By J. W. S. Cassels

The learn of certain instances of elliptic curves is going again to Diophantos and Fermat, and this present day it truly is nonetheless one of many liveliest facilities of study in quantity idea. This publication, addressed to starting graduate scholars, introduces easy conception from a modern perspective yet with an eye fixed to the historic historical past. The valuable element offers with curves over the rationals: the Mordell-Wei finite foundation theorem, issues of finite order (Nagell-Lutz), and so forth. The therapy is dependent via the local-global perspective and culminates within the description of the Tate-Shafarevich crew because the obstruction to a Hasse precept. In an introductory part the Hasse precept for conics is mentioned. The booklet closes with sections at the thought over finite fields (the "Riemann speculation for functionality fields") and lately built makes use of of elliptic curves for factoring huge integers. necessities are saved to a minimal; an acquaintance with the basics of Galois thought is thought, yet no wisdom both of algebraic quantity idea or algebraic geometry is required. The p-adic numbers are brought from scratch. Many examples and workouts are integrated for the reader, and people new to elliptic curves, whether or not they are graduate scholars or experts from different fields, will locate this a useful creation.

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