Large Sieve for Exceptional Cusp Forms

• Large sieve inequalities provide analytic bounds controlling weighted spectral averages of Fourier coefficients or Hecke eigenvalues for exceptional cusp forms.
• Trace formulas and decompositions of Kloosterman sums are key methodologies used to address challenges in the non-tempered or complementary spectrum.
• Recent advances yield optimal bounds that improve sieve applications in prime factorization, shifted convolution problems, and arithmetic distribution.

Large sieve inequalities for exceptional cusp forms encompass a spectrum of results in analytic number theory, where the focus lies on bounding averages of Fourier coefficients or Hecke eigenvalues over families of automorphic forms possessing small Laplace eigenvalues or, more generally, lying in the "exceptional" range of the automorphic spectrum. These inequalities play a crucial role in addressing obstructions arising from forms violating generalized Ramanujan conjectures, as well as in bounding bilinear and multilinear (e.g., Kloosterman sum) forms associated with trace formulas. Recent advances provide best-possible bounds for such exceptional spectra, refine the treatment of the non-tempered regime, and connect to optimal sieve methods and deep applications in prime factorization and distribution problems.

1. Definition and Fundamental Principles

Large sieve inequalities are analytic bounds that control weighted spectral averages of automorphic Fourier coefficients or Hecke eigenvalues, typically associated with automorphic forms on reductive groups such as GL(2), GL(3), or PGL_2 over number or function fields. In the context of exceptional cusp forms, these inequalities specifically address contributions from the portion of the spectrum corresponding to small Laplace eigenvalues (non-tempered, or "complementary series") which frequently create analytic obstacles in estimating exponential sums and moments.

A prototypical spectral large sieve for GL(3) (Theorem 2 of [Blomer–Buttcane]) takes the form: $$\int_{T\,Ω} \frac{1}{\mathcal{N}(\pi)} \Bigl|\sum_{N\leq n\leq 2N} a_n\,A_\pi(n,1)\Bigr|^2 d\pi \ll_{ε,Ω} (T^5 + T^2 N)^{1+ε} \sum_{N\leq n\leq 2N}|a_n|^2,$$ where the integration runs over a suitable spectral set $Ω$, $A_\pi(n,1)$ are normalized Hecke eigenvalues, and $\mathcal{N}(\pi)$ is an adjoint $L$-function factor, both for cusp forms and Eisenstein series. The terms $T^5$ and $T^2 N$ represent, respectively, the harmonic volume and the analytic length in the underlying Kuznetsov formula (Blomer et al., 2015).

In the setting of forms for congruence subgroups or over number fields, the large sieve targets the exceptional (complementary series) spectrum, as encapsulated by the Laplace eigenparameter $v_V$ in

$$\sum_{V(v_V,p_V)=(v,0)} X^{v_V}\Bigl|\sum_{N/2<|n|^2\le N} b_n\,c^a_V(n;v,0)\Bigr|^2 \ll_\varepsilon (1+X\,M_a\,N)^{\vartheta(q)} (1+O_\varepsilon(M_a\,N^{1+\varepsilon}))^{1-\vartheta(q)} |b|_N^2 \ln(2+M_aN),$$

where $\vartheta(q)\leq 2/9$ by the Kim–Shahidi bound (Watt, 2013).

2. Spectral Families and Exceptional Range

The exceptional spectrum of cusp forms refers to those automorphic representations with Laplace eigenvalues $\lambda_j \in (0,1/4)$, equivalently spectral parameter $\theta_j = \sqrt{1 - 4\lambda_j} > 0$, violating the generalized Ramanujan conjecture or Selberg eigenvalue conjecture. While most results are formulated for generic, tempered families, large sieve inequalities for exceptional forms must incorporate the delicate analysis of the complementary series, where archimedean (or non-archimedean) parameters approach the walls of the Weyl chamber or real line.

In GL(3), the relevant spectrum treated by (Blomer et al., 2015) includes:

• Genuine GL(3) Hecke–Maaß forms with spectral parameters $\boldsymbol{\mu} \sim i\mathbb{R}^3$.
• Maximal Eisenstein series attached to SL(2) forms.
• Minimal and degenerate Eisenstein spectra.
• Exceptional forms for which some $\mu_i$ are partially real, yielding small Laplace eigenvalues.

In the context of Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$, the large sieve sums are split between principal- and complementary-series, with the latter controlled via Kim–Shahidi's upper bound $\vartheta(q)\leq 2/9$, serving as an effective substitute for Selberg's conjecture (Watt, 2013). For PGL$_2(\mathbb{Z}[i])$, new advances yield optimal square-region large sieve inequalities, sharply treating small-parameter ("exceptional") forms (Qi, 13 Apr 2024).

3. Methodologies: Trace Formulas, Kloosterman Decomposition, and Frequency Structure

Spectral large sieve inequalities for exceptional cusp forms are derived via analytic trace formulas, most prominently the Kuznetsov formula, relating spectral and arithmetic (Kloosterman sum) data. The optimal bounds on the arithmetic side hinge on deep decompositions of higher-rank Kloosterman sums and precise analysis of integral transforms.

For GL(3), the driving engine is the decomposition of the long Weyl element Kloosterman sum $S(1,m,n,1;D_1,D_2)$ as a finite sum of products of classical GL(2) Kloosterman and Ramanujan sums (Theorem 5 in (Blomer et al., 2015)), leading to the best-possible bilinear bound: $$\sum_{D_1\leq X_1, D_2\leq X_2} \sum_{m,n\leq N} a_m\,\overline{b_n}\,S(1,m,n,1;D_1,D_2) \ll_\varepsilon (X_1 X_2)^\varepsilon \|a\|_2\,\|b\|_2 \left[X_1 X_2(X_1+X_2) + (N X_1 X_2)^{1/2}(X_1+X_2)^{3/2} + N X_1 X_2\right].$$ This arithmetic decomposition enables sharp spectral bounds upon coupling with Mellin-transform truncation and hybrid large sieve estimates.

