This book explores the theory of elliptic curves, integrating algebra, geometry, analysis, and number theory. Aimed at advanced undergraduates, it offers an accessible introduction to Diophantine equations and arithmetic geometry. Topics include group structures, key theorems, and applications in cryptography and Fermat's Last Theorem.
This book constitutes the thoroughly refereed post-proceedings of the International Conference on Cryptography and Lattices, CaLC 2001, held in Providence, RI, USA in March 2001. The 14 revised full papers presented together with an overview paper were carefully reviewed and selected for inclusion in the book. All current aspects of lattices and lattice reduction in cryptography, both for cryptographic construction and cryptographic analysis, are addressed.
"The Arithmetic of Elliptic Curves" explores advanced topics in elliptic curves, building on foundational theories like the Mordell-Weil theorem. This second volume delves into specialized subjects such as elliptic and modular functions, complex multiplication, elliptic surfaces, Néron models, and Tate's theories, enriching the understanding of this vast field.
Focusing on the arithmetic theory of elliptic curves, this book explores its modern formulation by integrating fundamental concepts from algebraic number theory and algebraic geometry. It highlights the variety of methods employed in the study of elliptic curves, making it a comprehensive resource for understanding this complex area of mathematics.
This advanced graduate-level book introduces diophantine geometry and includes a proof of the Mordell conjecture, appealing to both graduate students and professional mathematicians. Each section is filled with numerous exercises to enhance understanding.