*Umberto Zannier and David Masser*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153704
- eISBN:
- 9781400842711
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153704.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter describes some results of unlikely intersections in the case of multiplicative algebraic groups (also called “tori”) 𝔾ⁿᵐ, together with a sketch of some of the proofs.

This chapter describes some results of unlikely intersections in the case of multiplicative algebraic groups (also called “tori”) 𝔾ⁿᵐ, together with a sketch of some of the proofs.

*Umberto Zannier*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153704
- eISBN:
- 9781400842711
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153704.001.0001
- Subject:
- Mathematics, Geometry / Topology

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 ...
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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 the author at the Institute for Advanced Study in Princeton in May 2010.The book consists of four chapters and seven brief appendixes. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this to a relative case of the Manin–Mumford issue. The fourth chapter focuses on the André–Oort conjecture (outlining work by Pila).Less

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 the author at the Institute for Advanced Study in Princeton in May 2010.The book consists of four chapters and seven brief appendixes. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this to a relative case of the Manin–Mumford issue. The fourth chapter focuses on the André–Oort conjecture (outlining work by Pila).

*Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a ...
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This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with The Structure of Spherical Buildings and The Structure of Affine Buildings.Less

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with *The Structure of Spherical Buildings* and *The Structure of Affine Buildings*.