Linnik problem for Maass--Hecke cuspforms and effective multiplicity one theorem
Junehyuk Jung, Min Lee
TL;DR
This work addresses two quantitative questions about joint spectra of the Laplace–Beltrami operator and Hecke operators on arithmetic surfaces. It develops archimedean spectral large sieve inequalities, including a symmetric-square variant, to obtain a Linnik-type bound on the dimension of joint eigenspaces for Maass–Hecke cuspforms when only small primes are allowed in the Hecke data. It also proves an effective multiplicity-one result: for any fixed η>0, two Maass–Hecke cuspforms with the same Laplacian eigenvalue and identical Hecke data up to $n<\eta t$ are proportional for large $t$, improving earlier bounds via a QUE-based approach of Brook–Lindenstrauss. The paper extends these results to Maass–Hecke cuspforms on $\mathrm{PGL}_2$ over arbitrary number fields, and discusses conditional generalizations under standard conjectures. The combination of spectral large sieve techniques and quantum unique ergodicity underpins both the dimension bounds and the multiplicity control, yielding precise, effective statements with potential arithmetic applications.
Abstract
We investigate two related problems concerning the dimension of joint eigenspaces of the Laplace--Beltrami operator and a finite set of Hecke operators on $\mathbb{X}=\mathrm{PGL}_2(\mathbb{Z})\backslash \mathbb{H}$. First, we consider Linnik problem for Maass--Hecke cuspforms. We prove that the dimension of such a joint eigenspace, for Maass--Hecke cuspforms with eigenparameter in $[T, T+1]$, associated to Hecke operators $T_p$ with $p < (\log T)^α$ is $O_ε(T^{{\frac{4}α} + ε})$. For this, we prove a new form of spectral large sieve inequality for symmetric-squares of Maass--Hecke cuspforms, by exploiting the fact that the forms under consideration are unramified at every non-archimedean place. Second, we consider the effective multiplicity one problem, determining the minimal number of Hecke eigenvalues needed to distinguish two Maass--Hecke cusp forms with the same Laplace eigenvalue. We prove that for any fixed $η>0$, if two Maass--Hecke cuspforms, with eigenparameter $t$, share Hecke eigenvalues $λ_{φ_1}(n) = λ_{φ_2}(n)$ for all $n < ηt$, and $t$ is sufficiently large, then the forms are proportional. This improves the previously known best bound due to Huntley in 1991. Key ingredient for the improvement is the result by Brook and Lindenstrauss that classifies quantum limits of a joint eigenfunction of a Hecke operator and the Laplace--Beltrami operator on arithmetic hyperbolic surfaces. We also discuss generalizations of these results to Maass--Hecke cuspforms on $\mathrm{PGL}_2$ over arbitrary number field.