  We study "quadratic periods" on quaternionic Shimura varieties and formulate
an integral refinement of Shimura's conjecture regarding Petersson inner
products of automorphic forms that are related by the Jacquet-Langlands
correspondence. The main result is that this integral refinement is implied by
another conjecture (Conjecture D below) regarding integrality of theta lifts
between certain quaternionic unitary groups.
