Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, 2012
Progress in Mathematics Series, Vol. 298

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.

Chapter 1. Introduction.- Chapter 2. Review of Chains and Cochains.- Chapter 3. Review of Intersection Homology and Cohomology.- Chapter 4. Review of Arithmetic Quotients.- Chapter 5. Generalities on Hilbert Modular Forms and Varieties.- Chapter 6. Automorphic vector bundles and local systems.- Chapter 7. The automorphic description of intersection cohomology.- Chapter 8. Hilbert Modular Forms with Coefficients in a Hecke Module.- Chapter 9. Explicit construction of cycles.- Chapter 10. The full version of Theorem 1.3.- Chapter 11. Eisenstein Series with Coefficients in Intersection Homology.- Appendix A. Proof of Proposition 2.4.- Appendix B. Recollections on Orbifolds.- Appendix C. Basic adèlic facts.- Appendix D. Fourier expansions of Hilbert modular forms.- Appendix E. Review of Prime Degree Base Change for GL2.- Bibliography.

Award winning monograph of the 2011 Ferran Sunyer i Balaguer Prize competition Contains basic material on intersection cohomology, modular cycles and automorphic forms from the classical and adèlic points of view Appendices on orbifolds, Fourier expansions, and base change help to make the book self-contained Contains topics of interest for geometers and number theorists interested in locally symmetric spaces and automorphic forms Includes supplementary material: sn.pub/extras