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Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves

Part of the Progress in Mathematics Book Series (volume 334)

DOI: 10.1007/978-3-030-44329-0

ISBN: 978-3-030-44328-3

Publisher: Springer International Publishing - Springer Journals

Aug 22, 2020

1 - 10 of 10 Chapters

LitStream

Chapter 1Hermitian Vector Bundles over Arithmetic Curves

Bost, Jean-Benoît

Pages: 11 –23

[In this chapter, we gather some basic results concerning Hermitian vector bundles over arithmetic curves, namely affine schemes defined by rings of integers of number fields.]
LitStream

Chapter 2θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves

Bost, Jean-Benoît

Pages: 25 –48

[This chapter is devoted to the definitions and to some basic properties of the θ- invariants of (finite-rank) Hermitian vector bundles over arithmetic curves. Their extensions to infinite-dimensional Hermitian vector bundles established in the latter chapters of this monograph, as well as their...
LitStream

Chapter 3Geometry of Numbers and θ-Invariants

Bost, Jean-Benoît

Pages: 49 –76

[The results of the present chapter will not be used in the next chapters of this monograph.]
LitStream

Chapter 4Countably Generated Projective Modules and Linearly Compact Tate Spaces over Dedekind Rings

Bost, Jean-Benoît

Pages: 77 –106

[In this chapter, we denote by A a Dedekind ring (in the sense of Bourbaki, [21, Vii.2.1]; in other words, A is either a field or a Noetherian integrally closed domain of dimension 1), and we introduce the categories of (topological) modules CPA and CTCA attached to A.]
LitStream

Chapter 5Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves

Bost, Jean-Benoît

Pages: 107 –135

[In this chapter, we introduce several categories of infinite-dimensional Hermitian vector bundles over an arithmetic curve SpecOK defined by the ring of integers OK of a number field K.]
LitStream

Chapter 6θ-Invariants of Infinite-Dimensional Hermitian Vector Bundles: Definitions and First Properties

Bost, Jean-Benoît

Pages: 137 –154

[The last paragraphs of this chapter are again dedicated to concrete examples involving these invariants.]
LitStream

Chapter 7Summable Projective Systems of Hermitian Vector Bundles and Finiteness of θ-Invariants

Bost, Jean-Benoît

Pages: 155 –175

[We still denote by K a number field, by OK its ring of integers, and by π the morphism of schemes form SpecOK to Spec Z.]
LitStream

Chapter 8Exact Sequences of Infinite-Dim. Hermitian Vector Bundles and Subadditivity of Their θ-Invariants

Bost, Jean-Benoît

Pages: 177 –217

[In this chapter, we investigate the properties of short exact sequences and admissible short exact sequences of pro- and ind-Hermitian vector bundles.]
LitStream

Chapter 9Infinite-Dimensional Vector Bundles over Smooth Projective Curves

Bost, Jean-Benoît

Pages: 219 –236

[In this chapter, we discuss various results concerning infinite-rank vector bundles over C, which constitute “geometric” counterparts of the main results of this monograph devoted to infinite-rank vector bundles over the “compactified arithmetic curve” attached to a number field K.]
LitStream

Chapter 10Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization

Bost, Jean-Benoît

Pages: 237 –303

[The formalism presented in this monograph has been developed with a view toward applications to Diophantine geometry and transcendence theory. In Diophantine geometry, one of ten encounters situations that combine formal geometry over the integers and complex analytic geometry.]

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