Hilbert Modular Forms

1. Front Matter

   Pages I-VIII

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2. Introduction

   • Eberhard Freitag

   Pages 1-3

3. Hilbert Modular Forms

   • Eberhard Freitag

   Pages 5-71

4. Dimension Formulae

   • Eberhard Freitag

   Pages 73-131

5. The Cohomology of the Hilbert Modular Group

   • Eberhard Freitag

   Pages 133-202

6. Back Matter

   Pages 203-252

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Important results on the Hilbert modular group and Hilbert modular forms are introduced and described in this book. In recent times, this branch of number theory has been given more and more attention and thus the need for a comprehensive presentation of these results, previously scattered in research journal papers, has become obvious. The main aim of this book is to give a description of the singular cohomology and its Hodge decomposition including explicit formulae. The author has succeeded in giving proofs which are both elementary and complete. The book contains an introduction to Hilbert modular forms, reduction theory, the trace formula and Shimizu's formulae, the work of Matsushima and Shimura, analytic continuation of Eisenstein series, the cohomology and its Hodge decomposition. Basic facts about algebraic numbers, integration, alternating differential forms and Hodge theory are included in convenient appendices so that the book can be used by students with a knowledge of complex analysis (one variable) and algebra.

• Complex analysis
• Eisenstein series
• Selberg trace formula
• cusp form
• number theory
• reduction theory

• Mathematisches Institut, Universität Heidelberg, Heidelberg, Fed. Rep. of Germany

  Eberhard Freitag

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