Abelian Varieties and Number Theory.

Format:

Book

Author/Creator:

Jarden, Moshe.

Contributor:

Shaska, Tony.

Series:

Contemporary Mathematics

Contemporary Mathematics ; v.767

Language:

English

Subjects (All):

Abelian varieties.

Number theory.

Geometry, Algebraic.

Physical Description:

1 online resource (218 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry.The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc.The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.

Contents:

Cover

Title page

Contents

Preface

Bibliography

Abelian varieties and finitely generated Galois groups

1. Introduction

2. Probabilistic methods

3. Diophantine geometry

4. Combinatorics

5. Arithmetic

6. Analytic number theory

Acknowledgment

References

Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one

Notation

2. Isotrivial and special curves

3. Rössler's theorem on inseparable points on curves

4. The Mordell-Lang conjecture

5. Proof of Theorem 1.1

6. Corollaries and applications

Acknowledgments

The group of automorphisms of the Heisenberg curve

2. The triangle group approach

3. Restriction and lifting of automorphisms in covers

4. The Fermat curve

5. Automorphism group of the Heisenberg curve

6. Modular automorphisms

7. The curve ₃

Tate-Shafarevich groups of constant elliptic curves and isogeny volcanoes

2. Preliminaries

3. Calculation of the Selmer group

4. Calculation of \Sha

5. Concluding remarks

Endomorphisms of ordinary superelliptic Jacobians

2. Definitions and statements

3. The plan of the proof

4. Endomorphism algebras of abelian varieties

5. Abelian schemes

6. Proof of Theorem 3.1

7. Proof of Theorem 3.2

8. Superelliptic Jacobians

An introduction to Deuring's theory of constant reductions

Introduction

1. The Gauss valuation

2. Regular reduction

3. The inertia theorem

4. Reduction of genus

5. Good reduction

6. Almost all reductions

Purely inseparable points on curves

2. Semistable sheaves

3. Proof of Theorem 1.1

Acknowledgments.

References

Correspondences on curves in positive characteristic

2. Étale morphisms to the affine line

3. The main results and their proofs

4. Possible/impossible generalizations

The inverse Galois problem over ℂ( )

2. Rings of holomorphic functions

3. Patching of Galois groups

Explicit Hilbert's irreducibility theorem in function fields

2. Set-up and preliminaries

3. Separable polynomials

4. Inseparable polynomials

A comparison between obstructions to local-global principles over semi-global fields

1. Completions and henselizations of semi-global fields

2. Local points on -varieties

3. Local-global principles for rational points

The generic monodromy of Drinfeld modular varieties in special characteristic

1. Statement of the main result

2. Formal -modules, -divisible groups and deformations of Drinfeld modules

3. The result of Strauch

4. Proof of Theorem \autoref{Thm:Main}

5. An example

6. An application

Acknowledgements

Symmetric tensor products of octonion algebras

1. The associative case

2. Octonion algebras

3. Algebras of symmetric tensors

4. Central idempotents

5. Central simple algebras

Representations of étale fundamental groups and specialization of algebraic cycles

2. Representation-theoretic results

3. Specialization of representations of the étale fundamental group and injectivity of the restriction morphism on ¹

4. Motivic representations

5. Specialization of the ℓ-adic Abel-Jacobi filtration on Chow groups

6. Applications, examples

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470464233

1470464233

OCLC:

1252642642