Primes of the form x2 + ny2

Primes of the form x² + ny²

Talk given on Tuesday June 29th 2004 in the Institute for Experimental Mathematics (IEM).
Download: [dvi] [ps] [pdf]

In this talk we study ring class fields of orders in imaginary quadratic fields to determine which primes are of the form x² + ny², where x, y integers, for arbitrary n. We give certain examples how our result works in practise.

This was a part of a seminar on Class Field Theory given on Summer semester 2004 in the University of Duisburg-Essen, Campus Essen.

Organisation: G. Boeckle, I. Bouw, C. Diem, G. Frey.

Full program of the seminar:

20.04.04 1. Two introductionary examples (Organizational) Gebhard Boeckle

27.04.04 2. Global Fields Oscar Ledesma [ps] [pdf]

04.05.04 3. Local Fields Xavier Taixes [dvi] [ps] [pdf]

11.05.04 4. Ideles and ideal class group Guido Blady

18.05.04 5. The Herbrand quotient and the Class Field axiom Peter Beelen

25.05.04 6. The existence theorem and the reciprocity map Manuel Blicke

08.06.04 & 15.06.04 7. The main theorems of Class Field Theory Franziska Bittner

15.06.04 & 22.06.04 8. Class Field Theory and class groups Martin Moeller

29.06.04 9. Primes of the form x² + ny² Marios Magioladitis [dvi] [ps] [pdf]

06.07.04 10. Elliptic functions and complex multiplication Roger Oyono

13.07.04 11. Modular functions and modular polynomials Irene Bouw

20.07.04 12. The first main theorem of complex multiplication Bjoern Buth

27.07.04 13. Weber functions and imaginary quadratic fields with class number 1 Claus Diem

-- 14. The class equation: Formulas and the theorems of Deuring and Gross-Zagier --

References
1. D. Cox, Primes of the form x² + ny², Pure and Appl. Math., Wiley, 1989. This was the book mainly used. The talk presents the main theorems, specially theorems 9.2 and 9.4, stated by Cox in chapter two of his book. Click here for the web page for the book.
2. J. Neukirch, Algebraic Number Theory, Springer Verlag, 1999. English translation of the german book published in 1992.
3. J.S. Milne, Class Field Theory, Lecture notes v3.1, 1997. Available here.

Last update: July 1st, 2004