Quantum unique ergodicity for Hecke eigenfunctions on the sphere (Böcherer-Sarnak-Schulze-Pillot conjecture)

Brooks-Le Masson-Lindenstrauss (IMRN 2016) proved quantum ergodicity for joint eigenfunctions of the spherical Laplacian and averaging operators over rotations generating a free group, and stated as the open problem of that setting the conjecture of Böcherer-Sarnak-Schulze-Pillot (Comm. Math. Phys. 2003): the joint eigenfunctions of the Laplacian and the Hecke averaging operators attached to a maximal order in a definite quaternion division algebra over Q with nonzero Hecke eigenvalues (the canonical arithmetic basis of the sphere — the $\mathcal{O}^{\times}$-invariant eigenfunctions, corresponding under Jacquet-Langlands to holomorphic newforms of growing weight) satisfy quantum unique ergodicity, i.e. the measures $|\psi_\ell(x)|^2 d\sigma(x)$ converge weak-* to the uniform measure for every sequence of Hecke eigenfunctions, not merely along density-one subsequences. The conjecture reduces (Böcherer-Sarnak-Schulze-Pillot) to subconvex bounds for degree-8 triple-product L-functions in the weight aspect; the arithmetic QUE problem on hyperbolic surfaces is settled by dynamical and Fourier-based methods (Lindenstrauss; Soundararajan; Holowinsky-Soundararajan) that have no analogue on the sphere, whose geodesic flow is completely degenerate. The most recent progress (Sanchez Garza, 2026) proves quantitative QUE and shrinking-scale equidistribution conditionally on the generalized Lindelöf hypothesis; no unconditional result is known and the conjecture remains open.

Background

Let $M$ be a compact Riemannian manifold and $\Delta$ its Laplace-Beltrami operator. Quantum ergodicity (QE) is the statement that for an orthonormal basis of $\Delta$-eigenfunctions, the probability measures $|\psi_j|^2 d\mathrm{Vol}$ equidistribute along a subsequence of density one; quantum unique ergodicity (QUE) is the far stronger statement that they equidistribute along every sequence of eigenfunctions. A classical theorem of Snirel'man, Zelditch, and Colin de Verdière (see Zelditch 1987) states that QE holds for every orthonormal basis when the geodesic flow is ergodic; on negatively curved manifolds QUE is conjectured (Rudnick-Sarnak 1994) and open in general.

The sphere $S^2$ is the opposite extreme. The geodesic flow is completely degenerate (all geodesics are closed), and the Laplacian has large degeneracies: the eigenspace $\mathcal{H}_\ell$ of spherical harmonics of degree $\ell$ (restrictions to $S^2$ of homogeneous harmonic polynomials of degree $\ell$ on $\mathbb{R}^3$) has dimension $2\ell+1$. Zelditch (1992) showed that the standard basis of spherical harmonics fails QE, while a random orthonormal basis is QE with probability one, and in fact QUE with probability one (VanderKam 1997). Which explicitly describable bases equidistribute is therefore a genuine question.

The arithmetic candidate is given by Hecke operators. Identify $S^2$ with the trace-zero, norm-one elements of the real Hamilton quaternions; each quaternion $\gamma$ of positive reduced norm $\mathrm{Nrd}(\gamma)$ (the quaternion analogue of the determinant) acts on $S^2$ by conjugation, hence by a rotation. Averaging operators built from such rotations were introduced by Lubotzky-Phillips-Sarnak (1986) to study the distribution of points on the sphere. Fix a maximal order $\mathcal{O}$ (a lattice closed under multiplication) in the rational Hamilton quaternion division algebra $B$ (the algebra that remains a division algebra after tensoring with $\mathbb{Q}_2$ and with $\mathbb{R}$; more generally one may take any definite, i.e. ramified-at-infinity, quaternion division algebra over $\mathbb{Q}$ of class number one — meaning that it has a single class of left ideals — of discriminant in $\{2,3,5,7,13\}$), for instance the Hurwitz order. For each positive integer $n$ the Hecke operator $T_n$ averages a function over the finitely many rotations coming from elements $\gamma \in \mathcal{O}$ with $\mathrm{Nrd}(\gamma) = n$. The $T_n$ are self-adjoint, commute with each other and with $\Delta$, and all map into the $\mathcal{O}^{\times}$-invariant subspace, so each $\mathcal{H}_\ell$ splits into joint eigenspaces; the Hecke eigenfunctions are the $\mathcal{O}^{\times}$-invariant joint eigenfunctions, and by the theory of theta series (the classical construction packaging a harmonic polynomial on a quaternion order into a modular form) and the Jacquet-Langlands correspondence (a dictionary between automorphic forms on the quaternion algebra and on $\mathrm{GL}_2$) their joint eigenspaces are one-dimensional and correspond to holomorphic newforms (Hecke-eigen cusp forms on $\Gamma_0(N)$ not induced from lower level) of weight $2\ell+2$ and level equal to the discriminant.

