Arithmetic quantum unique ergodicity under reduced Hecke input: escape of mass at a single Hecke place, and Laplace spectral multiplicities, on arithmetic hyperbolic surfaces

The 2010 note of Brooks and Lindenstrauss on graph eigenfunctions and quantum unique ergodicity established that on compact congruence arithmetic hyperbolic surfaces, joint eigenfunctions of the Laplacian and a single Hecke operator equidistribute — measure rigidity applied to quantum limits invariant under one Hecke place — and posed four open directions. The literature has since closed one of them — complete proofs, in a stronger form covering joint quasimodes (approximate eigenfunctions) and including compact quotients of the product of two upper half planes, appeared in Inventiones in 2014 — and largely closed a second at the level of the entropy method: the single-place positive-entropy technique was extended to higher-rank arithmetic quotients by Shem-Tov, and the kernel construction behind it to affine buildings of arbitrary reduced type (the higher-dimensional analogues of the Hecke tree) by Anker–Rémy–Trojan. The surviving open core consists of two linked components with one objective — determining how far arithmetic quantum unique ergodicity persists when the imposed arithmetic symmetry is reduced: (A) on finite-volume congruence arithmetic surfaces such as the modular surface, joint eigenfunctions of the Laplacian and one Hecke operator are known to have quantum limits that are scalar multiples of the uniform measure, but escape of mass has never been ruled out, and it remains open whether one Hecke operator alone forces the scalar to be 1; (B) on compact congruence surfaces, no nontrivial uniform bound on the multiplicities of the Laplace spectrum is known, though such a bound would immediately imply quantum unique ergodicity for arbitrary orthonormal bases of Laplace eigenfunctions without any Hecke assumption.

Background

Let $\mathbb{H}$ denote the upper half plane with its hyperbolic metric, and for a discrete group $\Gamma$ of isometries let $M=\Gamma\backslash\mathbb{H}$ be the quotient hyperbolic surface, carrying the (positive) Laplacian $\Delta$. If $\varphi_j$ is a sequence of $L^2$-normalized eigenfunctions of $\Delta$ with eigenvalues $\lambda_j\to\infty$, the probability measures $|\varphi_j|^2\,d\mathrm{Vol}$ describe the spatial distribution of the corresponding quantum states in the high-energy limit; their weak-$*$ limits on $M$, when they exist, are called quantum limits. The quantum ergodicity theorem (Shnirelman, Zelditch, Colin de Verdière) states that when the geodesic flow on the unit cotangent bundle is ergodic — ergodicity meaning that the flow admits no invariant measurable subset of positive measure other than the whole space — almost every eigenfunction equidistributes, i.e. the measures $|\varphi_j|^2\,d\mathrm{Vol}$ converge weakly to the normalized volume measure along a subsequence of density one. The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak predicts that on compact negatively curved manifolds, all eigenfunction sequences equidistribute, i.e. the only quantum limit is the normalized volume measure. On a non-compact finite-volume surface a new phenomenon appears: a quantum limit can have total mass strictly less than 1, which is called escape of mass (mass drifts into the cusp — the non-compact end of the surface, where the injectivity radius tends to zero — along the sequence).

The arithmetic surfaces of interest here carry an additional structure: a commuting family of Hecke operators. The compact ones are obtained from a quaternion division algebra over $\mathbb{Q}$ that is split over $\mathbb{R}$: for an order $R$ in such an algebra (an order being a subring that is simultaneously a lattice in the algebra), the group $\Gamma$ of norm-one elements of $R$ acts on $\mathbb{H}$ with compact quotient $M=\Gamma\backslash\mathbb{H}$, a compact congruence surface. The non-compact members of the class are the finite-volume congruence surfaces, most prominently the modular surface $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathbb{H}$. For each prime $p$ outside a finite set of bad primes, the Hecke operator $T_p$ averages a function over the $p+1$ Hecke translates of each point; the Hecke points organize into a $(p+1)$-regular tree, and the Hecke operators commute with $\Delta$ and with one another. One may therefore restrict attention to joint eigenfunctions of $\Delta$ and $T_p$ for a single prime $p$ — a much weaker arithmetic constraint than joint eigenfunctionality for the whole Hecke algebra, which is the hypothesis of the classical arithmetic QUE results.

