Matrix Kloosterman Sums: Theory & Applications

• Matrix Kloosterman sums are higher-dimensional analogues of classical Kloosterman sums that encapsulate intricate arithmetic and geometric structures in automorphic forms and number theory.
• They are evaluated through trace formulas and cohomological methods, yielding explicit eigenvalue spectra and sharp uniform bounds essential for analytic investigations.
• Applications range from random matrix theory and cryptography to spectral graph theory, making these sums pivotal in both theoretical research and practical computations.

Matrix Kloosterman sums are higher-dimensional analogues of the classical Kloosterman sums, appearing in the analytic theory of automorphic forms, exponential sum estimates, representation theory, and number theory. Extending from scalar sums over finite fields or modular rings to matrix or group-valued settings, matrix Kloosterman sums encapsulate intricate arithmetic and geometric structures connected to the representation theory of GLₙ over finite fields, cohomological methods, and random matrix theory.

1. Definitions and Variants

1.1 Classical and Matrix Kloosterman Sums

The classical (scalar) Kloosterman sum over a finite field 𝔽_q is defined as

$$K(a, b; q) = \sum_{x\in 𝔽_q^×} \psi(ax + b x^{-1}),$$

where $\psi: 𝔽_q \to ℂ^×$ is a nontrivial additive character. For modular arithmetic, the definition is analogous with $x \in (\mathbb{Z}/q\mathbb{Z})^×$.

The most common matrix analogues (editor's term: "GLₙ–Kloosterman sums") generalize this in several distinct yet closely related ways:

• Sum over matrices: For $A,B\in M_n(\mathbb{F}_q)$,

$$K_n(A,B) = \sum_{X\in GL_n(\mathbb{F}_q)} \psi(\operatorname{Tr}(A X + X^{-1}B)).$$

This sum is over invertible $n\times n$ matrices, with the trace function as argument (Erdélyi et al., 2021).

• Rank-$k$ twisted matrix Kloosterman sums: For $k\geq 1$ and fixed multiplicative characters $\chi_i: \mathbb{F}_q^×\to ℂ^×$,

$$K_n(x; \chi) = \sum_{g_1\cdots g_k = x} \left(\prod_{i=1}^k \chi_i(\det g_i)\right)\psi(\operatorname{Tr}(g_1 + \cdots + g_k)),$$

where the sum is over $k$-tuples of invertible matrices in $GL_n(\mathbb{F}_q)$ with product $x$ (Zelingher, 2023, Yang, 4 Jan 2026).

• Matrix Kloosterman sums modulo prime powers: For $A,B \in M_n(\mathbb{Z}/p^r\mathbb{Z})$,

$$K_n(p^r;A,B) = \sum_{X \in GL_n(\mathbb{Z}/p^r\mathbb{Z})} \exp\left( \frac{2\pi i}{p^r} \operatorname{Tr}(AX + X^{-1}B) \right)$$

(Erdélyi et al., 2022).

1.2 Group and Lie-Type Refinements

• Long-word Kloosterman sums in the context of $SL_n$ or higher rank groups use Bruhat and Weyl group combinatorics for parametrization (notably in the $SL_3$ case (Kıral et al., 2020)), and are crucial in the analysis of trace formulas for higher rank groups.
• Exotic matrix Kloosterman sums generalize further via Deligne–Katz–Shintani correspondences and evaluation at Frobenius orbits in group extensions (Zelingher, 8 Jul 2025).

2. Spectral Theory and Matrix Structure

The spectral analysis of matrix Kloosterman sums investigates the eigenvalues and algebraic behavior of structured matrices with Kloosterman sum entries.

