Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds

1. Front Matter

   Pages i-xvii

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2. An Application of the Hard Lefschetz Theorem

   • Rainer Weissauer

   Pages 1-17

3. CAP-Localization

   • Rainer Weissauer

   Pages 19-52

4. The Ramanujan Conjecture for Genus two Siegel modular Forms

   • Rainer Weissauer

   Pages 53-73

5. Character identities and Galois representations related to the group GSp(4)

   • Rainer Weissauer

   Pages 75-173

6. Local and Global Endoscopy for GSp(4)

   • Rainer Weissauer

   Pages 175-210

7. A special Case of the Fundamental Lemma I

   • Rainer Weissauer

   Pages 211-238

8. A special Case of the Fundamental Lemma II

   • Rainer Weissauer

   Pages 239-269

9. The Langlands-Shelstad transfer factor

   • Rainer Weissauer

   Pages 271-290

10. Fundamental lemma (twisted case)

    • Rainer Weissauer

    Pages 291-320

11. Reduction to unit elements

    • Rainer Weissauer

    Pages 321-333

12. Appendix on Galois cohomology

    • Rainer Weissauer

    Pages 335-338

13. Appendix on Double Cosets

    • Rainer Weissauer

    Pages 339-354

14. Back Matter

    Pages 355-368

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This volume grew out of a series of preprints which were written and circulated - tween 1993 and 1994. Around the same time, related work was done independently by Harder [40] and Laumon [62]. In writing this text based on a revised version of these preprints that were widely distributed in summer 1995, I ?nally did not p- sue the original plan to completely reorganize the original preprints. After the long delay, one of the reasons was that an overview of the results is now available in [115]. Instead I tried to improve the presentation modestly, in particular by adding cross-references wherever I felt this was necessary. In addition, Chaps. 11 and 12 and Sects. 5. 1, 5. 4, and 5. 5 were added; these were written in 1998. I willgivea moredetailedoverviewofthecontentofthedifferentchaptersbelow. Before that I should mention that the two main results are the proof of Ramanujan’s conjecture for Siegel modular forms of genus 2 for forms which are not cuspidal representations associated with parabolic subgroups(CAP representations), and the study of the endoscopic lift for the group GSp(4). Both topics are formulated and proved in the ?rst ?ve chapters assuming the stabilization of the trace formula. All the remaining technical results, which are necessary to obtain the stabilized trace formula, are presented in the remaining chapters. Chapter 1 gathers results on the cohomology of Siegel modular threefolds that are used in later chapters, notably in Chap. 3. At the beginning of Chap.

• Ramanujan's conjecture
• Selberg trace formula
• automorphic representations
• cohomology
• cohomology group
• endoscopy
• homology
• l-adic representations

• Inst. Mathematik, Universität Heidelberg, Heidelberg, Germany

  Rainer Weissauer

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