Primes of the Form X2+ny2 : Fermat, Class Field Theory, and Complex Multiplication.

Cover -- Title Page -- Copyright -- Contents -- Preface to the First Edition -- Preface to the Second Edition -- Notation -- Introduction -- Chapter One: From Fermat to Gauss -- 1. Fermat, Euler and Quadratic Reciprocity -- A. Fermat -- B. Euler -- C. P = x2 + ny2 and Quadratic Reciprocity -- D. Beyond Quadratic Reciprocity -- E. Exercises -- 2. Lagrange, Legendre and Quadratic Forms -- A. Quadratic Forms -- B. P = x2 + ny2 and Quadratic Forms -- C. Elementary Genus Theory -- D. Lagrange and Legendre -- E. Exercises -- 3. Gauss, Composition and Genera -- A. Composition and the Class Group -- B. Genus Theory -- C. P = x2 + ny2 and Euler's Convenient Numbers -- D. Disquisitiones Arithmeticae -- E. Exercises -- 4. Cubic and Biquadratic Reciprocity -- A. Z[w] and Cubic Reciprocity -- B. Z[i] and Biquadratic Reciprocity -- C. Gauss and Higher Reciprocity -- D. Exercises -- Chapter Two: Class Field Theory -- 5. The Hilbert Class Field and P = x2 + ny2 -- A. Number Fields -- B. Quadratic Fields -- C. The Hilbert Class Field -- D. Solution of P = x2 + ny2 for Infinitely Many n -- E. Exercises -- 6. The Hilbert Class Field and Genus Theory -- A. Genus Theory for Field Discriminants -- B. Applications to the Hilbert Class Field -- 7. Orders in Imaginary Quadratic Fields -- A. Orders in Quadratic Fields -- B. Orders and Quadratic Forms -- C. Ideals Prime to the Conductor -- D. The Class Number -- E. Exercises -- 8. Class Field Theory and the Cebotarev Density Theorem -- A. The Theorems of Class Field Theory -- B. The Čebotarev Density Theorem -- C. Norms and Ideles -- D. Exercises -- 9. Ring Class Fields and p = x2 + ny2 -- A. Solution of p = x2 + ny2 for All n -- B. The Ring Class Fields of Z[√-27] and Z[√-64] -- C. Primes Represented by Positive Definite Quadratic Forms -- D. Ring Class Fields and Generalized Dihedral Extensions -- E. Exercises.

Chapter Three: Complex Multiplication -- 10. Elliptic Functions and Complex Multiplication -- A. Elliptic Functions and the Weierstrass r-function -- B. The J-invariant of a Lattice -- C. Complex Multiplication -- D. Exercises -- 11. Modular Functions and Ring Class Fields -- A. The J-function -- B. Modular Functions for Γo(m) -- C. The Modular Equation Φm(x, y) -- D. Complex Multiplication and Ring Class Fields -- E. Exercises -- 12. Modular Functions and Singular J-invariants -- A. The Cube Root of the J-function -- B. The Weber Functions -- C. J-invariants of Orders of Class Number 1 -- D. Weber's Computation of J (√-14) -- E. Imaginary Quadratic Fields of Class Number 1 -- F. Exercises -- 13. The Class Equation -- A. Computing the Class Equation -- B. Computing the Modular Equation -- C. Theorems of Deuring, Gross and Zagier -- D. Exercises -- Chapter Four: Additional Topics -- 14. Elliptic Curves -- A. Elliptic Curves and Weierstrass Equations -- B. Complex Multiplication and Elliptic Curves -- C. Elliptic Curves over Finite Fields -- D. Elliptic Curve Primality Tests -- E. Exercises -- 15. Shimura Reciprocity -- A. Modular Functions and Shimura Reciprocity -- B. Extended Ring Class Fields -- C. Shimura Reciprocity for Extended Ring Class Fields -- D. Shimura Reciprocity for Ring Class Fields -- E. The Idelic Approach -- F. Exercises -- References -- Additional References -- A. References Added to the Text -- B. Further Reading for Chapter One -- C. Further Reading for Chapter Two -- D. Further Reading for Chapter Three -- E. Further Reading for Chapter Four -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.