HBA Lecture Notes in Mathematics IMSc Lecture Notes in Mathematics Pietro Corvaja Integral Points on Algebraic Varieties An Introduction to Diophantine Geometry HBA Lecture Notes in Mathematics IMSc Lecture Notes in Mathematics Series Editor Sanoli Gun, Institute of Mathematical Sciences, Chennai, Tamil Nadu, India Editorial Board R. Balasubramanian, Institute of Mathematical Sciences, Chennai Abhay G. Bhatt, Indian Statistical Institute, New Delhi Yuri F. Bilu, Université Bordeaux I, France Partha Sarathi Chakraborty, Institute of Mathematical Sciences, Chennai Carlo Gasbarri, University of Strasbourg, Germany Anirban Mukhopadhyay, Institute of Mathematical Sciences, Chennai V. Kumar Murty, University of Toronto, Toronto D.S. Nagaraj, Institute of Mathematical Sciences, Chennai Olivier Ramaré, Centre National de la Recherche Scientifique, France Purusottam Rath, Chennai Mathematical Institute, Chennai Parameswaran Sankaran, Institute of Mathematical Sciences, Chennai Kannan Soundararajan, Stanford University, Stanford V.S. Sunder, Institute of Mathematical Sciences, Chennai About the Series The IMSc Lecture Notes in Mathematics series is a subseries of the HBA Lecture Notes in Mathematics series. This subseries publishes high-quality lecture notes of the Institute of Mathematical Sciences, Chennai, India. Undergraduate and graduate students of mathematics, research scholars, and teachers would find this bookseriesuseful.Thevolumesarecarefullywrittenasteachingaidsandhighlight characteristic features of thetheory. The books inthis series are co-published with Hindustan Book Agency, New Delhi, India. More information about this series at http://www.springer.com/series/15465 Pietro Corvaja Integral Points on Algebraic Varieties An Introduction to Diophantine Geometry 123 Pietro Corvaja Dipartimento di Matematica Universitàdegli studidi Udine Udine, Italy Thisworkisaco-publicationwithHindustanBookAgency,NewDelhi,licensedforsalein all countries in electronic form only. Sold and distributed in print across the world by Hindustan Book Agency, P-19 Green Park Extension, New Delhi 110016, India. ISBN: 978-93-80250-83-0 ©HindustanBook Agency 2016. 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Theregisteredcompanyaddressis:152BeachRoad,#22-06/08GatewayEast,Singapore189721,Singapore Contents Introduction.................................................... vii 1 Integral points on algebraic varieties........................ 1 1.1 Introducing the problem................................... 1 1.2 The conjecture of Lang-Vojta .............................. 4 1.3 Behaviour of integral points under morphisms................ 12 2 Diophantine approximation................................. 19 2.1 Diophantine approximation on the line ...................... 19 2.2 Higher dimensional Diophantine approximation .............. 25 2.3 Approximation to higher degree hypersurfaces................ 29 3 The theorems of Thue and Siegel ........................... 31 3.1 Thue’s equation .......................................... 31 3.2 Hyperelliptic curves and sums of two units................... 34 3.3 Siegel’s Theorem on curves ................................ 37 3.4 A Subspace Theorem approach to Siegel’s Theorem........... 40 4 Hilbert Irreducibility Theorem ............................. 45 4.1 Hilbert Irreducibility Theorem ............................. 45 4.2 Universal Hilbert Sequences................................ 49 4.3 Hilbert Irreducibility over algebraic groups................... 52 5 Integral points on surfaces.................................. 55 5.1 The Subspace Theorem approach........................... 57 5.2 Divisibility problems...................................... 63 5.3 Constructing integral points on surfaces ..................... 66 5.4 Higher dimensional results................................. 70 References...................................................... 73 v Introduction ThissurveyisintendedtobeaconcreteintroductiontoDiophantinegeometry; it originates in a three-week course delivered at the Institute for Mathematical Sciences in Chennai during the special year devoted to Number Theory. The leading theme is represented by the distribution of integral points on algebraicvarieties.Roughlyspeaking,aDiophantineequationshouldhaveonly finitelymanysolutionsinintegersunlessthereisageometricreasonexplaining their abundance. By ‘geometric’, we mean some property satisfied by the alge- braic varietyformedbythecomplexsolutionstothegivenequation(orsystem of equations). The celebrated Lang-Vojta’s conjecture formalizes this principle: it gives geometrical condition on an algebraic variety under which the set of integral points should be degenerate, i.e. contained in a finite union of proper closed subvarieties. Most of this text is devoted to explaining in concrete instances somefeaturesofthisconjecture,andtoprovingsomeparticularcases.Anowa- days classical theorem of Faltings and Vojta solved the conjecture for varieties which can be embedded into semi-abelian varieties. It contains e.g. the solu- tion to Mordell’s conjecture on rational points on compact hyperbolic curves, as well as Siegel’s finiteness theorem for integral points on open hyperbolic curves. We have chosen to focus on a recent different method, which relies on