Endoscopy for GSp(4) and the cohomology of Siegel modular threefolds

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This volume originated from a series of preprints circulated between 1993 and 1994, alongside independent work by Harder and Laumon. The text is based on a revised version of these preprints, which were widely distributed in summer 1995. Although there was an initial plan to reorganize the original content, it was ultimately decided to modestly improve the presentation by adding necessary cross-references. Additional chapters and sections were incorporated, written in 1998. The two main results include the proof of Ramanujan’s conjecture for Siegel modular forms of genus 2, specifically for forms that are not cuspidal representations associated with parabolic subgroups (CAP representations), and the analysis of the endoscopic lift for the group GSp(4). These topics are formulated and proved in the first five chapters, which assume the stabilization of the trace formula. The remaining chapters present the technical results required to obtain the stabilized trace formula. Chapter 1 compiles results on the cohomology of Siegel modular threefolds, which are utilized in subsequent chapters, particularly in Chapter 3. The content is structured to provide a comprehensive overview of the subject matter, enhancing the reader's understanding of these complex topics.