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arXiv · 1606.02267

Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras

Lior Silberman·Akshay Venkatesh
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Abstract

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $\Gamma \backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $\Gamma<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneered by Lindenstrauss

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Lior Silberman, Akshay Venkatesh. 2016-06-07. Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras. https://arxiv.org/abs/1606.02267

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