Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

Product description Review "Zannier's book is well written and a pleasure to read. . . . [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject. . . . [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry."---Joseph H. Silverman, Bulletin of the AMS"This book is indeed a great source of knowledge and inspiration for everybody interested in the unlikely intersection problems. The author must be commended for doing this job, and doing it so well."---Yuri Bilu, Mathematical Reviews Clippings About the Author Umberto Zannier is professor of mathematics at the Scuola Normale Superiore di Pisa in Pisa, Italy. He is the author of Lecture Notes on Diophantine Analysis and the editor of Diophantine Geometry. Excerpt. © Reprinted by permission. All rights reserved. Some Problems of Unlikely Intersections in Arithmetic and GeometryBy Umberto ZannierPRINCETON UNIVERSITY PRESSCopyright © 2012 Princeton University PressAll right reserved.ISBN: 978-0-691-15371-1 ContentsPreface.......................................................................................................................ixNotation and Conventions......................................................................................................xiIntroduction: An Overview of Some Problems of Unlikely Intersections..........................................................11 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture...................................................152 An Arithmetical Analogue....................................................................................................433 Unlikely Intersections in Elliptic Surfaces and Problems of Masser..........................................................624 About the André-Oort Conjecture........................................................................................96Appendix A Distribution of Rational Points on Subanalytic Surfaces by Umberto Zannier.........................................128Appendix B Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions by David Masser.....................136Appendix C Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof by David Masser.............................138Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence Approach by David Masser.............................140Appendix E A Transcendence Measure for a Quotient of Periods by David Masser..................................................143Appendix F Counting Rational Points on Analytic Curves: A Transcendence Approach by David Masser..............................145Appendix G Mixed Problems: Another Approach by David Masser...................................................................147Bibliography..................................................................................................................149Index.........................................................................................................................159Chapter OneUnlikely Intersections in Multiplicative Groups and the Zilber Conjecture As anticipated in the introduction, in this first chapter we shall describe some results of unlikely intersections in the case of multiplicative algebraic groups (also called "tori") [??]nm, together with a sketch of some of the proofs. (The important analogue for abelian varieties shall be discussed later in Chapter 3 with other methods.) Remark 1.0.1 Algebraic subgroups and cosets. Before going on, it shall be convenient to recall briefly the simple theory giving the structure of algebraic subgroups and cosets of [??]nm. (Simple proofs may be found, e.g., in [BG06], Ch. 3.) Every algebraic subgroup G of [??]nm may be defined by equations xa = 1 (on denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), where the vector a = (a1; ...; an) runs through a lattice Λ = ΛG [subset] Zn; of course it suffices to choose equations corresponding to a basis of Λ. The correspondence G ↔ ΛG is one-to-one. If ΛG has rank r then G has dimension n - r, and is irreducible if and only if ΛG is primitive (i.e., is a factor of Zn). We say that G is proper if G [not equal to]] [??]nm. Any irreducible algebraic subgroup G is also called a "subtorus" and, setting d := dim G, it becomes isomorphic (as an algebraic group) to Gdm, the isomorphism being induced by a suitable monomial change of coordinates xi → xai on [??]nm. Such a G may be also parametrized by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The torsion points in [??]nm are those whose coordinates are roots of unity; they are Zariski-dense in any algebraic subgroup G. Further, any coset of G in [??]nm