*(English)*Zbl 1110.11015

The theory of modular forms for congruence subgroups of the group ${\text{SL}}_{2}\left(\mathbb{Z}\right)$ is a fundamental toolkit in both algebraic number theory and arithmetic geometry of elliptic curves. Its richness, complexity, and crucial significance became particularly manifest through the spectacular approaches to finally prove Fermat’s Last Theorem, the Shimura-Taniyama-Weil Conjecture, the Serre Conjecture on the modularity of the Galois representations of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the Conjecture of Birch and Swinnerton-Dyer, and other related deep conjectures.

However, while the theoretical framework of modular forms has already progressed very well, in the course of the last half-century, its practical use in concrete situations appears to be rather difficult regarding the computational aspects.

The book under review addresses exactly this practical problem in the theory of modular forms. More precisely, it is mainly devoted to the contemporary question of how to use a computer to explicitly determine spaces of modular forms.

In this vein, this is a highly non-traditional book on modular forms, and it is absolutely novel and unique of its kind in the relevant literature.

The author, a leading expert in the field of computational arithmetic, presents here the first comprehensive textbook about algorithms for computing with modular forms, together with an accompanying introduction to the underlying theory of modular forms. The text emerged from notes for a well-tried course the author taught at several universities between 2004 and 2006, and all the algorithms explained here have been successfully implemented and tested by the author himself. As a result, various theoretical formulas and constructions for modular forms have been turned into precise computational recipes, whose realizations can now be found as part of the author’s free open source computer algebra system SAGE (cf.: *W. Stein*, SAGE: Software for Algebra and Geometry Experimentation,

As this is intended to be a graduate-level textbook, the reader is assumed to have acquired some familiarity with modular forms beforehand, though most of the basic definitions and facts are systematically recalled and compiled, thereby making the text largely self-contained and accessible for less experienced readers. Also, there are always precise hints for complementary reading concerning the respective theoretical background material from algebraic number theory, complex analysis, geometry (of elliptie curves), linear algebra, and other related areas.

The main text of the book is divided into eleven chapters, complemented by an appendix.

Chapter 1 recalls the foundational material about congruence subgroups of ${\text{SL}}_{2}\left(\mathbb{Z}\right)$, modular forms on the upper-half plane, $q$-expansions, and applications of modular forms throughout mathematics.

Chapter 2 discusses in greater detail modular forms of level one, including Eisenstein series, the cusp form ${\Delta}$, dimension formulas for spaces of level-1 modular forms, Miller bases, Hecke operators, and the Ramanujan tau function. The second part of this chapter deals with explicit and fast algorithms to compute Hecke operators, Fourier coefficients, constant terms of Eisenstein series, and Bernoulli numbers.

Chapter 3 turns to modular forms of higher level, with the focus on forms of weight 2 to begin with. After a geometric description of cuspidal modular forms as differentials on modular curves, modular symbols are introduced and then applied to derive practical methods for explicitly computing cusp forms of weight 2 via basis forms and eigenforms.

Chapter 4 introduces Dirichlet characters as a basic tool for both constructing Eisenstein series and decomposing spaces of modular forms. The better part of this chapter is dedicated to a detailed study of how to effectively represent and compute with Dirichlet characters.

Chapter 5 gives methods of how to compute the Eisenstein subspace of modular forms via generalized Bernoulli numbers attached to a Dirichlet character and an integer. In this context, a new analytic algorithm is presented.

Chapter 6 describes various theoretical dimension formulas for spaces of modular forms, with a particular focus on modular forms for the congruence subgroups ${{\Gamma}}_{0}\left(N\right)$ and ${{\Gamma}}_{1}\left(N\right)$.

Chapter 7 is independent of the others in that it deals with algorithms in linear algebra over exact fields. These algorithms are not only interesting in their own right, but are indeed of crucial importance to actual implementations of algorithms for computing with modular forms.

Chapter 8 is declared to be the most important part of the entire book. In fact, this chapter generalizes the methods cf Chapter 3 to modular forms of higher weight and arbitrary level, where the emphasis is on computing with general modular symbols and Hecke operators. The modular symbols formalism, which is a (co)homological tool in its nature, is very fundamental to general algorithms for computing with modular forms.

This is demonstrated in Chapter 9, where the central algorithm derived in Chapter 8 is applied to computing with “newforms” à la Atkin-Lehner-Li. This leads to practical methods for decomposing spaces of modular forms using Dirichlet characters, and then computing bases for the spaces of Hecke eigenforms for each subspace through several approaches.

Computing cusp forms, congruenees between “newforms”, and generating Hecke algebras are further topics of crucial significance in current research that are touched upon in this chapter.

Chapter 10 is somewhat more advanced than the others in that it is about computing period maps associated to “newforms”. Apart from heavily building on Chapters 8 and 9, this chapter assumes some familiarity of the reader with Abelian varieties. The main focus of this chapter is a discussion of (new) tricks for speeding convergence of certain infinite series occurring in algorithms for computing period maps.

Each chapter comes with numerous instructive examples and exercises, and detailed solutions to most of the exercises form the content of the concluding Chapter 11.

Finally, there is an appendix of about 50 pages to the present text, written by P. E. Gunnells and titled “Computing in Higher Rank”. Its goal is to describe some of the recently developed computational techniques for more general automorphic forms attached to reductive algebraic groups containing an arithmetic subgroup ${\Gamma}$. This points to the computational aspects (and problems) of the famous Langlands program, and the author surveys some of the ideas and achievements in this direction. The topics in this appendix are closely related to the main themes of the book, thereby helping see its key ideas in a much wider (and much more advanced) context. After giving some background on automorphic forms and the cohomology of arithmetic groups, the author turns to topological tools used to compute the cohomology of an arithmetic group, and describes then the computation of the generalized Hecke operators on the respective top-degree cohomology group. This less formal exposition comes with numerous illustrating examples, methodological remarks, and references to the relevant research literature for detailed reading.

All together, the book under review is a brilliant text of highest originality, richness, and importance. Written by an utmost active and creative researcher in the field, it offers a wealth of explicit methods and practical recipes in computer arithmetic, most of which are very recent and absolutely new. The author’s computational methods and their appertaining software are certainly pioneering with a view to further research in the field of modular forms and related areas, and that is why this unique text must be seen as an indispensable source for every algebraic number theorist or arithmetic geometer in these days.

Due to the author’s vivid and lucid style of exposition, his mastery in the field, and the minimum of assumed prerequisites, this textbook for graduate students should be even easily accessible to advanced undergraduates, or to interested mathematicians in general.

##### MSC:

11F11 | Holomorphic modular forms of integral weight |

11-02 | Research monographs (number theory) |

11Y16 | Algorithms; complexity (number theory) |

11F67 | Special values of automorphic $L$-series, etc |

11F55 | Groups and their modular and automorphic forms (several variables) |

11F75 | Cohomology of arithmetic groups |