Lectures on Elliptic Curves (London Mathematical Society Student Texts, Vol. 24) 1st Edition by J. W. S. Cassels (PDF)

Description

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the “Riemann hypothesis for function fields”) and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.

User’s Reviews

Editorial Reviews: Review ‘… an excellent introduction … written with humour.’ Monatshefte für Mathematik Book Description A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a beautiful book. In less than 150 pages it conveys so much, from Mordell’s theorem to the Tate-Schafarevich group. Well organized, beautifully written, a gem of a book – a must read, buy it.

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