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

ISBN-13: 9783540893059 / Angielski / Miękka / 2009 / 374 str.

Rainer Weissauer

Opis

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.

Spis treści

An Application of the Hard Lefschetz Theorem.- CAP-Localization.- The Ramanujan Conjecture for Genus two Siegel modular Forms.- Character identities and Galois representations related to the group GSp(4).- Local and Global Endoscopy for GSp(4).- A special Case of the Fundamental Lemma I.- A special Case of the Fundamental Lemma II.- The Langlands-Shelstad transfer factor.- Fundamental lemma (twisted case).- Reduction to unit elements.- Appendix on Galois cohomology.- Appendix on Double Cosets.

Blurb

The geometry of modular curves and the structure of their cohomology groups have been a rich source for various number-theoretical applications over the last decades. Similar applications may be expected from the arithmetic of higher dimensional modular varieties. For Siegel modular threefolds some basic results on their cohomology groups are derived in this book from considering topological trace formulas.