A first course in modular forms. (English) Zbl 1062.11022

Graduate Texts in Mathematics 228. Berlin: Springer (ISBN 0-387-23229-X/hbk). xv, 436 p. (2005).
On September 19, 1994, Andrew Wiles obtained a complete proof of the so-called Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves defined over \(\mathbb Q\) with a key ingredient supplied by joint work with R. Taylor on ring-theoretic properties of certain Hecke algebras [A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029)]. By means of this fundamental result, A. Wiles was able to complete the proof of Fermat’s Last Theorem after some 350 years since its formulation.
The conjecture stating that all rational elliptic curves arise from certain modular forms in a specific manner was first suggested by Taniyama in the 1950s. The precise formulation of that conjecture was given by Shimura, and in 1967 A. Weil provided some stronger evidence for the conjecture, which then became famous under the name “Modularity Conjecture” or “Shimura-Taniyama-Weil Conjecture”. The fact that the modularity conjecture should imply Fermat’s last theorem was first observed by G. Frey (1986), and affirmatively settled by K. Ribet shortly thereafter. In fact, Ribet’s result required only that the modular conjecture be true for semi-stable elliptic curves in order to deduce Fermat’s last theorem, and that is why Wiles finally proved the latter by verifying the modular conjecture for this class of elliptic curves.
Finally, the entire modularity conjecture was completely proved in 1999 by C. Breuil, B. Conrad, F. Diamond and R. Taylor [On the modularity of elliptic curves over \(\mathbb Q\): wild 3-adic exercises, J. Am. Math. Soc. 14, No. 4, 843–939 (2001; Zbl 0982.11033)], making the former Shimura-Taniyama-Weil conjecture into what is now called the modularity theorem.
The textbook under review provides a modern introduction to the theory of modular forms, with the aim to explain the modularity theorem to beginning graduate students and advanced undergraduates. This ambitious program, with the first author being one of the co-architects of the general modularity theorem, is carried out in as down-to-earth a way as possible. Although some of the difficulties in studying the advanced theory of modular forms lie in the material itself, the authors believe that a more expansive narrative with many exercises will help students into the subject, and that is exactly the style they have chosen. As for the algebraic aspects of modular forms, indispensible to understand their role in number theory, the authors have tried to discuss those without assuming a basic knowledge in algebraic geometry or algebraic number theory. Instead, they have compiled (and cited) all the necessary facts from these areas wherever needed, without letting them take over the text. Thus the minimal prerequisites are undergraduate semester courses in linear algebra, abstract algebra, real analysis, complex analysis, general topology, and elementary number theory. With regard to the precise contents, the book comprises nine chapters entitled as follows:
1. Modular Forms, Elliptic Curves, and Modular Curves; 2. Modular Curves as Riemann Surfaces; 3. Dimension Formulas; 4. Eisenstein Series; 5. Hecke Operators; 6. Jacobians and Abelian Varieties; 7. Modular Curves as Algebraic Curves; 8. The Eichler-Shimura Relation and \(L\)-functions; 9. Galois Representations.
At the end of the book, there is a section providing hints and answers to the numerous exercises added to each of the 70 paragraphs of the entire text. The headlines of the single chapters are aptly chosen, and they indicate the respective contents without any ambiguity.
May it therefore suffice to mention that the topics covered include Hecke operators and Atkin-Lehner theory, Hecke eigenforms and their arithmetic properties, Jacobians of modular curves, Abelian varieties associated to Hecke eigenforms, elliptic and modular curves modulo \(p\), the Eichler-Shimura relation, Galois representations associated to elliptic curves and to Hecke eigenforms, Galois representations and modularity, Fourier transforms, Mellin transforms, and many other basic notions and results. In the course of the text, the modularity theorem, that is the Leitmotiv of the book, is stated in its various forms. The authors thoroughly relate these various statements to each other and touch upon their applications to number theory, in particular to Fermat’s last theorem.
All in all, this is the first comprehensive introduction to the recent modularity theorem, the former Shimura-Taniyama-Weil conjecture, and a masterly introduction to the allied theory of modular forms as well. The authors have managed to write an utmost user-friendly textbook on a highly advanced classical and contemporary topic of mathematical research, without compromising rigor or mathematical depth. Written in a very comprehensible, detailed, lucid and instructive manner, this unique textbook is widely self-contained and perfectly suitable for self-study by beginners. Moreover, this book is an excellent guide to the relevant research literature, leading the reader right to the forefront of research in the field. But also experts and teachers will get a lot of methodological inspiration from the authors’ approach, and many useful ideas for efficient teaching. Finally, the present new textbook may be seen as a timely completion of the related great book “Invitation to the Mathematics of Fermat-Wiles” by Y. Hellegouarch [(San Diego, CA: Academic Press) (2002; Zbl 1036.11001), translation from the French original by L. Schneps], the French original of which was first published in [Enseignment de Mathématiques, Paris: Masson (1997; Zbl 0887.11003)].


11F11 Holomorphic modular forms of integral weight
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11F25 Hecke-Petersson operators, differential operators (one variable)
11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11D41 Higher degree equations; Fermat’s equation