Hecke-Maass Cusp Form

Updated 20 October 2025

Hecke-Maass cusp forms are real-analytic eigenfunctions of the hyperbolic Laplacian on arithmetic quotients that generalize classical modular forms.
They are analyzed using amplification techniques and lattice point counting to derive sup-norm bounds and understand mass distribution.
Their study advances insights into subconvexity, quantum chaos, and automorphic L-functions through trace formulas and computational methods.

A Hecke-Maass cusp form is a real-analytic eigenfunction of the hyperbolic Laplacian on arithmetic quotients of the upper half-plane (or higher-rank symmetric spaces), invariant under a congruence subgroup, and simultaneously a Hecke eigenfunction for almost all Hecke operators. These nonholomorphic automorphic forms generalize classical modular forms and play a central role in the spectral theory of automorphic representations and analytic number theory. Their spectral and arithmetic properties, including eigenvalue distributions, sup-norm bounds, and quantum statistical behavior, are the subject of deep analytic and arithmetic investigations, and underpin advances in the subconvexity problem, quantum chaos, and the theory of automorphic $L$ -functions.

1. Definition and Basic Properties

Let $X_0(N) = \Gamma_0(N)\backslash\mathbb{H}$ , where $\Gamma_0(N) \subseteq \mathrm{SL}_2(\mathbb{Z})$ is a congruence subgroup and $\mathbb{H}$ is the upper half-plane. A Maass cusp form $f$ for level $N$ is a smooth, real- or complex-valued function on $X_0(N)$ satisfying:

Automorphy: $f(\gamma z) = f(z)$ for all $\gamma \in \Gamma_0(N)$ ,
Laplacian eigenvalue: $\Delta f = \lambda f$ where $\Delta = -y^2(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})$ and $\lambda = \frac14 + t^2$ for some $t \in \mathbb{R}$ ("spectral parameter"),
Cuspidality: The constant term in the Fourier expansion of $f$ at each cusp vanishes,
Hecke eigenfunction: $f$ is an eigenfunction for the (commuting) family of Hecke operators $T_n$ , with eigenvalues $\lambda_f(n)$ for $(n,N)=1$ .

The normalized Hecke-Maass cusp forms are typically $L^2$ -normalized: $\|f\|_2 = 1$ .

2. Supremum Norms and Distribution of Mass

Quantification of the sup-norm (maximum modulus) of Hecke–Maass cusp forms as a function of geometric (level) or spectral parameters reveals the distribution of "mass" and reflects arithmetic complexity. For an $L^2$ -normalized Hecke–Maass cusp newform $f$ of square-free level $N$ and Laplacian eigenvalue $\lambda$ , the optimal available bound in the level aspect is

$\|f\|_\infty \ll_{\lambda, \varepsilon} N^{-1/6+\varepsilon}$

for any $\varepsilon>0$ (Harcos et al., 2011). This result, achieved via amplification techniques and geometric (lattice point counting) arguments, indicates that the mass of $f$ becomes more evenly distributed ("spread out") on $X_0(N)$ as $N \rightarrow \infty$ .

The bound is analogous in spirit to the Weyl or Burgess exponents for subconvexity problems in $L$ -function theory, indicating natural analytic barriers. The level aspect exponent of $-1/6$ emerges via refined counting of lattice points satisfying arithmetic and geometric constraints, and reflects the interaction between the underlying arithmetic (as encoded in the Hecke algebra and level structure) and the geometry of the modular surface.

Key ingredients:

Amplification method (from the Iwaniec–Sarnak eigenvalue aspect, but in level aspect), constructing weighted sums using Hecke operators to focus analytic mass,
Precise lattice point counting in the upper half-plane, utilizing uniform lower bounds for $\operatorname{Im}(z) \gg N^{-1}$ , enabled by the square-free assumption and Atkin–Lehner theory,
Lattice counting bounds: for a Euclidean lattice $M$ with minima $\lambda_1 \leq \lambda_2$ and disc of radius $R$ , $\#(M \cap D) \ll 1 + \frac{R}{\lambda_1} + \frac{R^2}{\lambda_1 \lambda_2}$ ,
Disentangling matrix types (upper-triangular, parabolic, general) and, for parabolic cases, reduction to Pell-type equations.

3. Hecke Eigenvalues, Ramanujan Conjecture, and Distribution

The arithmetic content of Hecke-Maass cusp forms is encapsulated by their Hecke eigenvalues $\lambda_f(n)$ , which are the Fourier coefficients in the expansion of $f(z)$ along the real line and encode substantial analytic and arithmetic information.

For a primitive Hecke-Maass cusp form $\phi$ of level $N$ and Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$ , there exists a prime $p \nmid N$ such that the Satake parameters $(\alpha_p, \beta_p)$ satisfy $|\alpha_p|=|\beta_p|=1$ , i.e., the Ramanujan conjecture holds locally at $p$ , and

$p \ll (N(1+|t_{\phi}|))^{0.27332}$

(Luo et al., 2014). This explicit upper bound for the least Ramanujan prime strengthens classical (ineffective) results. Moreover, the set of unramified Ramanujan primes for a fixed $\phi$ has natural density at least $34/35$.

Probability and distributional results for $\lambda_f(n)$ and their incidence of small or large values are numerous:

Infinitely many primes $p$ with $|\lambda_f(p)| < 1$ and infinitely many with $|\lambda_f(p)| > \sqrt{2}$ (Kumari et al., 2022).
For the sequence of coefficients $\{A(m,1)\}$ of self-dual Maass cusp forms for $\mathrm{GL}_3$ , sign changes occur with frequency at least $X^{5/6-\varepsilon}$ in $m \leq X$ under standard $L$ -function subconvexity assumptions (Jääsaari, 2022).
Asymptotic and density bounds for small and large eigenvalues, as well as Omega results for large oscillations (Kumari et al., 2022).

