Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change

1. Front Matter

   Pages i-xiii

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2. Introduction

   • Jayce Getz, Mark Goresky

   Pages 1-19

3. Review of Chains and Cochains

   • Jayce Getz, Mark Goresky

   Pages 21-28

4. Review of Intersection Homology and Cohomology

   • Jayce Getz, Mark Goresky

   Pages 29-39

5. Review of Arithmetic Quotients

   • Jayce Getz, Mark Goresky

   Pages 41-55

6. Generalities on Hilbert Modular Forms and Varieties

   • Jayce Getz, Mark Goresky

   Pages 57-89

7. Automorphic Vector Bundles and Local Systems

   • Jayce Getz, Mark Goresky

   Pages 91-110

8. The Automorphic Description of Intersection Cohomology

   • Jayce Getz, Mark Goresky

   Pages 111-134

9. Hilbert Modular Forms with Coefficients in a Hecke Module

   • Jayce Getz, Mark Goresky

   Pages 135-150

10. Explicit Construction of Cycles

    • Jayce Getz, Mark Goresky

    Pages 151-166

11. The Full Version of Theorem 1.3

    • Jayce Getz, Mark Goresky

    Pages 167-177

12. Eisenstein Series with Coefficients in Intersection Homology

    • Jayce Getz, Mark Goresky

    Pages 179-182

13. Back Matter

    Pages 183-256

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In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.

• Fourier coefficients
• Hecke operators
• Hilbert modular varieties
• automorphic forms
• intersection cohomology

• , Department of Mathematics, McGill University, Montreal, Canada

  Jayce Getz

• , School of Mathematics, Institute for Advanced Study, Princeton, USA

  Mark Goresky

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