The study of quantum limits in this arithmetic setting combines two inputs. The first is the microlocal lift (in the Wolpert variant), which lifts each eigenfunction $\varphi_j$ to a function $\Phi_j$ on the unit cotangent bundle $\Gamma\backslash\mathrm{SL}_2(\mathbb{R})$ so that the measures $|\Phi_j|^2\,d\mathrm{Vol}$ become asymptotically invariant under the geodesic flow and Hecke eigenfunctions lift to Hecke eigenfunctions. The second is entropy, the exponential rate at which the flow separates nearby trajectories as averaged against an invariant measure: for a geodesic-flow-invariant measure, positive metric entropy on almost every ergodic component (the ergodic components being the indecomposable pieces into which the measure decomposes under the flow) is the key hypothesis of the measure classification theorems of Einsiedler, Katok and Lindenstrauss and of Lindenstrauss. Anantharaman proved that every quantum limit on a compact negatively curved manifold has positive average entropy; Bourgain and Lindenstrauss strengthened this to positive entropy on almost every ergodic component for quantum limits invariant under the full Hecke algebra. Lindenstrauss's classification of invariant measures then implies arithmetic QUE on compact congruence surfaces for joint eigenfunctions of $\Delta$ and all Hecke operators; on finite-volume surfaces such as the modular surface, the same argument shows only that every quantum limit is a scalar multiple of the uniform measure, and Soundararajan eliminated escape of mass — for joint eigenfunctions of the full Hecke algebra — to complete QUE there; Holowinsky and Soundararajan did the same for holomorphic Hecke eigenforms (the holomorphic analogues of Maass forms).

The 2010 note of Brooks and Lindenstrauss showed that for the entropy step, a single Hecke operator suffices: for joint eigenfunctions of $\Delta$ and one Hecke operator on a compact congruence surface, every quantum limit has positive entropy on almost every ergodic component, so QUE holds for such sequences. The engine is a discrete multitemporal wave equation — a wave equation with several simultaneous time parameters — on the $(p+1)$-regular tree of Hecke points at the single place $p$, with Chebyshev-polynomial solutions, used to construct a convolution kernel supported on a Hecke ball whose matrix coefficients are uniformly small yet whose eigenvalue on $T_p$-eigenfunctions is bounded below; combined with a counting estimate for Hecke points inside geodesic tubes, this rules out concentrated low-entropy limits. The note sketched the same circle of ideas for compact irreducible quotients of $\mathbb{H}\times\mathbb{H}$, where the $\mathrm{SL}_2(\mathbb{R})$-action on the second factor replaces $T_p$, and listed four future directions. Of these, the full proofs of the two sketched results appeared in the authors' 2014 Inventiones paper (in a stronger form covering joint quasimodes, i.e. approximate joint eigenfunctions, of the Laplacian and a single Hecke operator), and the single-place technique was extended to higher rank by Shem-Tov, who proved positive entropy on almost every ergodic component for single-place sequences on quotients of $\mathrm{SL}_n(\mathbb{R})$ by $\mathrm{SL}_n(\mathbb{Z})$ and by division-algebra lattices, and single-place arithmetic QUE on compact quotients associated to division algebras of prime degree; Anker, Rémy and Trojan generalized the kernel construction to affine buildings of arbitrary reduced type (the higher-dimensional analogues of the regular tree, on which Hecke orbits at a non-archimedean place organize in higher rank) and the positive-entropy theorem to arbitrary split simple real Lie groups (the split real forms, such as $\mathrm{SL}_n(\mathbb{R})$). Two directions, however, remain open and constitute the problem recorded here: whether escape of mass can be ruled out at a single Hecke place in the presence of a cusp, and whether the multiplicities of the Laplace spectrum on compact congruence surfaces admit uniform bounds. The two are linked by a common objective, determining arithmetic QUE when the Hecke input is minimal or absent: for the second, Lindenstrauss's classification converts bounded multiplicities into QUE for every orthonormal basis of Laplace eigenfunctions on a compact congruence surface, with no Hecke assumption at all.