For an explicitly constructed $q \times q$ matrix $B_{(q)} = (S(m,n; q))_{1\leq m,n\leq q}$, where $S(m,n;q)$ is the classical Kloosterman sum modulo $q$:

• The spectrum of $B_{(q)}$ consists of only three eigenvalues $\{0, +q, -q\}$, with explicit multiplicities depending on arithmetic invariants,

$$\text{mult}(0) = q - \varphi(q), \quad \text{mult}(+q) = \frac{\varphi(q) + \widetilde{\varphi}(q)}{2}, \quad \text{mult}(-q) = \frac{\varphi(q) - \widetilde{\varphi}(q)}{2},$$

where $\varphi(q)$ is Euler's totient function and $$\widetilde{\varphi}(q) = \#\{1\leq k\leq q: (k,q)=1, k^2\equiv -1 \bmod q\}$$ (Ushiroya, 2018).

• The spectral simplicity is underpinned by the convolution identity

$B_{(q)}^2 = q \cdot A_{(q)},$

where $A_{(q)}$ is the Ramanujan sum matrix, and leads to closed expressions for traces and characteristic polynomials. Explicit exponential vector eigenfunctions realizing these eigenvalues are constructed in terms of roots of unity.

For $n\geq1$, such structure can be generalized to block-diagonal, Jordan-normal, and more general conjugacy classes, leveraging the representation theory of $GL_n(\mathbb{F}_q)$ and symmetric functions (Yang, 4 Jan 2026).

3. Evaluation, Bounds, and Cohomological Purity

3.1 Explicit Evaluation

Through reduction to Jordan block structure and use of Green’s polynomials $Q_\lambda^\mu(t)$, matrix Kloosterman sums are evaluated in closed form:

$$K_{dr}(x; \chi) = (-1)^{(k-1)dr} q^{(k-1)dr(dr-1)/2} \sum_{\lambda\vdash r} \frac{Q_\lambda^\mu(q^d)}{z_\lambda}(-1)^{(k-1)\ell(\lambda)}\prod_{j=1}^{\ell(\lambda)} K_{d\lambda_j}(\chi; \xi),$$

where $x$ is a generalized Jordan matrix and $\xi$ the root of the characteristic polynomial (Zelingher, 2023, Yang, 4 Jan 2026).

3.2 Bounds

Sharp uniform bounds for matrix Kloosterman sums have been established:

• For regular semisimple $a\in GL_n(\mathbb{F}_q)$,

$|K_n(a)| \leq 2^n q^{n^2/2}$

(Erdélyi et al., 2021).

• Under genericity and regularity, matrix Kloosterman sums modulo $p^k$ admit "square-root" bounds,

$|K_n(p^k;A,B)| \ll_n p^{kn^2/2},$

generalizing Weil's bound for $n=1$ (Erdélyi et al., 2022).

3.3 Purity and Cohomological Interpretation

Matrix Kloosterman sums are cohomologically interpreted via the trace formula as the Frobenius action on étale cohomology groups attached to certain Artin–Schreier sheaves. For $a\in GL_n(\mathbb{F}_q)$:

• The compactly supported cohomology $H^i_c$ of the associated local systems is "pure" of arithmetic weight, guaranteeing no cancellation between cohomological degrees (Erdélyi et al., 2022).
• The precise structure of this cohomology underlies square-root cancellation and enables geometric decompositions of the parameter space according to spectral data.

4. Representation Theory, Symmetric Functions, and Special Values

Matrix Kloosterman sums are closely related to other analytic and algebraic quantities via the apparatus of symmetric function theory:

• Their connection to Hall–Littlewood and modified Hall–Littlewood polynomials forms a bridge between exponential sum theory and symmetric function identities (Zelingher, 2023, Zelingher, 8 Jul 2025).
• Via the trace formula and the explicit Frobenius eigenvalues appearing in Kloosterman sheaves (Katz, Deligne), matrix Kloosterman sums are evaluated in terms of symmetric functions of these eigenvalues.
• Their multiplicative and direct-sum properties extend naturally from representation-theoretic decompositions into orbits and Jordan types.

In the context of nonabelian ("exotic") generalizations and Bessel–Speh representations, these sums realize explicit identities relating spectral data of automorphic representations to values of matrix Kloosterman sums (Zelingher, 8 Jul 2025).

