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PART I: COMPUTATIONAL THEORY 1. Introduction 2. Modular Forms and Modular Symbols 3. Computing the Space of Modular Symbols 4. Hecke Operators and Eigenforms PART II: CURVE ALGORITHMS & TABLES 5. Periods and L-series of Modular Forms 6. Constructing Elliptic Curves from Modular Forms 7. Computing Invariants and Curve Parameters 8. Description of the Tables
PART I: COMPUTATIONAL THEORY
PART II: CURVE ALGORITHMS & TABLES
Algorithms for Modular Elliptic Curves by John E. Cremona is a classic research text and practical reference that details the explicit computation of modular elliptic curves over the rational numbers. This renowned volume guides readers through modular symbols, L-functions, period lattices, and conductor computations without losing sight of explicit algorithmic implementation. Designed for graduate students, computational number theorists, and arithmetic geometers, this foundational text bridges abstract modularity theorems and practical computer algebra. Cremona provides step-by-step mathematical algorithms and explicit computational strategies for calculating curve parameters and proving the Birch and Swinnerton-Dyer conjecture for specific curves. Whether you are studying elliptic curves for the first time or building computational software, this book offers an indispensable roadmap. Recognized worldwide as the foundation for the famous Cremona Database, it remains one of the best computational elliptic curve books for self-study. By balancing rigorous number theory with concrete computer algorithms, it serves as an essential resource for mastering modular curves with confidence.
Title: Algorithms for Modular Elliptic Curves by John Cremona Publisher: Cambridge University Press Year: 1992 Pages: 351 Type: PDF Language: English ISBN-10 #: 0521418135 ISBN-13 #: 978-0521418133 License: External Educational Resource Amazon: Amazon
The author John E. Cremona is a distinguished British mathematician and Emeritus Professor of Mathematics at the University of Warwick, internationally celebrated for his groundbreaking contributions to computational number theory and arithmetic geometry. He is best known for developing the comprehensive Cremona Database, which catalogs rational elliptic curves up to large conductors. Building on decades of research, Cremona authored Algorithms for Modular Elliptic Curves to share the exact algorithms used to compute and classify modular forms and elliptic curves. His dedication to mathematical software and clear computational exposition has made this text a classic reference for arithmetic computations that remains vital to number theorists worldwide.
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