The book "The Arithmetic of Elliptic Curves" by Joseph H. Silverman is a comprehensive introduction to the arithmetic properties of elliptic curves, a field that has seen significant progress over the past two decades. The second edition, updated and expanded, aims to remain a useful resource for both students and researchers. Key updates include new sections on the variation of the trace of Frobenius and additional exercises. The book covers fundamental topics such as algebraic varieties, algebraic curves, the geometry of elliptic curves, formal groups, elliptic curves over finite fields, local and global fields, integral points, and computational aspects. It also discusses advanced topics like complex multiplication, modular functions, and cryptography. The author emphasizes an algebraic approach, using minimal machinery from algebraic geometry, and includes numerous exercises to aid understanding. The book is structured to be accessible to readers with a background in algebraic number theory and basic complex analysis, making it a valuable resource for those interested in the arithmetic theory of elliptic curves.The book "The Arithmetic of Elliptic Curves" by Joseph H. Silverman is a comprehensive introduction to the arithmetic properties of elliptic curves, a field that has seen significant progress over the past two decades. The second edition, updated and expanded, aims to remain a useful resource for both students and researchers. Key updates include new sections on the variation of the trace of Frobenius and additional exercises. The book covers fundamental topics such as algebraic varieties, algebraic curves, the geometry of elliptic curves, formal groups, elliptic curves over finite fields, local and global fields, integral points, and computational aspects. It also discusses advanced topics like complex multiplication, modular functions, and cryptography. The author emphasizes an algebraic approach, using minimal machinery from algebraic geometry, and includes numerous exercises to aid understanding. The book is structured to be accessible to readers with a background in algebraic number theory and basic complex analysis, making it a valuable resource for those interested in the arithmetic theory of elliptic curves.