5. Advanced Applications and Distributional Phenomena

5.1 Analytic Number Theory and Automorphic Forms

Matrix Kloosterman sums are central in:

• The spectral theory of automorphic forms for $GL_n$ and the analytic study of trace formulas, including the Bruggeman–Kuznetsov formula for $SL_3(\mathbb{Z})$ (Kıral et al., 2020, Buttcane, 2011).
• Quantum ergodicity and rates of equidistribution for linear toral mappings, with matrix-valued sums allowing for nontrivial sub-square-root cancellation (Ostafe et al., 2021).
• Explicit constructions of Ramanujan graphs and multigraphs, via adjacency matrices whose nontrivial eigenvalues are Kloosterman sums, realizing combinatorial and spectral analogues of deep number-theoretic conjectures (Fleming et al., 2010).

5.2 Random Matrix Statistics and Cryptography

Recent work clarifies that, when normalized, the values of matrix Kloosterman sums become equidistributed in the sense of Sato–Tate—their traces, or higher symmetric functions, match those of Haar-distributed elements in compact Lie groups such as $SU(n)$ or $Sp(2n)$. This underpins:

• Statistical testing for cryptography by detecting algebraic bias in pseudorandom sequences, outperforming standard frequency and run-based tests (Yang, 4 Jan 2026).
• A spectral framework for distinguishing random versus highly structured outputs in cryptographic settings.

5.3 Exponential Sums Modulo Prime Powers

In the arithmetic setting $\mathbb{Z}/p^r\mathbb{Z}$, matrix Kloosterman sums encode information on solution sets to modular equations, with optimal bounds depending subtly on the rank and regularity of the argument matrices. This enables their use in effective equidistribution problems on expanding horospheres (Erdélyi et al., 2022).

6. Connections to Higher Rank Groups and Trace Formulas

For $SL_n$, the theory of matrix Kloosterman sums generalizes via Weyl group actions and Bruhat decompositions:

• In $SL_3$, fine stratifications yield decompositions of long-word Kloosterman sums into products of classical Kloosterman sums, facilitating explicit evaluation in the geometric side of trace formulas (Kıral et al., 2020, Buttcane, 2011).
• Analytic machinery such as the Mellin–Barnes inversion and the Kuznetsov formula provides nontrivial cancellation and error term analysis, essential for mean-square and subconvexity problems in automorphic $L$-functions.

7. Open Problems and Future Directions

• Efficient computational approaches to Green’s polynomials for large $n$ and their role in evaluating matrix Kloosterman sums (Yang, 4 Jan 2026).
• Full characterization of Sato–Tate equidistribution for general conjugacy/Jordan types.
• Further exploration of matrix Kloosterman sum bounds in post-quantum cryptographic settings and their correlation structure across different block types.
• Applications to explicit enumeration and asymptotics in zero-trace invertible matrices, and further combinatorial applications in matrix rings (Li et al., 2011).
• The extension of these techniques and phenomena to more general reductive groups and moduli spaces.

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Key References:

• (Ushiroya, 2018) Eigenvalues of Matrices whose Elements are Ramanujan Sums or Kloosterman Sums
• (Erdélyi et al., 2021) Matrix Kloosterman sums
• (Erdélyi et al., 2022) The purity locus of matrix Kloosterman sums
• (Yang, 4 Jan 2026) Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography
• (Zelingher, 2023) On matrix Kloosterman sums and Hall-Littlewood polynomials
• (Erdélyi et al., 2022) Matrix Kloosterman sums modulo prime powers
• (Zelingher, 8 Jul 2025) On exotic matrix exponential sums and Bessel-Speh functions
• (Li et al., 2011) Gauss sums over some matrix groups
• (Ostafe et al., 2021) Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity
• (Kıral et al., 2020, Buttcane, 2011) Parametrizations, spectral theory, and trace formulas for higher-rank Kloosterman sums
• (Fleming et al., 2010) Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs