Husemoller Elliptic Curves

Erscheinungsjahr 2013
ISBN: 978-1-4757-5119-2
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 350 Seiten, Web PDF

Reihe: Mathematics and Statistics

The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A general outline ofmuch ofthe material can be found in Tate's colloquium lectures reproduced as an article in Inven tiones [1974]. The first part grew out of Tate's 1961 Haverford Philips Lectures as an attempt to write something for publication c10sely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This inc1udes parts of the Introduction and the first six chapters The aim ofthis part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate teturned to Haverford to give again, in revised form, the originallectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan forthe book.

Zielgruppe

Graduate

Autoren/Hrsg.

Husemoller, Dale

Inhaltsverzeichnis

to Rational Points on Plane Curves.- 1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve.- 2 Plane Algebraic Curves.- 3 Elliptic Curves and Their Isomorphisms.- 4 Families of Elliptic Curves and Geometric Properties of Torsion Points.- 5 Reduction mod p and Torsion Points.- 6 Proof of Mordell’s Finite Generation Theorem.- 7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields.- 8 Descent and Galois Cohomology.- 9 Elliptic and Hypergeometric Functions.- 10 Theta Functions.- 11 Modular Functions.- 12 Endomorphisms of Elliptic Curves.- 13 Elliptic Curves over Finite Fields.- 14 Elliptic Curves over Local Fields.- 15 Elliptic Curves over Global Fields and ?-Adic Representations.- 16 L-Function of an Elliptic Curve and Its Analytic Continuation.- 17 Remarks on the Birch and Swinnerton-Dyer Conjecture.- Appendix Guide to the Exercises.