Applications of Arthur's multiplicity formula to Siegel modular forms

Abstract

We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one. Contents 1. Introduction 1 Acknowledgments 4 2. Arthur's multiplicity formula and supplementary results 4 2.1. Local A-parameters and local A-packets 5 2.2. Global A-parameters and global A-packets 6 2.3. Arthur's multiplicity formula 8 2.4. Moeglin's results 8 2.5. Adams-Johnson packets 10 3. Siegel modular forms and holomorphic cusp forms 13 3.1. Siegel modular forms 13 3.2. Lowest weight modules 14 3.3. Automorphic representations generated by Siegel modular forms 16 3.4. Proof of Lifting Theorem 17 3.5. Arthur's multiplicity formula for holomorphic cusp forms 21 Appendix A. The hypothesis Langlands group and Arthur's character 23 A.1. Hypothesis Langlands group 23 A.2. Component groups 23 A.3. Formal definition of Arthur's characters 24 References 25