Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication

PREFACE TO THE SECOND EDITION

The philosophy of the second edition is to preserve as much of the original text as possible. The major changes are:

• A new §15 on Shimura reciprocity has been added, based on work of Peter Stevenhagen and Alice Gee [A10, A11, A23] and Bumkyo Cho [A6].
• The fifteen sections are now organized into four chapters:
  • – The original §§1–13, which present a complete solution of p = x² + ny², now constitute Chapters One, Two and Three.
  • – The new Chapter Four consists of the original §14 (on elliptic curves) and the new §15 (on Shimura reciprocity).
• An “Additional References” section has been added to supplement the original references [1]–[112]. This section is divided into five parts:
  • – The first part consists of references [A1]–[A24] that are cited in the text. These references (by no means complete) provide updates to the book.
  • – The remaining four parts give some references (also not complete) for further reading that are relevant to the topics covered in Chapters One, Two, Three and Four.
• The expanded Notation section now includes all notation used in the book. Specialized notation is listed according to the page where it first appears.

The other changes to the text are very minor, mostly to enhance clarity, improve formatting, and simplify some of the proofs. One exception is the addition of new exercises: at the end of §12, Exercise 12.31 shows how Ramanujan could have derived Weber's formula for (thanks to Heng Huat Chan), and at the end of §14, ...