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Abstract

From the preface: This book grew out of three series of lectures given at the summer school on ``Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004.\par The lectures were organized in three series that are reflected in the title of this book both by their numbering and their content. \par The first series treats the classical one-variable theory and some of its many applications in number theory, algebraic geometry and mathematical physics: Don Zagier, Elliptic modular forms and their applications (1--103).\par The second series, which has a more geometric flavor, gives an introduction to the theory of Hilbert modular forms in two variables and to Hilbert modular surfaces. In particular, it discusses Borcherds products and some geometric and arithmetic applications: Jan Hendrik Bruinier, Hilbert modular forms and their applications (105--179).\par The third gives an introduction to Siegel modular forms, both scalar- and vector-valued, especially Siegel modular forms of degree 2, which are functions of three complex variables. It presents a beautiful application of the theory of curves over finite fields to Siegel modular forms by providing evidence for a conjecture of Harder on congruences between elliptic and Siegel modular forms: Gerard van der Geer, Siegel modular forms and their applications (181--245).\par Günter Harder came forward with this conjecture in a colloquium lecture in Bonn in 2003. He kindly allowed us to include his notes for this colloquium talk in Bonn on the subject: A congruence between a Siegel and an elliptic modular form (247--262). \par Even though the three lecture series are strongly connected, each of them is self contained and can be read independently of the others.\par The lectures will be reviewed individually.