4. Quantum Limits, Moments, and Value Distribution

Hecke-Maass cusp forms are central to quantum unique ergodicity (QUE) and quantum chaos research. The quantum ergodicity theorem, in this context, asserts that almost all high energy (large eigenvalue) Hecke–Maass forms become equidistributed in their $L^2$ -mass distribution. Conditional on approaches such as the generalized Lindelöf hypothesis (GLH), the fourth moment (Gaussian value) is attained with a power-saving error: $\|f\|_4^4 = (2\pi/3) + O(T^{-\delta})$ where $T$ is the (large) spectral parameter (Buttcane et al., 2016). This supports the Random Wave Conjecture (RWC), predicting that high-energy eigenfunctions behave like Gaussian random waves; moments (both diagonal and joint) converge to those of independent normal random variables (Hua et al., 2 May 2024).

Vanishing of smooth cubic moments (with polynomial decay rate $T^{-1/12+\epsilon}$ ) is established for Hecke–Maass cusp forms, indicating that odd-order central moments (for instance, the third moment) vanish asymptotically (Huang, 2022). The analysis relies critically on approximate functional equations for triple product and Rankin--Selberg $L$ -values, application of Watson's formula, the Kuznetsov trace formula, and sharp moment estimates for $L$ -functions. For Eisenstein series, fourth and cubic moment asymptotics have been established via regularization and truncation, with main term behavior matching X the Gaussian moment, illuminating the continuous spectrum's quantum statistics (Huang et al., 14 Oct 2025).

Conditional results confirm that higher moments and joint moments for distinct, orthogonal Hecke–Maass cusp forms factorize as in independent Gaussian variables, lending further support for the random wave and statistical independence conjectures (Hua et al., 2 May 2024, Guo, 6 Oct 2024).

5. Computational Aspects and Explicit Constructions

Rigorous computation of Hecke–Maass cusp forms has seen significant algorithmic advancements. For squarefree level and trivial character, explicit versions of the Selberg trace formula with Hecke operators, as in Strömbergsson's approach, allow one to construct quadratic forms encoding spectral data and numerically solve associated generalized eigenvalue problems (Seymour-Howell, 2022). This yields both Laplace and Hecke eigenvalues with explicit error control, enabling large scale computation and verification of spectral properties such as Poissonian spacing, Ramanujan–Petersson bounds, and Sato–Tate distribution.

Lifting constructions provide explicit non-tempered examples in orthogonal groups: Maass cusp forms can be lifted via generalized theta lifts (analyzed via Borcherds' expansion) to higher rank orthogonal groups $O(1,8n+1)$ , resulting in globally irreducible, Hecke-eigen, cusp forms whose local non-archimedean components are non-tempered, and whose standard $L$ -functions factor as products of symmetric squares of the underlying $\mathrm{GL}_2$ cusp form and shifted Riemann zeta functions (Li et al., 2018).

6. Multiplicity One, Uniqueness, and Large Sieve Inequalities

Effective multiplicity one theorems for Hecke–Maass cusp forms connect the analytic uniqueness of the eigenfunctions to their arithmetic data. If two forms of eigenparameter $t$ agree on all Hecke eigenvalues $\lambda(n)$ for $n < \eta t$ , then they are proportional for sufficiently large $t$ and any fixed $\eta > 0$ (Jung et al., 22 Feb 2025). The proof relies on refined spectral large sieve inequalities for symmetric square lifts and the measure-classification result (quantum unique ergodicity for joint eigenfunctions of Laplace and finitely many Hecke operators) established by Brook and Lindenstrauss.

Regarding the Linnik problem, the paper shows that the dimension of the joint eigenspace for Maass–Hecke cuspforms with eigenparameter in $[T, T+1]$ , associated to Hecke operators $T_p$ with $p < (\log T)^\alpha$ , is $O_\epsilon(T^{4/\alpha+\epsilon})$ . This is achieved via new spectral large sieve inequalities for symmetric square $L$ -functions and combinatorial amplification arguments using the multiplicativity of Hecke eigenvalues. Extensions to general number fields are also established, with analogous dimension bounds and effective uniqueness criteria.

7. Higher Rank and Generalizations

Hecke–Maass cusp forms have natural extensions to higher rank, such as $\mathrm{GL}_n$ , where analogous analytic and arithmetic phenomena persist (with new geometric and combinatorial complexity). Results for sup-norms in the eigenvalue aspect for $\mathrm{SL}_3$ Hecke–Maass forms achieve power-saving exponents over the generic convexity barrier, deploying an amplified pre-trace formula, Weyl chamber analysis, and counting of matrices with prescribed determinantal divisors (Holowinsky et al., 2014).

Spectral Kuznetsov trace formulas and relative trace formulas on higher rank groups (Tsuzuki, 2021) enable the paper of automorphic periods, nonvanishing $L$ -values, and spectral averages in the level aspect, distinguishing automorphic representations via analytic invariants.

Advances in explicit Voronoi summation formulae at general cusps (using the local theory of $p$ -adic Whittaker functions and adelic methods) have provided new explicit bounds for sums of Fourier coefficients and illuminated Atkin–Lehner relations (Assing et al., 2019).

Hecke–Maass cusp forms are foundational to modern analytic number theory and automorphic representation theory, serving as the analytic avatars of deep arithmetic information, and their sup-norms, eigenvalue statistics, and quantum statistical properties continue to guide developments at the frontier of the Langlands program and quantum chaos.