Preface InJuly2013anLMS-EPSRCShortInstructionalCourseon‘O-minimalityand diophantinegeometry’washeldintheSchoolofMathematicsattheUniversity ofManchester.Thisvolumeconsistsoflecturenotesfromthecoursestogether with several other surveys. The motivation behind the short course was to introduce participants to some of the ideas behind Pila’s recent proof of the André-Oort conjecture for products of modular curves. The underlying ideas are similar to an earlier proof by Pila and Zannier of the Manin-Mumford conjecture(whichhasinfactlongbeenatheorem,originallyduetoRaynaud) andcombiningtheresultsofthevariouscontributionshereleadstoaproofof thisconjectureincertaincases.Thebasicstrategyhasthreemainingredients: the Pila-Wilkie theorem, bounds on Galois orbits, and functional transcen- dence results. Each of the topics was the focus of a course. Wilkie discussed o-minimality and the Pila-Wilkie theorem without assuming any background in mathematical logic. (The argument given here is, in fact, slightly different from that given in the original paper, at least in the one-dimensional case.) Habegger’s course focused on the Galois bounds and on the completion of the proof (of certain cases of Manin-Mumford) from the various ingredients. AndPila’slecturescoveredfunctionaltranscendence,alsotouchingonvarious recentrelatedworkbyZilber.Wehavealsoincludedsomefurtherlecturenotes byWilkiecontainingaproofoftheo-minimalityoftheexpansionofthereal field by restricted analytic functions, which is sufficient for the application of Pila-Wilkie to Manin-Mumford. At the short course there were also three guest lectures. Yafaev spoke on very recent breakthroughs on the functional transcendence side in the setting of general Shimura varieties. Masser spoke on some other results (‘relative Manin-Mumford’) that can be obtained by a similar strategy. Jones discussed improvements to the Pila-Wilkie theorem. Unfortunately, Yafaev was unable to contribute to this volume. During the week of the course, tutorials were given by Daw and Orr. For this volume, xi Downloaded from https:/www.cambridge.org/core. Columbia University Libraries, on 17 Jun 2017 at 07:49:32, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316106839.001 xii Preface Orr has written a survey of abelian varieties which contains a proof of the functional transcendence result necessary for the application in Habegger’s course. Daw has contributed an introduction to Shimura varieties which we hope will prove valuable to those who wish to go on to study the general André-Oortconjecture.Finally,wearepleasedtoincludeapaperbyTsimer- man in which he gives a proof of Ax’s theorem on the functional case of Schanuel’sconjectureviao-minimality. WewouldliketothanktheLondonMathematicalSocietyandtheEngineer- ingandSciencesResearchCouncilforfundingthecourse,andtheSchoolof MathematicsattheUniversityofManchesterforhostingthemeeting.Andwe aregratefultothespeakersandtutorsatthemeetingandtothecontributorsto thisvolume. Downloaded from https:/www.cambridge.org/core. Columbia University Libraries, on 17 Jun 2017 at 07:49:32, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316106839.001 1 The Manin-Mumford Conjecture, an elliptic Curve, its Torsion Points & their Galois Orbits P.Habegger Abstract Thisisanextendedwrite-upofmyfivehourlecturecourseinJuly2013 on applications of o-minimality to problems in Diophantine Geometry. The course covered arithmetic properties of torsion points on elliptic curvesandhowtheycombinewiththePila-WilkiePointCountingTheo- remandtheAx-Lindemann-WeierstrassTheoremtoproveaspecialcase oftheManin-MumfordConjecture. 1 Overview Thesenotesareawrite-upofmylecturecoursetitledDiophantineApplications whichwaspartoftheLMS-EPSRCShortInstructionalCourse–O-Minimality andDiophantineGeometryinManchester,July2013.Thepurposeoftheshort coursewastopresentrecentdevelopmentsinvolvingtheinteractionofmethods fromModelTheorywithproblemsinNumberTheory,mostnotablytheAndré- Oort and Manin-Mumford Conjectures, to an audience of students in Model TheoryandNumberTheory. AttheheartofthisconnectionisthepowerfulPila-WilkieCountingTheo- rem[26].Itgivesupperboundsforthenumberofrationalpointsonsetswhich aredefinableinano-minimalstructure. TheManin-MumfordConjectureconcernsthedistributionofpointsoffinite order on an abelian variety with respect to the Zariski topology. We give a rathergeneralversionofthisconjecture,lateronweoftenworkinthesituation O-MinimalityandDiophantineGeometry,ed.G.O.JonesandA.J.Wilkie.Publishedby CambridgeUniversityPress.(cid:2)c CambridgeUniversityPress2015. 1 Downloaded from https:/www.cambridge.org/core. Columbia University Libraries, on 17 Jun 2017 at 07:49:32, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781316106839.002