Elliptic Modular Forms and Their Applications

Abstract

Foreword These notes give a brief introduction to a number of topics in the classical theory of modular forms. Some of theses topics are (planned) to be treated in much more detail in a book, currently in preparation, based on various courses held at the Collège de France in the years 2000–2004. Here each topic is treated with the minimum of detail needed to convey the main idea, and longer proofs are omitted. Classical (or " elliptic ") modular forms are functions in the complex upper half-plane which transform in a certain way under the action of a discrete subgroup Γ of SL(2, R) such as SL(2, Z). From the point of view taken here, there are two cardinal points about them which explain why we are interested. First of all, the space of modular forms of a given weight on Γ is finite dimen-sional and algorithmically computable, so that it is a mechanical procedure to prove any given identity among modular forms. Secondly, modular forms occur naturally in connection with problems arising in many other areas of mathematics. Together, these two facts imply that modular forms have a huge number of applications in other fields. The principal aim of these notes – as also of the notes on Hilbert modular forms by Bruinier and on Siegel modular forms by van der Geer – is to give a feel for some of these applications, rather than emphasizing only the theory. For this reason, we have tried to give as many and as varied examples of interesting applications as possible. These applications are placed in separate mini-subsections following the relevant sections of the main text, and identified both in the text and in the table of contents by the symbol ♠ . (The end of such a mini-subsection is correspond-ingly indicated by the symbol ♥ : these are major applications.) The subjects they cover range from questions of pure number theory and combinatorics to differential equations, geometry, and mathematical physics. The notes are organized as follows. Section 1 gives a basic introduction to the theory of modular forms, concentrating on the full modular group 2 D. Zagier Γ 1 = SL(2, Z). Much of what is presented there can be found in standard textbooks and will be familiar to most readers, but we wanted to make the exposition self-contained. The next two sections describe two of the most im-portant constructions of modular forms, Eisenstein series and theta series. Here too most of the material is quite standard, but we also include a number of concrete examples and applications which may be less well known. Sec-tion 4 gives a brief account of Hecke theory and of the modular forms arising from algebraic number theory or algebraic geometry whose L-series have Eu-ler products. In the last two sections we turn to topics which, although also classical, are somewhat more specialized; here there is less emphasis on proofs and more on applications. Section 5 treats the aspects of the theory con-nected with differentiation of modular forms, and in particular the differential equations which these functions satisfy. This is perhaps the most important single source of applications of the theory of modular forms, ranging from irrationality and transcendence proofs to the power series arising in mirror symmetry. Section 6 treats the theory of complex multiplication. This too is a classical theory, going back to the turn of the (previous) century, but we try to emphasize aspects that are more recent and less familiar: formulas for the norms and traces of the values of modular functions at CM points, Borcherds products, and explicit Taylor expansions of modular forms. (The last topic is particularly pretty and has applications to quite varied problems of num-ber theory.) A planned seventh section would have treated the integrals, or " periods, " of modular forms, which have a rich combinatorial structure and many applications, but had to be abandoned for reasons of space and time. Apart from the first two, the sections are largely independent of one another and can be read in any order. The text contains 29 numbered " Propositions " whose proofs are given or sketched and 20 unnumbered " Theorems " which are results quoted from the literature whose proofs are too difficult (in many cases, much too difficult) to be given here, though in many cases we have tried to indicate what the main ingredients are. To avoid breaking the flow of the exposition, references and suggestions for further reading have not been given within the main text but collected into a single section at the end. Notations are standard (e.g., Z, Q, R and C for the integers, rationals, reals and complex numbers, respectively, and N for the strictly positive integers). Multiplication precedes division hierarchically, so that, for instance, 1/4π means 1/(4π) and not (1/4)π. The presentation in Sections 1–5 is based partly on notes taken by Chris-tian Grundh, Magnus Dehli Vigeland and my wife, Silke Wimmer-Zagier, of the lectures which I gave at Nordfjordeid, while that of Section 6 is partly based on the notes taken by John Voight of an earlier course on complex multiplication which I gave in Berekeley in 1992. I would like to thank all of them here, but especially Silke, who read each section of the notes as it was written and made innumerable useful suggestions concerning the exposition. And of course special thanks to Kristian Ranestad for the wonderful week in Nordfjordeid which he organized.