Periods of Quaternionic Shimura Varieties. I.

Atsushi Ichino, Kyoto University, Japan, and Kartik Prasanna, University of Michigan, Ann Arbor, MI.

Introduction Quaternionic Shimura Varieties Unitary and Quaternionic Unitary Groups Weil Representations The Rallis Inner Product Formula and the Jacquet-Langlands Correspondence Schwartz Functions Explicit Form of the Rallis Inner Product Formula The Main Conjecture on the Arithmetic of Theta Lifts Appendix A. Abelian Varieties, Polarizations and Hermitian Forms Appendix B. Metaplectic Covers of Symplectic Similitude Groups Appendix C. Splittings: The Case $\textrm{dim}_{B}V =2$ and $\textrm{dim}_{B}W =1$ Appendix D. Splittings for the Doubling Method: Quaternionic Unitary Groups Bibliography