Böcherer-Sarnak-Schulze-Pillot (2003) conjectured mass equidistribution (QUE) for this arithmetic basis, and separately conjectured that its moments become Gaussian (the random-wave prediction). The analytic heart is a formula of Watson-Ichino type (after Watson's unpublished thesis and Ichino's published trilinear-form identity) relating triple correlations $\int_{S^2} \varphi\, \psi\, \psi' \, d\sigma$ of Hecke eigenfunctions to central values of degree-8 L-functions (the Rankin-Selberg-type L-function attached to a triple of automorphic representations, $\mathrm{GL}_2 \times \mathrm{GL}_2 \times \mathrm{GL}_2$). Proving QUE reduces to bounding these central values beyond the convexity bound (the elementary estimate any L-function satisfies via its functional equation and Cauchy-Schwarz) in the archimedean weight aspect as $\ell \to \infty$, uniformly in the test function; this required subconvexity is not known. Böcherer-Sarnak-Schulze-Pillot proved the third-moment case of the Gaussian conjecture unconditionally and showed that QUE follows from such subconvex estimates, hence from GRH.

On hyperbolic arithmetic surfaces the analogous arithmetic QUE problem is settled: Lindenstrauss (2006) classified the invariant measures of the microlocal lift (the phase-space lift of the eigenfunction measures to the unit cotangent bundle) with positive entropy, and Soundararajan (2010) excluded escape of mass; for holomorphic forms, mass equidistribution was proved by Holowinsky-Soundararajan (2010) via Fourier-expansion-based sieve and weak-subconvexity arguments. None of these tools is available for the sphere: there is no hyperbolic dynamics to classify and no Fourier expansion on the quaternionic side.

The source paper, Brooks-Le Masson-Lindenstrauss (2016), proves QE (density-one subsequences only) for joint eigenfunctions of $\Delta$ and an averaging operator over a finite set of rotations generating a free subgroup of $\mathrm{SO}(3)$ (continuous test functions), a quantitative $O(1/\log \ell)$ rate when the rotation entries are algebraic numbers (using the Bourgain-Gamburd spectral gap for finitely generated subgroups of $\mathrm{SU}(2)$), and a new proof of quantum ergodicity for large regular graphs. These are free-group toy models that exclude the arithmetic Hecke operators; the paper's introduction states explicitly that the Böcherer-Sarnak-Schulze-Pillot QUE conjecture, the arithmetic problem, remains open. No unconditional result is known; the newest work on the conjecture (Sanchez Garza, 2026) is conditional on the generalized Lindelöf hypothesis.

Problem Statement

Prove or refute Conjecture 1 of Böcherer-Sarnak-Schulze-Pillot (2003), the open problem restated by Brooks-Le Masson-Lindenstrauss (2016) in connection with their quantum ergodicity theorem for averaging operators on the sphere.

Let $B$ be a definite quaternion division algebra over $\mathbb{Q}$ of class number one (that is, with a single class of left ideals for its maximal orders), of discriminant $N \in \{2,3,5,7,13\}$; the target is the primary instance, the Hamilton quaternion algebra with the Hurwitz order $\mathcal{O}$ (discriminant $N = 2$), and it suffices to settle the conjecture for this instance; the remaining discriminants are not required. Identify $S^2$ with the trace-zero, reduced-norm-one elements of $B \otimes \mathbb{R}$ (the reduced norm $\mathrm{Nrd}$ is the quaternion analogue of the determinant), so that each $\gamma \in \mathcal{O}$ acts on $S^2$ by the rotation $x \mapsto \gamma x \gamma^{-1}$. For each positive integer $n$ define the Hecke averaging operator

