Quasi-Projective and Formal-Analytic Arithmetic Surfaces.

Format:

Book

Author/Creator:

Bost, Jean-Benoît.

Contributor:

Bost, Jean-Benoit

Charles, Francois

Series:

Annals of Mathematics Studies

Annals of Mathematics Studies ; v.224

Language:

English

Subjects (All):

Arakelov theory.

Diophantine analysis.

Physical Description:

1 online resource (265 pages)

Edition:

1st ed.

Place of Publication:

Princeton : Princeton University Press, 2026.

Summary:

A milestone in the geometric understanding of algebraization theorems that also provides an introduction to Arakelov geometry Motivated by questions of transcendental number theory, arithmetic, and Diophantine geometry, this book provides a thorough study of a new kind of mathematical object--formal-analytic arithmetic surfaces.

Contents:

Cover

Contents

Introduction

0.1 Formal-analytic surfaces as analogues of formal surfaces or germs of analytic surfaces along a projective curve

0.2 Formal-analytic surfaces over Spec Z: Definition

0.3 Formal-analytic surfaces over Spec Z: Further definitions and main results

0.4 Contents of the memoir

0.5 Acknowledgments

0.6 Conventions, notation, and general results on arithmetic surfaces

PART I. Projective curves in analytic surfaces, Nori's finiteness theorems, and pseudoconcavity

1. Projective surfaces, pseudoconcave analytic surfaces, and étale fundamental groups

1.1 Degree of morphisms between projective surfaces and the Hodge index inequality

1.2 Applications: Connectedness theorems and Nori's theorem on étale fundamental groups of nodal curves in smooth surfaces

1.3 Analytic maps from analytic thickening of projective curves to algebraic surfaces

1.4 Germs of pseudoconcave analytic surfaces

2. CNB-divisors and fibered analytic surfaces

2.1 Compact, connected, big and nef effective divisors in analytic surfaces

2.2 Application to analytic and algebraic varieties fibered over a projective curve

2.3 Examples and complements-The universal meromorphic map : ⇢ alg

part II. bΔ Green functions on Riemann surfaces and intersection theory on quasi-projective arithmetic surfaces

3. Green functions with ∞ and 21 regularity and arithmetic intersection numbers

3.1 The Arakelov degree of 0-cycles and the height of 1-cycles

3.2 Green functions with ∞ regularity and *-products on Riemann surfaces

3.3 Arakelov divisors on arithmetic surfaces and arithmetic intersection numbers

3.4 Green functions with L21 regularity on Riemann surfaces

3.5 Arakelov intersection theory with L21 Green functions on integral normal arithmetic surfaces

4. Green functions with bΔ regularity.

4.1 The spaces bΔ(M) and Mcp(M)

4.2 Green functions with bΔ regularity and *-products

4.3 Examples

4.4 Functoriality of bΔ Green functions

4.5 Application to intersection theory on arithmetic surfaces

5. The Archimedean overflow Ex ( : ( , ) → )

5.1 The invariant Ex ( , )

5.2 The invariant Ex ( : ( , ) → )

5.3 Green functions for the diagonal of a Riemann surface

5.4 Overflow and Green functions for the diagonal

PART III. Formal-analytic arithmetic surfaces, pseudoconcavity, and finiteness theorems

6. Formal-analytic arithmetic surfaces and arithmetic intersection numbers

6.1 Definitions

6.2 Arakelov divisors and intersection numbers on f.-a. arithmetic surfaces

6.3 Pseudoconcavity and Pseudoconvexity

6.4 The arithmetic surfaces Ṽ(D̄(0

1), )

6.5 Proof of the generic triviality of (ṽ(d̄(0, 1), )) in the pseudoconcave case

7. Maps from formal-analytic arithmetic surfaces to arithmetic schemes

7.1 Morphisms from formal-analytic arithmetic surfaces to - schemes

7.2 Morphisms to arithmetic surfaces, arithmetic intersection numbers, and overflow

7.3 Meromorphic maps from f.-a. arithmetic surfaces to proper arithmetic schemes

8. Pseudoconcave formal-analytic arithmetic surfaces I: Degree bounds, algebraicity, and the field ℳ(Ṽ)

8.1 The invariant ( ). Degree bounds on morphisms between arithmetic surfaces

8.2 Algebraicity of maps from pseudoconcave f.-a. arithmetic surfaces to arithmetic schemes

8.3 The field of meromorphic functions on a pseudoconcave f.-a. arithmetic surface

9. Pseudoconcave formal-analytic arithmetic surfaces II: the algebra (ṽ)

fundamental groups of arithmetic surfaces

9.1 A finiteness rresult for the algebra (ṽ) and a structure theorem for morphisms to affine arithmetic surfaces.

9.2 Arithmetic Lefschetz-Nori theorems on étale fundamental groups

9.3 Applications: Arithmetic surfaces with finite étale fundamental group and integral models of modular curves

Bibliography

Notation index

Index.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Bost, Jean-Benoît Quasi-Projective and Formal-Analytic Arithmetic Surfaces

ISBN:

9780691287898

OCLC:

1601989124