Density Hypothesis for Congruence Subgroups

• Density Hypothesis for Congruence Subgroups is a framework that quantitatively estimates automorphic representation multiplicities by extending prior spherical analyses.
• It employs specialized test functions in the Selberg trace formula to control non-tempered and non-spherical spectral contributions with uniform bounds.
• The method connects spectral gap control, arithmetic volume growth, and lattice properties, impacting automorphic forms and expander graph constructions.

The density hypothesis for congruence subgroups addresses the quantitative distribution of automorphic representations—specifically non-tempered and non-spherical types—across congruence covers of irreducible uniform lattices in $\mathrm{PSL}_2(\mathbb{R})^d$. The central aim is to establish uniform upper bounds on the multiplicities of such representations, thereby extending the scope of the earlier spherical density hypothesis to a broader set of spectral types. This is achieved by precise control with respect to both the level (i.e., the volume of the congruence cover) and key spectral parameters.

1. Formulation of the Density Hypothesis

Given an irreducible uniform arithmetic lattice $\Gamma \subset \mathrm{PSL}_2(\mathbb{R})^d$ and its congruence subgroups $\Gamma(\mathfrak{a})$, the density hypothesis predicts that for any family of non-tempered representations $\mathcal{B}$ (or for a single $\pi$), the sum of multiplicities in $L^2(\Gamma(\mathfrak{a})\setminus G)$ satisfies: $$(\mathcal{B}, \Gamma(\mathfrak{a})) \leq C\cdot\mathrm{vol}(\Gamma(\mathfrak{a}) \backslash G)^{2/p(\mathcal{B})+\varepsilon}$$ where $p(\mathcal{B})$ is the matrix coefficient integrability parameter ($p(\pi) = 2$ for tempered, $p(\pi) > 2$ for non-tempered), and $C$ depends only on $\Gamma$ and $\varepsilon$. This framework generalizes prior multiplicity bounds, relating spectral geometry to arithmetic volume growth. The key improvement is uniformity not only in level but also in spectral parameters.

2. Spectral Parameters and Representation Types

To precisely track representation contributions, the paper introduces a product parameter: $T(\pi) = \prod_{j=1}^d T(\pi_j)$ with

$$T(\pi_j) = \begin{cases} 1 & \text{if } \pi_j \simeq \pi_s,\, s\in(0,\tfrac{1}{2}) \ 1 + |r| & \text{if } \pi_j \simeq \pi_{1/2+ir} \ |m| & \text{if } \pi_j \simeq D_m \end{cases}$$

where $D_m$ is a discrete series of weight $m$, and $\pi_{1/2+ir}$ is a spherical principal series. The parameter $T(\pi)$ approximates the square root of the shifted Casimir eigenvalue and is crucial for bounding multiplicities when spectral parameters become large.

3. Extension to Non-Spherical Non-Tempered Representations

Previous density hypothesis results often focused on representations with trivial $K$-type (the spherical case). The present work develops test functions for the Selberg trace formula tailored to non-spherical cases, accommodating discrete series of arbitrary (possibly large) weight. These test functions are used to localize on exceptional complementary series and discrete series representations and to control non-tempered contributions uniformly.

The main result states: $$(\pi, \Gamma(\mathfrak{a})) \leq C \cdot (T(\pi) \cdot \mathrm{vol}(\Gamma(\mathfrak{a}) \backslash G))^{2/p(\pi)+\varepsilon}$$ This is achieved for all irreducible lattices and for congruence subgroups $\Gamma(\mathfrak{a})$ arising from arithmetic constructions (e.g., quaternion-algebraic models), confirming the hypothesis uniformly in both level and spectral parameter.

4. Methodological Innovations

• Test Function Design: Specialized test functions in the Selberg trace formula are constructed to concentrate on prescribed spectral types, not limited to the spherical situation.
• Spectral Localization: The approach estimates the density of representations with high weight or large spectral parameter, using polynomial bounds in $T(\pi)$.
• Orbital Integral Analysis: The non-spherical trace formula requires refined control of orbital integrals associated with non-trivial $K$-types, achieved via explicit uniform estimates.
• Uniformity and Level Growth: The bounds remain valid as the level increases, which is encoded by the volume term $\mathrm{vol}(\Gamma(\mathfrak{a}) \backslash G)$.

5. Applications and Consequences

The established bounds have several immediate implications:

• Spectral Gap Control: Uniform control of non-tempered and non-spherical multiplicities implies strong spectral gap results in sequences of congruence covers.
• Exceptional Eigenvalue Density: The proportion of non-tempered or exceptional eigenvalues does not grow faster than the indicated power of the volume, supporting conjectures regarding sparse exceptional spectra in arithmetic manifolds.
• Automorphic Forms and Expander Graphs: The density bounds directly impact equidistribution theorems, diophantine problems, and constructions of expander graphs; strong bounds enable optimal rates of convergence and mixing in associated dynamical systems.
• Generalization Beyond Spherical Case: The methods open the possibility for further extensions to other groups, including higher-rank and co-compact models, as well as for refined analysis of representation-theoretic and geometric quantities.

6. Mathematical Formulations

Key formulas from the paper include:

Representation Type	$T(\pi_j)$ Definition	Multiplicity Bound Formula
Spherical principal series	$1$ or $1 + |r|$	$$(\pi, \Gamma(\mathfrak{a})) \leq C(T(\pi) \cdot \mathrm{vol})^{2/p(\pi)+\varepsilon}$$
Discrete series	$|m|$	Same as above
Family of representations $\mathcal{B}$	$T(\mathcal{B}) = \max T(\pi)$	$$(\mathcal{B}, \Gamma(\mathfrak{a})) \leq C (T(\mathcal{B})^2 \cdot \mathrm{vol})^{2 / p(\mathcal{B}) + \varepsilon}$$

These explicitly quantify how the spectral and arithmetic inputs interact.

7. Significance within Automorphic Representation Theory

This work verifies the strong form of the Sarnak–Xue density hypothesis for congruence covers in $\mathrm{PSL}_2(\mathbb{R})^d$ beyond the spherical setting—all exceptional spectral types are controlled uniformly. The result unifies spectral, arithmetic, and geometric analysis of lattices and their congruence subgroups, and augments the foundation for future studies on multiplicity one, explicit spectral gap bounds, and the fine structure of automorphic spectra in higher dimensional locally symmetric spaces (Kelmer, 26 Sep 2025).