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The Decomposition of Primes in Selected Torsion Point Fields : Lecture Notes in Computer Science, - Clemens Adelmann

The Decomposition of Primes in Selected Torsion Point Fields

Lecture Notes in Computer Science,

Paperback Published: July 2009
ISBN: 9783540420354
Number Of Pages: 148

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Introduction.- Decomposition laws.- Elliptic curves.- Elliptic modular curves.- Torsion point fields.- Invariants and resolvent polynomials.- Appendix: Invariants of elliptic modular curves; L-series coefficients a p; Fully decomposed prime numbers; Resolvent polynomials; Free resolution of the invariant algebra.

Introductionp. 1
Decomposition Lawsp. 5
Foundations of Prime Ideal Decompositionp. 5
Decomposition in Abelian Extensionsp. 11
Density Statementsp. 22
Elliptic Curvesp. 25
Defining Equationsp. 25
Addition on Elliptic Curvesp. 26
Division Polynomialsp. 28
Torsion Pointsp. 30
l-adic Representationsp. 32
Reduction Modulo p and L-Seriesp. 35
Elliptic Modular Curvesp. 41
Modular Curvesp. 41
Modular Formsp. 45
Hecke Operatorsp. 48
The Eichler-Shimura Formulap. 54
Torsion Point Fieldsp. 59
The Groups GL(2,n)p. 59
Subfields of k(En)p. 66
Defining Polynomials of Subfields of k(En)p. 73
Decomposition in 2- and 3-Torsion Point Fieldsp. 76
Decomposition in 4-Torsion Point Fieldsp. 80
Decomposition in pm-Torsion Point Fieldsp. 85
Invariants and Resolvent Polynomialsp. 87
Foundations of Galois Theoryp. 87
Procedure for the Description of Subfieldsp. 90
Invariant Algebrasp. 91
Application to 4-Torsion Point Fieldsp. 100
Appendicesp. 107
Invariants of Elliptic Modular Curvesp. 109
L-series Coefficients app. 111
Fully Decomposed Prime Numbersp. 119
Resolvent Polynomialsp. 127
Free Resolution of the Invariant Algebrap. 131
Referencesp. 134
Referencesp. 135
Symbolsp. 137
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540420354
ISBN-10: 3540420355
Series: Lecture Notes in Computer Science,
Audience: General
Format: Paperback
Language: English
Number Of Pages: 148
Published: July 2009
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 0.86
Weight (kg): 0.23