This textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.
Author(s): Mihran Papikian
Series: Graduate Texts in Mathematics 296
Publisher: Springer
Year: 2023
Language: English
Pages: 526
Preface
Acknowledgements
Contents
Notation and Conventions
1 Algebraic Preliminaries
1.1 Polynomials
Exercises
1.2 Modules over Polynomial Rings
Exercises
1.3 Algebraic Extensions
Exercises
1.4 Trace and Norm
Exercises
1.5 Inseparable Extensions
Exercises
1.6 Finite Fields
Exercises
1.7 Central Simple Algebras
Exercises
2 Non-Archimedean Fields
2.1 Valuations
Exercises
2.2 Completions
Exercises
2.3 Extensions of Valuations
Exercises
2.4 Hensel's Lemma
Exercises
2.5 Newton Polygon
Exercises
2.6 Ramification and Inertia Group
Exercises
2.7 Power Series
2.7.1 Convergence
2.7.2 Weierstrass Preparation Theorem
2.7.3 Weierstrass Factorization Theorem
2.7.4 Newton Polygon of Power Series
2.7.5 Formal Substitutions
Exercises
2.8 Extensions of Valuations of Global Fields
Exercises
3 Basic Properties of Drinfeld Modules
3.1 Additive Polynomials
Exercises
3.2 Definition of Drinfeld Modules
Exercises
3.3 Morphisms
Exercises
3.4 Module of Morphisms
3.4.1 Anderson Motive of a Drinfeld Module
3.4.2 Embeddings into the Twisted Laurent Series Ring
Exercises
3.5 Torsion Points
Exercises
3.6 Torsion Points in Terms of Anderson Motives
Exercises
3.7 Weil Pairing
3.7.1 Exterior Product of Anderson Motives
3.7.2 Weil Pairing via Explicit Formulas
3.7.3 Adjoint of a Drinfeld Module
Exercises
3.8 Isomorphisms
Exercises
4 Drinfeld Modules over Finite Fields
4.1 Endomorphism Algebras
Exercises
4.2 Characteristic Polynomial of the Frobenius
Exercises
4.3 Isogeny Classes
Exercises
4.4 Supersingular Drinfeld Modules
Exercises
5 Analytic Theory of Drinfeld Modules
5.1 Additive Power Series
Exercises
5.2 Lattices and Drinfeld Modules
Exercises
5.3 Applications of Analytic Uniformization
Exercises
5.4 Carlitz Module and Zeta-Values
Exercises
5.5 Fields Generated by Lattices of Drinfeld Modules
Exercises
6 Drinfeld Modules over Local Fields
6.1 Reductions of Drinfeld Modules
Exercises
6.2 Tate Uniformization
Exercises
6.3 Galois Action on Torsion Points
Exercises
6.4 Rational Torsion Submodule
Exercises
6.5 Formal Drinfeld Modules
Exercises
7 Drinfeld Modules Over Global Fields
7.1 Carlitz Cyclotomic Extensions
Exercises
7.2 Rational Torsion Submodule
Exercises
7.3 Division Fields: Examples
7.3.1 General Linear Group
7.3.2 Special Linear Group
7.3.3 Borel Subgroup
7.3.4 Split Cartan Subgroup
7.3.5 Non-split Cartan Subgroup
7.3.6 Normalizer of a Split Cartan Subgroup
7.3.7 Normalizer of a Non-split Cartan Subgroup
7.3.8 Boston-Ose Theorem
Exercises
7.4 Division Fields: A Reciprocity Theorem
Exercises
7.5 Complex Multiplication
Exercises
7.6 Mordell-Weil Theorem and Class Number Formula
7.6.1 Poonen's Theorem
7.6.2 Taelman's Theorem
7.6.3 Anderson's Theorem
Exercises
A Drinfeld Modules for General Function Rings
Exercises
B Notes on Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
References
Index
