The Arithmetic of Elliptic Curves, Second Edition by Joseph H. Silverman is a comprehensive text on the arithmetic of elliptic curves, covering both theoretical and computational aspects. The book is part of the Graduate Texts in Mathematics series and includes a detailed preface, acknowledgments, and a thorough introduction to the subject. The second edition includes updates, expansions, and corrections to the first edition, with additional chapters on algorithmic aspects of elliptic curves and Szpiro’s conjecture. The book is structured into chapters that cover algebraic varieties, algebraic curves, the geometry of elliptic curves, formal groups, elliptic curves over finite and local fields, and global fields. It also includes appendices on further topics such as complex multiplication, modular functions, and duality theory. The text is aimed at graduate students and researchers in number theory and algebraic geometry, providing a self-contained introduction to the basic arithmetic properties of elliptic curves. The book includes numerous exercises and references to additional literature, making it a valuable resource for both learning and research. The author acknowledges the contributions of many individuals who provided feedback and corrections, and thanks those who helped in the development of the book. The text is written in a clear and accessible style, with a focus on both theoretical foundations and practical applications, particularly in cryptography. The book is an essential resource for anyone interested in the study of elliptic curves and their arithmetic properties.The Arithmetic of Elliptic Curves, Second Edition by Joseph H. Silverman is a comprehensive text on the arithmetic of elliptic curves, covering both theoretical and computational aspects. The book is part of the Graduate Texts in Mathematics series and includes a detailed preface, acknowledgments, and a thorough introduction to the subject. The second edition includes updates, expansions, and corrections to the first edition, with additional chapters on algorithmic aspects of elliptic curves and Szpiro’s conjecture. The book is structured into chapters that cover algebraic varieties, algebraic curves, the geometry of elliptic curves, formal groups, elliptic curves over finite and local fields, and global fields. It also includes appendices on further topics such as complex multiplication, modular functions, and duality theory. The text is aimed at graduate students and researchers in number theory and algebraic geometry, providing a self-contained introduction to the basic arithmetic properties of elliptic curves. The book includes numerous exercises and references to additional literature, making it a valuable resource for both learning and research. The author acknowledges the contributions of many individuals who provided feedback and corrections, and thanks those who helped in the development of the book. The text is written in a clear and accessible style, with a focus on both theoretical foundations and practical applications, particularly in cryptography. The book is an essential resource for anyone interested in the study of elliptic curves and their arithmetic properties.