Periods of quaternionic Shimura varieties. I

TL;DR

This paper investigates quadratic periods on quaternionic Shimura varieties, proposing an integral refinement of Shimura's conjecture on Petersson inner products, linked to a conjecture on theta lift integrality between quaternionic unitary groups.

Contribution

It introduces an integral refinement of Shimura's conjecture and connects it to a new conjecture on theta lift integrality, advancing understanding of automorphic forms on quaternionic Shimura varieties.

Findings

Integral refinement of Shimura's conjecture formulated

Connection established between the refinement and theta lift integrality conjecture

Main result shows the refinement follows from the theta lift conjecture

Abstract

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.