Periods of Quaternionic Shimura Varieties. I.

Formulates a new conjecture about quadratic periods of automorphic forms on quaternion algebras, an integral refinement of Shimura's algebraicity conjectures on these periods. The book also provides a strategy to attack the conjecture by reformulating it in terms of integrality properties of the theta correspondence for quaternionic unitary groups.

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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.