This is a carefully written introduction to the (very few) existing results on transcendence and algebraic independence of values of modular forms, based on a series of mini-courses delivered at a conference organized by a group of young French mathematicians in 2003. The book consists of four chapters. The first one gives a survey of modular forms with special emphasis on objects relevant to transcendence proofs, such as the Rankin-Cohen brackets, quasi-modular and quasi holomorphic modular forms. The second chapter is devoted to Nesterenko’s result on algebraic independence of values of certain Eisenstein series. Chapter 3 presents a proof of an earlier result (the “St. Etienne theorem” on transcendence of J(q)) from a more conceptual and geometric perspective. The final chapter offers an introduction to Hilbert modular forms in the classical language (including generalizations of the Rankin Cohen brackets). Various methods of construction of such forms are made explicit for the field Q(√5).

Reviewer:

jnek