Some Problems of Unlikely Intersections in Arithmetic and Geometry (Annals of Mathematics Studies) - Softcover

Zannier, Umberto

Softcover

ISBN 10: 069115371X ISBN 13: 9780691153711

Verlag: Princeton University Press, 2012

Inhaltsangabe

This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010.

Über diesen Titel

Über die Autorin bzw. den Autor

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.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Some Problems of Unlikely Intersections in Arithmetic and Geometry

By Umberto Zannier

PRINCETON UNIVERSITY PRESS

Copyright © 2012 Princeton University Press
All right reserved.
ISBN: 978-0-691-15371-1

Contents

Preface.......................................................................................................................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.........................................................................................................................159

Chapter One

Unlikely 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") [??]ⁿ_m, 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 [??]ⁿ_m. (Simple proofs may be found, e.g., in [BG06], Ch. 3.)

Every algebraic subgroup G of [??]ⁿ_m may be defined by equations x^a = 1 (on denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), where the vector a = (a₁; ...; a_n) runs through a lattice Λ = Λ_G [subset] Zⁿ; of course it suffices to choose equations corresponding to a basis of Λ. The correspondence G ↔ Λ_G is one-to-one. If Λ_G has rank r then G has dimension n - r, and is irreducible if and only if Λ_G is primitive (i.e., is a factor of Zⁿ). We say that G is proper if G [not equal to]] [??]ⁿ_m.

Any irreducible algebraic subgroup G is also called a "subtorus" and, setting d := dim G, it becomes isomorphic (as an algebraic group) to G^d_m, the isomorphism being induced by a suitable monomial change of coordinates x_i → x^ai on [??]ⁿ_m. Such a G may be also parametrized by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The torsion points in [??]ⁿ_m are those whose coordinates are roots of unity; they are Zariski-dense in any algebraic subgroup G. Further, any coset of G in [??]ⁿ_m may be defined by equations x^a = c_a, a [member of] Λ_G, for suitable nonzero constants c_a; this coset is a torsion coset (i.e., t a translate of G by a torsion point) if and only if the c_a are roots of unity (which amounts to the fact that it contains a torsion point). Any torsion coset of an irreducible G is a component of an algebraic subgroup of the same dimension.

As usual, we shall denote by [l] the multiplication-by-l map on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is not too di_cult to prove that if X is a nonempty irreducible subvariety of [??]ⁿ_m such that [l]X [subset] X for an integer l > 1, then X is a torsion coset (and conversely). (See, for instance, [Zan09], Theorem 4.6. A proof may be also obtained by projection, from the simpler case of hypersurfaces, for instance similarly to the argument in [BZ95]; see further Remark 1.1.1 in the next section for some details.)

1.1 Torsion points on subvarieties of [??]ⁿ_m

Let us start with Lang's original problem of torsion points on plane curves in [??]²_m and its generalization to higher dimensions, i.e., describing the torsion points on a subvariety X [subset] [??]ⁿ_m. Before offering a general statement in this direction, let us recall that equations in roots of unity go back to long ago; for example, P. Gordan studied in [Gor77] the equation cos 2πx + cos 2πy + cos 2πz = -1, in rationals x; y; z, with the purpose of classifying finite subgroups of PGL₂(C). Other equations of similar shape arise in the enumeration of polytopes satisfying suitable conditions. See [CJ76] for a brief description of this, and also for a general theory of trigonometric diophantine equations (in the authors' terminology), aimed to reduce every mixed diophantine equation (in angle-variables and usual variables) to a usual one. Such paper was also partly inspired by H.B. Mann's [Man65], classifying solutions in roots of unity to linear equations with rational coe_cients. (We shall see later that these results are relevant also for Lang's issue, and represent one of the possible tools to achieve a complete solution of it.)

Coming back to Lang's formulation, as mentioned in the introduction, work of M. Laurent [Lau84] and independently of Sarnak-Adams [SA94] led to the following general result:

Theorem 1.1. Torsion points theorem. Let Σ be any set of torsion points in [??]ⁿ_m([bar.Q]). The Zariski closure of Σ is a finite union of torsion cosets.

This may be reformulated by saying that

The torsion points in a subvariety X [subset] [??]ⁿ_m all lie and are Zariski-dense in a finite number of torsion cosets contained in X. In particular, if X is...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2012
• Sprache: Englisch
• ISBN 10: 069115371X
• ISBN 13: 9780691153711
• Einband: Taschenbuch
• Anzahl der Seiten: 176
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar