Arithmetic QUE through Hecke packets

Arithmetic QUE through Hecke packets

Definitions and mathematical statement

An arithmetic locally symmetric space has the form $\Gamma\backslash G/K_0$, with $\Gamma$ arithmetic and $K_0$ maximal compact. In the finite-volume source setting, take normalized joint Laplace/Hecke eigenfunctions $u_j$ with spectral parameters tending to high energy. Their probability densities are $|u_j|^2\,d\operatorname{vol}$. The stated conclusion is \[ |u_j|^2\,d\operatorname{vol}\ \rightharpoonup\ \frac{d\operatorname{vol}}{\operatorname{vol}(\Gamma\backslash G/K_0)}. \] The phrase "arithmetic quantum limit" includes the specified joint-eigenfunction and spectral regime. The catalogue does not delimit all higher-rank or noncompact variants; those require source-specific hypotheses, including exclusion of escape of mass. It is not a blanket statement for arbitrary eigenfunctions on arbitrary arithmetic quotients.

Formulation and concept references: [S1].

Short English statement

Do arithmetic high-energy joint eigenfunctions have only the uniform-volume distribution as a quantum limit?

Research attempt

Editorial note: a necessary scope distinction.

Section 6.1, Conjecture 39 of the supplied AIM list was inspected, including its definition of arithmetic quantum limits. Its brief formulation does not expressly impose irreducibility of the lattice or growth of energy in every factor of a reducible quotient. With arbitrary arithmetic products and only total high energy, the conclusion fails by the construction below. However, the catalogue explicitly retains the source's spectral regime. Accordingly this is treated as a scope obstruction, not a claimed counterexample to an intended irreducible or nondegenerate version of arithmetic QUE. The original wording is preserved. [S1]

Current frontier.

Shem-Tov--Silberman's recent theorem proves arithmetic QUE for congruence quotients arising from norm-one quaternion groups over a number field, including products of hyperbolic two- and three-spaces. Their Section 2.1 uses one number field in constructing the lattice, and Section 3 explicitly uses its irreducibility. A Cartesian product of two independent arithmetic surfaces is not that irreducible lattice setting. Their theorem therefore does not eliminate the scope issue just identified. [E1] The broader measure-rigidity and mass-escape questions remain distinct parts of the formulation source's discussion. [S1]

Attack route.

Test the universal statement under products before trying to invoke measure rigidity. A fixed-energy factor can retain a nonuniform density while another supplies all the growing energy. The proposed repair is factorwise equidistribution; we prove exactly when that repairs a pure-tensor sequence. On an irreducible lattice this factorization is absent, so the argument cannot replace an arithmetic rigidity theorem.

Attempt: a product obstruction.

Let $X_1,X_2$ be compact connected arithmetic hyperbolic surfaces from indefinite quaternion division algebras and torsion-free congruence groups, with probability volumes $m_1,m_2$. Use the commuting self-adjoint Hecke family and the Laplacian; their eigenspaces have a simultaneous real orthonormal eigenbasis. This compact arithmetic setup is included in the source framework. [S1][E1] Choose normalized real joint eigenfunctions $\phi_j$ on $X_1$ with $\lambda_j\to\infty$, and a fixed real nonconstant normalized joint eigenfunction $\psi$ on $X_2$, with eigenvalue $\mu>0$. On the product set \[ u_j(x,y)=\phi_j(x)\psi(y). \] The product group and arithmetic lattice give an arithmetic locally symmetric quotient. The product Hecke family and invariant differential operators act jointly diagonally on $u_j$, and \[ \|u_j\|_{L^2(m_1m_2)}=1, \qquad -\Delta u_j=(\lambda_j+\mu)u_j, \qquad \lambda_j+\mu\longrightarrow\infty. \] These form a subsequence of a complete joint product eigenbasis, not merely arbitrary Laplace eigenfunctions without Hecke symmetry.

The continuous test $a(x,y)=\psi(y)^2-1$ has volume mean zero, but \begin{equation}\label{wsref-0-32-30-30-30-33-33-32-36-3a-6f-62-73-74-72-75-63-74-69-6f-6e} \int a|u_j|^2\,dm_1dm_2=\int\psi^4\,dm_2-1>0 \end{equation} for every $j$. Cauchy--Schwarz gives $\int\psi^4\ge1$. Equality forces $\psi^2=1$ almost everywhere, hence everywhere by continuity. A real continuous function valued in $\{-1,1\}$ on a connected surface is constant, contradicting $\mu>0$. Compactness yields subsequential weak limits of these probability measures, and (20003326:obstruction) excludes volume for each such limit. There is no mass escape and no need to know QUE on the first factor.

Thus total high energy plus joint Hecke symmetry is not enough when independent product quotients and a fixed-energy marginal are admitted. If irreducibility or factorwise high energy is part of the source convention, this example is excluded. It must not then be advertised as disproving the catalogue target.

The precise pure-tensor repair.

Suppose $Y_1,Y_2$ are compact and $v_j,w_j$ are normalized with \[ |v_j|^2m_1\rightharpoonup m_1,\qquad |w_j|^2m_2\rightharpoonup m_2. \] Then $|v_j(x)w_j(y)|^2m_1m_2\rightharpoonup m_1m_2$. For a test $b(x)c(y)$ the integral is a product of two convergent integrals. Finite sums of such functions form a uniformly dense algebra on the compact product, and probability mass one bounds the approximation error independently of $j$. This proves the assertion by Stone--Weierstrass without an infinite series interchange. Conversely, convergence of product measures forces convergence of both marginals by testing functions of a single variable. Hence marginal equidistribution is necessary and sufficient for these pure tensors. No comparison of the two energy growth rates is needed once both marginal conclusions are available.

This proof also shows what remains for joint-eigenspace combinations. For $u_j=\sum_\alpha c_{j,\alpha}v_{j,\alpha}w_{j,\alpha}$, the density contains off-diagonal terms \[ c_{j,\alpha}\overline{c_{j,\beta}} v_{j,\alpha}\overline{v_{j,\beta}} w_{j,\alpha}\overline{w_{j,\beta}}. \] Marginal diagonal QUE alone does not control their sums, especially if multiplicity grows. A theorem for every joint eigenfunction needs that additional information or an argument excluding such multiplicity effects. No bound on these cross terms is supplied here.

Stress test.

The example uses compact congruence quotients and genuine joint eigenvectors, so neither cuspidal normalization nor lack of Hecke symmetry explains the failure. Its cause is reducibility with a fixed-energy factor. The 2026 proof uses irreducibility at the step unavailable in this construction. [E1] On a noncompact product, the probability/tightness argument would additionally require exclusion of mass escape; it was not silently extended there. The original catalogue's source-regime qualification is respected by keeping the result scoped, rather than adopting the broadest possible interpretation as a purported major counterexample.

Outcome.

Outcome: Exploratory approach with unresolved gap. An explicit fixed positive test discrepancy proves failure of the unrestricted product formulation. A necessary-and-sufficient marginal criterion has also been proved for pure tensors. These elementary results identify hypotheses that an exact QUE formulation must retain; no historical novelty is claimed. The general source-qualified arithmetic QUE target, particularly on irreducible quotients, is not resolved by this construction.

Sources

• S1 AIM workshop problem list, Emerging Applications of Measure Rigidity. https://aimath.org/WWN/measrigid/measrigid.pdf. Catalogue formulation source.
• E1 Z. Shem-Tov and L. Silberman, Arithmetic quantum unique ergodicity for products of hyperbolic 2- and 3-spaces, Journal d'Analyse Mathématique 158 $2026$, 413--448; arXiv:2206.05955, Theorem 1, Sections 2.1 and 3 (quaternionic construction and the irreducibility step). https://arxiv.org/html/2206.05955.