Periods of Quaternionic Shimura Varieties. I.

ISBN-13: 9781470448943

Atsushi Ichino;Kartik Prasanna

Opis

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.

Spis treści

• 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 $\textrm_V =2$ and $\textrm_W =1$
• Appendix D. Splittings for the Doubling Method: Quaternionic Unitary Groups
• Bibliography