Spectral Theory of Kloosterman Sums

• Spectral theory of Kloosterman sums is the study of oscillatory exponential sums via trace formulas and harmonic analysis on locally symmetric spaces.
• It employs explicit decomposition techniques and integral transforms to link classical, half-integral, and higher-rank sums with precise spectral data.
• Applications include achieving power-saving estimates in prime geodesic theorems and bounding automorphic L-functions through refined analytic methods.

The spectral theory of Kloosterman sums encompasses their role as fundamental objects encoding the off-diagonal behavior in automorphic trace formulas, their explicit characterization via harmonic analysis on locally symmetric spaces, and the deep connections with subconvex bounds for automorphic $L$-functions. This theory has evolved to address single-variable classical sums, half-integral weight analogues, and higher-rank generalizations on groups such as $\mathrm{GL}_n$, $\mathrm{Sp}(2n)$, and congruence subgroups with complex modulus structure.

1. Kloosterman Sums in Classical and Half-Integral Weight Settings

The classical Kloosterman sum for $SL(2,\mathbb{Z})$ is defined as

$S(m,n;c) = \sum_{\substack{a\pmod{c}\(a,c)=1}} e\left(\frac{am+\overline{a}n}{c}\right),$

with $\overline{a}$ the inverse mod $c$. These sums arise naturally on the geometric side of the Kuznetsov trace formula, which expresses averages of automorphic Fourier coefficients in terms of weighted sums of Kloosterman sums.

Half-integral weight analogues, such as those studied in the Kohnen plus-space (see "Modular invariants for real quadratic fields and Kloosterman sums" (Andersen et al., 2018)), introduce the half-integral weight multiplier $\nu_\theta$ and impose arithmetic constraints on the exponents. The variant Kloosterman sum in the Kohnen plus-space for $k=\lambda + 1/2$ is

$$S_k^+(m,n;c) := e(-k/4) \sum_{d~(\bmod~c), (d,c)=1} \left(\frac{c}{d}\right) d^{2k} e\left(\frac{m\overline{d} + n d}{c}\right) \times \begin{cases} 1 & 8|c \ 2 & 4||c \end{cases}$$

where $(c/d)$ is the extended Kronecker symbol. These sums enter the Kuznetsov formula in the plus-space:

$$\sum_{4|c>0} \frac{S_k^+(m,n;c)}{c} ~ \phi \left( \frac{4\pi\sqrt{mn}}{c} \right) = \mathcal{M} + \mathcal{H} + \mathcal{E}$$

with explicit spectral terms $\mathcal{M}$ (Maass forms), $\mathcal{H}$ (holomorphic forms), and $\mathcal{E}$ (Eisenstein series) encoding the spectral decomposition.

2. Kuznetsov Trace Formula and Spectral Decomposition

The Kuznetsov formula provides the bridge between spectral data and sums of Kloosterman sums. For $SL(2,\mathbb{Z})$, the trace formula takes the form

$$\begin{aligned} &\sum_j \frac{\overline{\rho_j}(m) \rho_j(n)}{\cosh(\pi t_j)}~\widehat{\phi}(t_j) + \frac{1}{4\pi}\int_{-\infty}^{\infty} \frac{\tau(m,1/2+it) \tau(n,1/2+it)}{|\zeta(1+2it)|^2}~\widehat{\phi}(t)\,dt \ &= \delta_{m,n}~\widetilde{\phi}(0) + \sum_{c=1}^{\infty} \frac{S(m,n;c)}{c}~\phi\left(\frac{4\pi\sqrt{mn}}{c}\right) \end{aligned}$$

where spectral transforms of test functions, Bessel function kernels, and explicit weightings encode the harmonic analysis of $L^2$ automorphic forms (Balkanova et al., 2018).

For congruence subgroups and nebentypus, explicit formulas connect Kloosterman sums associated to Atkin-Lehner cusps and Fourier coefficients of Eisenstein series via generalized double-coset parametrizations and character twists (Kiral et al., 2017).

3. Higher-Rank Generalizations: $\mathrm{GL}_3$, $\mathrm{Sp}(2n)$, and Beyond

Kloosterman sums have deep generalizations for higher-rank groups, notably in the context of trace formulas for $\mathrm{GL}_3$ and $\mathrm{Sp}(2n)$. On $\mathrm{GL}_3$, the long Weyl element Kloosterman sum is

$S(m_1,m_2,n_1,n_2;D_1,D_2) = \sum_{\substack{ B_1,B_2(\bmod D_1,D_2)\C_1,C_2(\bmod D_1,D_2)\ (B_i,C_i,D_i)=1\ D_1C_2 + B_1B_2 + D_2C_1 \equiv 0~\bmod D_1D_2 }} e\left(\cdots\right)$

where the phase involves nontrivial congruence conditions. For coprime moduli, the sum factorizes into classical GL(2) Kloosterman products. Global decomposition into products of lower-rank Kloosterman sums enables best-possible bilinear bounds and spectral large sieve inequalities (Blomer et al., 2015).

