Thisbookgrewoutoflecturesgivenatthesummerschoolon“ModularForms and their Applications” at the Sophus Lie Conference center in Nordfjordeid in June 2004. This center, set beautifully in the fjords of the west coast of Norway, has been the site of annual summer schools in algebra and algebraic geometry since 1996.
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The schools are a joint e?ort between the universities in Bergen, Oslo, Tromsø and Trondheim. They are primarily aimed at graduate students in Norway, but also attract a large number of students from other parts of the world. The theme varies among central topics in contemporary mathematics, but the format is the same: three leading experts give indep- dentbutconnectedseriesoflectures,andgiveexercisesthatthestudentswork on in evening sessions. In 2004 the organizing committee consisted of Stein Arild Strømme (Bergen), Geir Ellingsrud and Kristian Ranestad (Oslo) and Alexei Rudakov (Trondheim). We wanted to have a summer school that introduced the s- dents both to the beauty of modular forms and to their varied applications in other areas of mathematics, and were very fortunate to have Don Zagier, Jan Bruinier and Gerard van der Geer give the lectures. The lectures were organizedin three series that are re?ected in the title of this book both by their numbering and their content. The ?rst series treats the classical one-variabletheory and some of its many applications in number theory, algebraic geometry and mathematical physics. Thesecondseries,whichhasamoregeometric?avor,givesanintroduction tothetheoryofHilbertmodularformsintwovariablesandtoHilbertmodular surfaces. In particular, it discusses Borcherds products and some geometric and arithmetic applications
