The Decomposition of Primes in Torsion Point Fields Book

1	Introduction	1
2	Decomposition Laws	5
2.1	Foundations of Prime Ideal Decomposition	5
2.2	Decomposition in Abelian Extensions	11
2.3	Density Statements	22
3	Elliptic Curves	25
3.1	Defining Equations	25
3.2	Addition on Elliptic Curves	26
3.3	Division Polynomials	28
3.4	Torsion Points	30
3.5	l-adic Representations	32
3.6	Reduction Modulo [p] and L-Series	35
4	Elliptic Modular Curves	41
4.1	Modular Curves	41
4.2	Modular Forms	45
4.3	Hecke Operators	48
4.4	The Eichler-Shimura Formula	54
5	Torsion Point Fields	59
5.1	The Groups GL(2,n)	59
5.2	Subfields of k(E[subscript n])	66
5.3	Defining Polynomials of Subfields of k(E[subscript n])	73
5.4	Decomposition in 2- and 3-Torsion Point Fields	76
5.5	Decomposition in 4-Torsion Point Fields	80
5.6	Decomposition in p[superscript m]-Torsion Point Fields	85
6	Invariants and Resolvent Polynomials	87
6.1	Foundations of Galois Theory	87
6.2	Procedure for the Description of Subfields	90
6.3	Invariant Algebras	91
6.4	Application to 4-Torsion Point Fields	100
App. A	Invariants of Elliptic Modular Curves	109
App. B	L-series of Coefficients a[subscript p]	111
App. C	Fully Decomposed Prime Numbers	119
App. D	Resolvent Polynomials	127
App. E	Free Resolution of the Invariant Algebra	131
	References	134
	Symbols	137
	Index	141