Drinfeld Modules

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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