Similarly, for symplectic groups $\mathrm{Sp}(2n)$, Felber's framework generalizes the Kloosterman sum to matrix moduli:

$$K_n(Q,T;C) = \sum_{g \in X(C)} \exp\left(2\pi i\operatorname{tr}(A C^{-1}Q + C^{-1}D T)\right),$$

with X(C) a double-coset space defined via prescribed matrices in $\mathrm{Sp}_{2n}$, and the sum appearing in the Petersson/Kuznetsov formula for Siegel modular forms (Felber, 18 Dec 2025).

Local and global decompositions for $\mathrm{Sp}(4)$ employ stratification by torus action and advanced stationary phase estimates (Adolphson–Sperber, Dąbrowski–Fisher) to extract nontrivial bounds for general Weyl elements, with power savings over the trivial bounds (Man, 2020).

4. Power-Saving and Subconvexity Bounds

Sums of Kloosterman sums are tightly controlled by spectral methods, often exceeding what is possible by character sum or Burgess-type bounds. For half-integral weight, Andersen–Duke establish

$$\sum_{4|c \le x} \frac{S_k^+(m,n;c)}{c} \ll x^{1/6} + (dd')^{2/9}(vw)^{1/3}~(mnx)^\varepsilon$$

using a delicate analysis involving the modular plus-space Kuznetsov formula, Waldspurger correspondence, and Young's hybrid subconvexity for twisted $L$-functions (Andersen et al., 2018).

For symplectic groups (e.g. $\mathrm{Sp}(2n)$),

$$K_n(Q,T;C) \ll_n c_1^{n - 1/2} (c_1,2Q,2T)^{3/2} \prod_{i=2}^n c_i^{n-i+1},$$

with error terms directly impacting rates of equidistribution in high-rank geometric applications (Felber, 18 Dec 2025). Man's bounds for $\mathrm{Sp}(4)$ provide local and global estimates of the form $p^{A_w(r,s)}$, with exponents smaller than the trivial $p^{r+s}$, using orbit stratification and stationary phase (Man, 2020).

In the classical setting, spectral methods allow explicit error estimations in number-theoretic theorems such as the prime geodesic theorem. Uniform subconvex estimates for relevant $L$-functions directly translate into power savings for Kloosterman sum averages (Balkanova et al., 2018).

5. Structural Decomposition and Analytic Techniques

Recent advances exploit explicit decomposition and parametrization of Kloosterman sets. Bott–Samelson–inspired stratifications split long-word Kloosterman sets into fine cells parametrized by products of classical sums and residue class analysis (Kıral et al., 2020). This simplifies analytic handling within the Bruggeman–Kuznetsov formulas and enables closed-form expressions for divisor sums—generalizing Ramanujan-type identities to higher-rank settings.

Analytic ingredients include:

• Mellin–Barnes integral representations of Whittaker functions for evaluating oscillatory weights in trace formulas (Buttcane, 2011).
• Mean-value and large sieve inequalities for spectral sums, optimized using hybrid archimedean and arithmetic methods (Young, 2015, Blomer et al., 2015).
• p-adic stationary phase and nondegeneracy criteria for bounding critical orbit contributions in group schemes over local fields (Man, 2020).

6. Applications and Outlook

Spectral theory of Kloosterman sums underpins error-term analysis in prime geodesic counting, nonvanishing of automorphic $L$-functions, and mass equidistribution phenomena in arithmetic quotients. The explicit control of Kloosterman sums—both in classical and higher-rank contexts—yields optimal or near-optimal spectral large sieve inequalities, which are pivotal for moment estimates, subconvexity, and beyond endoscopy proposals (Blomer et al., 2015, Cai et al., 4 May 2025, Balkanova et al., 2018, Felber, 18 Dec 2025).

Active directions include:

• Sharp square-root cancellation for higher-rank symplectic sums.
• The explicit development of Kuznetsov-type trace formulas for $\mathrm{Sp}(2n)$ and other reductive groups.
• Structural generalizations of fine decomposition techniques to $\mathrm{SL}_4$, $\mathrm{GL}_n$, and non-split forms (Kıral et al., 2020).

The regularity and cancellation phenomena inherent in Kloosterman sums, when viewed through the lens of harmonic analysis and spectral theory, remain a central theme in modern analytic and arithmetic research.