$$T_n f(x) \;=\; \frac{1}{|\mathcal{O}^{\times}|} \sum_{\gamma \in \mathcal{O},\ \mathrm{Nrd}(\gamma)=n} f(\gamma \cdot x),$$

where $\mathcal{O}^{\times}$ is the finite unit group of $\mathcal{O}$ (of order $24$ for the Hurwitz order); for the Hamilton algebra $T_{2n}$ acts as $T_1$ or $T_2$ depending on the 2-adic valuation of $n$, so the nontrivial Hecke indices are the odd $n$ (Böcherer-Sarnak-Schulze-Pillot define $T_n$ for odd $n$ only; the normalization above is the one used by Sanchez Garza). The $T_n$ are self-adjoint and commute with each other and with the spherical Laplacian $\Delta$.

Let $\mathcal{H}_\ell$ be the space of spherical harmonics of degree $\ell$ (restrictions to $S^2$ of homogeneous harmonic polynomials of degree $\ell$ on $\mathbb{R}^3$), of dimension $2\ell+1$, with $\Delta$-eigenvalue $\ell(\ell+1)$ in the standard convention. Every $T_n$ maps $\mathcal{H}_\ell$ into the $\mathcal{O}^{\times}$-invariant subspace $\mathcal{H}_\ell^{\mathcal{O}^{\times}}$, because right multiplication by a unit permutes $\{\gamma \in \mathcal{O} : \mathrm{Nrd}(\gamma) = n\}$; in particular $T_1$ is the projection onto $\mathcal{H}_\ell^{\mathcal{O}^{\times}}$. The Hecke eigenfunctions are the joint eigenfunctions of $\Delta$ and all the $T_n$ that lie in $\mathcal{H}_\ell^{\mathcal{O}^{\times}}$, equivalently those with nonzero Hecke eigenvalues (a joint eigenspace with some eigenvalue $\lambda_n \neq 0$ is mapped by $T_n$ into the invariant subspace), equivalently those with $T_1 \psi = \psi$. This restriction is the sources' own: Böcherer-Sarnak-Schulze-Pillot impose it from the outset (their theta-series construction is nonzero only for eigenforms invariant under the unit group), and Sanchez Garza (2026) likewise defines the Hecke eigenfunctions on the invariant space $L^2(\mathcal{O}^{\times} \backslash S^2)$. It is also essential: the orthogonal complement of $\mathcal{H}_\ell^{\mathcal{O}^{\times}}$ in $\mathcal{H}_\ell$ is annihilated by every $T_n$ (it is the joint eigenspace with all eigenvalues zero), so its elements are joint eigenfunctions only in the trivial all-zero sense; they carry no arithmetic, and they include the standard non-equidistributing harmonics (Zelditch 1992). By the Jacquet-Langlands correspondence (a dictionary between automorphic forms on the quaternion algebra and on $\mathrm{GL}_2$) the nontrivial joint eigenspaces are one-dimensional and correspond to holomorphic newforms (Hecke-eigen cusp forms on $\Gamma_0(N)$ not induced from lower level) of weight $2\ell+2$ and level $N$.

Conjecture (QUE for the arithmetic basis of the sphere). For every sequence of $L^2$-normalized Hecke eigenfunctions $\psi_\ell \in \mathcal{H}_\ell^{\mathcal{O}^{\times}}$ with $\ell \to \infty$, the probability measures $|\psi_\ell(x)|^2\,d\sigma(x)$ converge weak-* to the normalized rotation-invariant measure; equivalently, for every continuous $a : S^2 \to \mathbb{C}$,

$$\int_{S^2} a(x)\,|\psi_\ell(x)|^2\,d\sigma(x) \;\longrightarrow\; \frac{1}{4\pi}\int_{S^2} a\,d\sigma \qquad (\ell \to \infty).$$

(In the normalization of Böcherer-Sarnak-Schulze-Pillot, who write $dv$ for the rotation-invariant measure of total mass $2\pi$ on $S^2$ and integrate against continuous functions: $\mu_{\psi_\nu} = |\psi_\nu|^2\,dv \to dv/2\pi$.)

The conjecture is known to follow from subconvex bounds for the degree-8 triple-product L-functions attached to triples of the corresponding newforms, and hence from GRH or the generalized Lindelöf hypothesis; quantitative rates and shrinking-scale versions under GLH are known (Sanchez Garza 2026).

References

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