Frequency-concentration and "sparse Fourier transform" phenomena allow new bounds for exceptional Maass forms. When the arithmetic weight sequence $(a_n)$ exhibits concentration in Fourier space—quantified via a representing measure $\mu$ with small mass and Diophantine integral—the large sieve saving parameter $X$ can be taken much larger than classical limits, yielding critical-range improvements for bilinear and multilinear forms (Pascadi, 5 Apr 2024). This is reflected in inequalities of the type: $$\sum_{\lambda_j<1/4} X^{\theta_j} \Bigl|\sum_{n\sim N}a_n\,\rho_{ja}(an)\Bigr|^2 \ll (qaNX)^{\varepsilon}(qaN)(1+aN/q)A^2$$ for appropriate $A$, depending on the concentration properties of $\mu$.

4. Role of Exceptional Spectra and Eigenvalue Bounds

The exceptional or complementary series spectrum represents the main analytic obstruction in achieving power-saving in moments of exponential sums or automorphic $L$-functions. In classical settings, the existence of small eigenvalues limits the allowable length and saving in large sieve inequalities. The Kim–Sarnak bound provides an unconditional limit $\theta_j \leq 7/32$ for Maass forms, while Kim–Shahidi offer $\vartheta(q)\leq 2/9$ for complex forms, allowing effective, though not optimal, control of "exceptional" contributions (Watt, 2013, Pascadi, 5 Apr 2024). If Selberg’s conjecture were true ($\vartheta(q)=0$), all complementary-series terms would vanish.

Modern approaches, such as those in (Pascadi, 5 Apr 2024), use weight sequences with concentrated Fourier support to exploit cancellation beyond traditional methods, yielding new large sieve bounds that propagate through trace formulas to significant improvements in sieve-theoretic applications (e.g., greatest prime factor estimates for $n^2+1$).

5. Key Analytic Tools and Bridging Propositions

Major technical ingredients in the derivation and application of these large sieve inequalities include:

• Global decompositions of higher-rank Kloosterman sums into products of lower-rank objects (Blomer et al., 2015).
• Precise truncation of two-dimensional Mellin transforms to restrict parameter regions contributing in integral transforms.
• Hybrid large sieve bounds for Dirichlet-type sums, essential in bridging spectral and arithmetic sides (Lemma 1, (Blomer et al., 2015)).
• Measure-theoretic representations of weight sequences to accommodate sparse Fourier structure, enabling greater saving parameters $X$ (Pascadi, 5 Apr 2024).
• Innovations in complex Bessel kernel representations that sharpen off-diagonal estimates and absorption of small-parameter contributions (Qi, 13 Apr 2024).

The table below summarizes salient spectral large sieve inequalities involving exceptional forms:

Group/Setting	Exceptional Range	Large Sieve Bound/Term
GL(3) (Blomer-Buttcane) (Blomer et al., 2015)	$\mu$ not purely imaginary	$(T^5+T^2N)^{1+\epsilon}\sum |a_n|^2$ over spectrum $\mu\in T\Omega$
$\mathrm{PGL}_2(\mathbb{Z}[i])$ (Qi) (Qi, 13 Apr 2024)	$(k,p)$ small	$(K^4+N^2)N^{\epsilon}\sum |a_N|^2$ on square $|k|,|p|\leq K$
SL(2), Maass exceptional (Watt) (Watt, 2013)	$v\in (0,2/9)$	$(1+X\,M_a\,N)^{\vartheta(q)}$ weighted by $X^{v_V}$
Maass forms, sparse weights (Pascadi, 5 Apr 2024)	$\theta_j>0$	$(qaNX)^{\epsilon}(qaN)(1+aN/q)A^2$ for $X$ per measure $\mu$

6. Applications and Significance in Analytic Number Theory

Large sieve inequalities for exceptional cusp forms have yielded advances in bounding bilinear and multilinear forms in Kloosterman sums, crucial for shifted-convolution problems and high-moment bounds of automorphic $L$-functions. They underpin sieve-theoretic results such as improved thresholds for the greatest prime factor of $n^2+1$, applications to exponent of distribution for primes and smooth numbers in arithmetic progressions, and push past classical bottlenecks when employing weights with spectral concentration (Pascadi, 5 Apr 2024).

Their best-possible nature, demonstrated for GL(3) and cubic metaplectic settings (Blomer et al., 2015, Dunn, 19 Mar 2024), affirms that reduction in analytic obstructions from exceptional forms is feasible under optimal arithmetic decompositions and hybrid analytic-arithmetic techniques.

7. Innovations and Current Research Directions

Recent studies have introduced:

• Concise and absolutely convergent double-integral expressions for complex Bessel transforms, enabling sharper off-diagonal sum estimates and optimal square-parameter bounds (Qi, 13 Apr 2024).
• Measure-based large sieve inequalities leveraging sparse or concentrated Fourier transforms, unlocking previously inaccessible saving regimes for exceptional Maass forms (Pascadi, 5 Apr 2024).
• Comprehensive decomposition and factorization techniques translating higher-rank arithmetic sums into amenable lower-rank objects with known bounds (Blomer et al., 2015).

A plausible implication is that further progress in large sieve inequalities, particularly regarding exceptional cusp forms, demands deeper arithmetical analysis of trace formula constituents, balanced analytic techniques that exploit spectral sparsity, and ongoing refinement of eigenvalue bounds in automorphic spectra. These advances continue to impact Diophantine applications, distribution questions, and the overall understanding of spectrum-arithmetic interplay in modern analytic number theory.