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Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces

Zvi Shem-Tov, Lior Silberman

TL;DR

The paper proves Arithmetic QUE for congruence quotients of products of hyperbolic spaces, showing that Hecke--Maass eigenfunctions become equidistributed in the limit. The authors develop a novel amplifier construction based on split primes in a number-field setting, coupled with a Diophantine lemma and positive-entropy bounds, to rule out limit measures supported on proper homogeneous submanifolds. A measure-rigidity framework then forces any limit to be a homogeneous (indeed $G$-invariant) measure, by excluding all lower-dimensional components via an induction on dimension. The work extends AQUE beyond rank-one settings to products of $\mathbb{H}^{(2)}$ and $\mathbb{H}^{(3)}$, including new cases like $\mathrm{SL}_2(\mathbb{Z}[i])\backslash \mathbb{H}^{(3)}$, and provides tools to handle arithmetic subgroups defined over number fields with complex places. The results contribute a robust method for non-concentration of mass and solidify the interplay between dynamics, automorphic forms, and algebraic groups in higher-dimensional locally symmetric spaces.

Abstract

We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $Γ\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.

Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces

TL;DR

The paper proves Arithmetic QUE for congruence quotients of products of hyperbolic spaces, showing that Hecke--Maass eigenfunctions become equidistributed in the limit. The authors develop a novel amplifier construction based on split primes in a number-field setting, coupled with a Diophantine lemma and positive-entropy bounds, to rule out limit measures supported on proper homogeneous submanifolds. A measure-rigidity framework then forces any limit to be a homogeneous (indeed -invariant) measure, by excluding all lower-dimensional components via an induction on dimension. The work extends AQUE beyond rank-one settings to products of and , including new cases like , and provides tools to handle arithmetic subgroups defined over number fields with complex places. The results contribute a robust method for non-concentration of mass and solidify the interplay between dynamics, automorphic forms, and algebraic groups in higher-dimensional locally symmetric spaces.

Abstract

We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients . An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.
Paper Structure (20 sections, 23 theorems, 64 equations)

This paper contains 20 sections, 23 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Statement of result
  3. Context: QUE Conjectures for locally symmetric spaces
  4. Discussion
  5. Sketch of the proof of \ref{['thm:main2']}
  6. Organization of the paper
  7. Notations and background
  8. Unit groups of quaternion algebras
  9. Factoring $G_\infty$ over a number field; complex conjugation
  10. The real group
  11. $p$-adic and adelic groups
  12. Hecke operators
  13. Homogeneity
  14. Positive Entropy
  15. Diophantine Lemma
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\psi_j\in L^2(Y)$ be a sequence of normalized joint eigenfunctions of both the ring of invariant differential operators and of the Hecke operators. Assume that the Laplace-eigenvalues $\lambda_j\to \infty$. Then the probability measures $\left| {\psi_j} \right|^2(y)\mathop{}\!\mathrm{d}\!\matho

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Example 3
  • Example 4
  • Proposition 5
  • Definition 6
  • Remark 7
  • Lemma 8
  • Corollary 9
  • Proposition 10
  • ...and 38 more