Problem Statement

The problem consists of two linked components sharing one objective: to determine how far arithmetic quantum unique ergodicity on arithmetic hyperbolic surfaces persists when the imposed Hecke symmetry is reduced — in component A to a single operator in the finite-volume (non-compact) setting, and in component B to none at all in the compact setting. A complete resolution settles both components.

Component A (escape of mass at a single Hecke place). Let $X=\Gamma\backslash\mathbb{H}$ be a finite-volume congruence arithmetic surface — the base case being the modular surface $\Gamma=\mathrm{PSL}_2(\mathbb{Z})$ — and fix a prime $p$ outside the finite set of bad primes, with Hecke operator $T_p$. Consider any sequence $\{\varphi_j\}$ of $L^2$-normalized joint eigenfunctions of $\Delta$ and $T_p$ (on the modular surface: Maass cusp forms — $L^2$-eigenfunctions of $\Delta$ that decay rapidly in the cusp — that are eigenfunctions of this single $T_p$). It is known — by the $p$-adic propagation technique of Brooks–Lindenstrauss combined with Lindenstrauss's classification of invariant measures, and by Shem-Tov's higher-rank Theorem 1.2 for non-degenerate sequences (non-degeneracy is the applicability condition of the Silberman–Venkatesh microlocal lift: the sequence's spectral parameters stay regular, i.e. bounded away from the walls of the Weyl chamber) — that any weak-$*$ limit of the probability measures $|\varphi_j|^2\,d\mathrm{Vol}$ is a scalar multiple of the normalized uniform measure on $X$. Prove that the scalar equals 1: that no escape of mass occurs and the sequence equidistributes (arithmetic QUE at a single Hecke place). Equivalently: show that for such sequences the $L^2$-mass does not leak into the cusp. A construction of an explicit single-place sequence with a quantum limit of total mass strictly less than 1 would resolve the component in the negative direction. (The same escape-of-mass question for single-place sequences on non-cocompact higher-rank arithmetic quotients, such as quotients of $\mathrm{SL}_n(\mathbb{R})$ by $\mathrm{SL}_n(\mathbb{Z})$ with $n$ prime, is open in exactly the same way and is recorded in the literature as such; it is context only and is not required here, and a resolution of the two-dimensional base case completes this component.)

Component B (uniform Laplace spectral multiplicities on compact congruence surfaces). Let $M=\Gamma\backslash\mathbb{H}$ be a compact congruence arithmetic surface arising from a quaternion division algebra over $\mathbb{Q}$, and let $m(\lambda)$ denote the multiplicity of the eigenvalue $\lambda$ of $\Delta$ on $M$. Prove, or refute, the conjecture that $m(\lambda)$ is bounded by a constant depending only on $M$. By Lindenstrauss's measure classification together with the known positive-entropy results, uniform boundedness of $m(\lambda)$ would imply arithmetic QUE for every orthonormal basis of Laplace eigenfunctions on $M$ with no Hecke-invariance assumption; a direct proof (or refutation) of QUE for arbitrary Laplace-eigenfunction sequences on $M$ resolves the same component. No uniform bound on $m(\lambda)$ is currently known on any fixed compact congruence surface.

Both components admit either a positive resolution (proof) or a negative resolution (explicit counterexample), each of which would settle the corresponding component; the common objective is a complete determination of the extent to which arithmetic QUE survives the reduction or removal of the Hecke hypothesis.

References

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