Drinfeld Modules

Mihran Papikian. Drinfeld Modules
Series
Title
Copyright
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