, including important contributions of the author and U. Zannier.NEWLINENEWLINEThis book is a welcomed addition to the limited set of volumes on this topic which are available so far (by Mordell, Lang, Serre, Vojta, Hindry-Silverman, Bombieri-Gubler and by Zannier). It should be made available to a large audience.