Quantum variance on quaternion algebras, I

We determine the quantum variance of a sequence of families of automorphic forms on a compact quotient arising from a non-split quaternion algebra. Our results compare to those obtained by Luo--Sarnak, Zhao, and Sarnak--Zhao on the modular curve, whose method required a cusp. Our method uses the theta correspondence to reduce the problem to the estimation of metaplectic Rankin--Selberg convolutions. We apply it here to the first non-split case.

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