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Full Description
This book formulates a new conjecture about quadratic periods of automorphic forms on quaternion algebras, which is an integral refinement of Shimura's algebraicity conjectures on these periods. It also provides a strategy to attack this conjecture by reformulating it in terms of integrality properties of the theta correspondence for quaternionic unitary groups. The methods and constructions of the book are expected to have applications to other problems related to periods, such as the Bloch-Beilinson conjecture about special values of $L$-functions and constructing geometric realizations of Langlands functoriality for automorphic forms on quaternion algebras.
Contents
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