theSubspaceTheoreminDiophantineapproximation,andmakesnouseofthe theory of algebraic groups. This work is not meant to supersede any previous standard textbook on Diophantine geometry, such as the classical books by Lang [40], Serre [48], Vo- jta [55] and the more recent ones by Hindry and Silverman [35] and Bombieri and Gubler [8], to which the reader is referred. The main goal of this work is to rapidly introduce the impatient reader to some concrete problems in Dio- phantine geometry, especially those involving integral points, to present some recent results not available in textbooks and to show some new viewpoints on classical material. At some points, we preferred to replace proofs by a detailed analysis of particular cases, referring to the papers quoted in the references for completeproofs.Insomeinstances,wedecidedtoproveageneralresultonlyin vii viii Introduction special cases, thinking that a simpler proof in a particular but significant case can be more illuminating than the more complicated proof of the most general statement. Needless to say, we have omitted many (if not most) central topics in Dio- phantine geometry, such as: local-to-global principles, arithmetic on elliptic curvesandabelianvarieties,asymptoticestimates(asappearinge.g.inManin’s conjecture). Even the investigation of rational points on algebraic varieties has been almost omitted, in favor of the case of integral points. In the first chapter, we introduce the general problem of the distribution of integralandrationalpointsonalgebraicvarieties;Lang-Vojta’sconjecture,the central objective of this work, is formulated and discussed. Also, we provide some useful facts on the behaviour of integral points under morphisms. In the second chapter, we present without proofs the main tools from Dio- phantine approximation theory, and show some relations among them. In par- ticular, we present different formulations of the Subspace Theorem, which will bethemaintoolforprovingthedegeneracyresultsforintegralpointsappearing in the subsequent chapters. The third chapter contains a complete proof of Siegel-Mahler theorem on integralpointsoncurves,aswellasdifferentapproachestoapreviousresultof Thue. The fourth chapter is devoted to the celebrated Hilbert Irreducibility The- orem; we prove a generalized version of Hilbert Irreducibility Theorem by ap- pealing to Siegel’s theorem on curves. The last chapter is devoted to the analysis of integral points on surfaces. It is perhaps the most original part, containing also very recent results not yet published in any textbook. Acknowledgments The author is very grateful to the Institute of Mathematical Sciences in Chen- nai,andespeciallytoProf.Balasubramanianandhiscollaborators,forinviting himontheoccasionoftheNumberTheoryYear2010-11.Duringasubsequent visit to the Institute, the author had the opportunity of lecturing on Hilbert irreducibility theorem, which constitues the topic of the fourth chapter of this text. Discussions with the audience in Chennai were much stimulating, and its encouragement to write up these notes was crucial. TheauthorisalsopleasedtothankMartinoBuchini,Jung-KyuCanci,Um- berto Zannier, Michel Waldschmidt and an anonymous referee, who read pre- vious versions of these notes, detected some inaccuracies and suggested many improvements. About the Author PietroCorvajaisFullProfessorofGeometryattheDipartimentoDiMathematica Einformatica at the Università degli studi di Udine, Italy. His research topics include arithmetic geometry, Diophantine approximation and the theory of tran- scendental numbers. ix Chapter 1 Integral points on algebraic varieties 1.1 Introducing the problem Our main concern will be the investigation of the solutions in in- tegers to systems of algebraic equations. Namely, given polynomials f1(X1,...,XN),...,fk(X1,...,XN) ∈ Z[X1,...,XN], we consider the solu- tions (x1,...,xN)∈ZN to the equations ⎧ ⎪⎨f1(x1,...,xN) = 0 . . . . . . ⎪⎩ . . . fk(x1,...,xN)= 0 The complex solutions to the above system form an affine algebraic variety, defined over the field Q of rational numbers. Hence the problem is rephrased as studying the integral points on algebraic varieties. One can consider the analogous question for rational points and can work also on projective varieties. Given homogeneous polynomials f1(X0,...,XN),...,fk(X0,...,XN) ∈ Z[X0,...,XN], their common zero set in PN is a projective algebraic variety. The solutions (x0,...,xN) ∈ QN+1 to the system fi(x0,...,xN) = 0 (i = 1,...,k) correspond to rational points (x0 :...:xN)∈PN(Q) on such variety. The aim of this survey is to show relations between the geometry of (com- plex) algebraic varieties and the distribution of integral or rational points on it. The following examples show that, to achieve this goal, it is necessary to allow extensions of finite degree of the field of definitions and/or of the ring of integers. (i) Consider the conic C ⊂ P2 of equation (in homogeneous coordinates) X2+ Y2 =3Z2.Itadmitsnorationalpoints,andneverthelessitisisomorphic(as an abstract complex curve) to the projective line P1, whose ration√al points are Zariski-dense. If we work over the number field Q(i), or Q( 3), this © Springer Science+Business Media Singapore 2016 and Hindustan Book Agency 2016 1 P. Corvaja, Integral Points on Algebraic Varieties, HBA Lecture Notes in Mathematics, DOI 10.1007/978-981-10-2648-5_1