Amplification via Non-Abelian Characters

• The paper presents an amplification argument using non-abelian characters to achieve a power-saving of c⁻¹/₁₂ in bilinear Kloosterman sum bounds, breaking classical limits in analytic number theory.
• The technique leverages the representation theory of SL₂(Z/cZ) through character orthogonality and combinatorial group estimates to refine results beyond traditional Weil and Fourier bounds.
• This method not only sharpens L-function moment asymptotics and large sieve inequalities but also paves the way for constructing infinite-dimensional, traceable non-abelian representations.

An amplification argument with non-abelian characters is an analytic technique leveraging the representation theory of non-abelian finite groups, specifically through character orthogonality and convolution, to obtain power-saving bounds in bilinear sums, especially when the natural Fourier methods stagnate. Recent advances show the utility of these methods in bounding Type II (bilinear) sums involving Kloosterman sums at composite moduli, breaking longstanding barriers in analytic number theory, with ramifications for $L$-function moment asymptotics and large sieve inequalities.

1. Context: Bilinear Kloosterman Sums and Classical Barriers

Type II (bilinear) sums of Kloosterman sums,

$$S = \sum_{m \in I}\sum_{n \in J} \alpha_m \beta_n \; \mathrm{Kl}(am, n; c),$$

with $$\mathrm{Kl}(m, n; c) = \sum_{x \in (\mathbb{Z}/c\mathbb{Z})^\times} e\left( \frac{m x + n x^{-1}}{c} \right)$$, are central in analytic number theory, notably in studies of automorphic $L$-functions and equidistribution. Previous approaches rely either on the Weil bound ($\mathrm{Kl}(\cdot) \ll \sqrt{c}$) or Fourier analysis on the abelian group $(\mathbb{Z}/c\mathbb{Z})^\times$ (Pólya–Vinogradov), which yield “trivial” bounds of order $\min(c, \sqrt{MNc})$ up to $c^{o(1)}$. When $MN \approx c$, surpassing these bounds has been classically difficult, especially for composite $c$ (notably near-primes $pq$, squares $p^2$). The recent non-abelian amplification argument saves a power $c^{-1/12}$ at critical range $M, N \asymp \sqrt{c}$ for such $c$, pushing beyond previous limits (Pascadi, 11 Nov 2025).

2. Non-Abelian Group-Theoretic Framework

The amplification builds on Fourier analysis over the non-abelian group $G = \mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$, which acts by Möbius transformations on the projective line $\mathbb{P}^1(\mathbb{Z}/c\mathbb{Z})$. The permutation representation

$$\rho_c: G \to \mathrm{U}(V_c), \quad V_c = \mathbb{C}^{\mathbb{P}^1(\mathbb{Z}/c\mathbb{Z})},$$

possesses character $$\chi_c(g) = \#\{ u \in \mathbb{P}^1: g\cdot u = u \}$$. The relevant subrepresentation, after orthogonal projection by Möbius inversion to “sift” coprimality and oldforms, is

$\rho_c^\circ = \sum_{d \mid c} \mu(c/d) P_c(d),$

with $P_c(d)$ the projection to functions constant on $\Gamma_c(d)$-orbits. This $\rho_c^\circ$ is of dimension $\asymp c$ and decomposes into irreducibles with dimensions $\gg c^{1-o(1)}$.

3. Amplifier Construction and Character Orthogonality

With smoothing and abelian Fourier transforms in $m, n$, the Kloosterman matrix is reduced to an operator norm $\| F(\rho_c^\circ) \|$, for

$$F(g) = \frac{1}{H_1H_2} \sum_{|h_1| \leq H_1, |h_2| \leq H_2} \alpha_{h_1} \beta_{h_2} \mathbf{1}_{g=T^{h_1} S T^{h_2}},$$

where $T, S$ are standard generators. The non-abelian amplifier is inserted as follows:

$A(\rho') = \sum_{n \in N} \chi' (n) \chi(n),$

where $N \triangleleft G$ is a normal subgroup (typically $N = \Gamma_c(d)$), and $\chi', \chi$ are traces of irreducible representations. By exploiting character orthogonality and positivity, for even $q$,

$$\|F(\rho)\|_{S^q}^q \leq \frac{|G|}{ \sum_{n \in N} |\chi(n)|^2 } \sum_{g_1 \dots g_q \in N} F(g_1) F(g_2^{-1}) \cdots F(g_q^{-1}) \chi(g_1\cdots g_q).$$

Irreducibility and subgroup normality underpin the bound, reducing the problem to explicit evaluations on $N$ and combinatorial counts in $\mathrm{PSL}_2(\mathbb{Z}/d\mathbb{Z})$.

4. Analytical Ingredients and Combinatorial Estimates

Three principal analytic inputs:

• Character sum over $\Gamma_c(d)$: By Clifford theory, for $\chi \subset \rho_c^\circ$,

$$\sum_{n \in \Gamma_c(d)} |\chi(n)|^2 \gg c^{3-o(1)}/d,$$

delivering a large denominator for the amplifier.

• Pointwise bounds for characters: For $g \in (\pm 1)\cdot \Gamma_c(d)$, $|\chi_c(g)| \ll f$, with $f^2 \mid cd$, $f \leq \sqrt{cd}$; so typically large on-diagonal, small off-diagonal.
• Solution counting in $\mathrm{PSL}_2(\mathbb{Z}/d\mathbb{Z})$: For $q=6$,

$$\# \{ (h_1, \dots, h_6): T^{h_1}S\cdots T^{h_6}S = I \} \ll H_2^2 + \frac{(H_1 H_2)^2}{d},$$

saving a factor of $d$ over the naive $H_1^3 H_2^3$ total.

5. Main Bilinear Bounds and their Extensions

Merging the preceding arguments yields for intervals $I, J$ of lengths $M, N \leq c$, with $(m, n, c) = 1$,

$$\sum_{m \in I,\, n \in J} \alpha_m \beta_n \mathrm{Kl}(a m, n; c) \ll \| \alpha \| \| \beta \|\, c^{1+o(1)} \left( \frac{d M^3 N}{c^3} + \frac{f M^2}{c^2} + \frac{f}{d^2} \right)^{1/6}.$$

For critical $c = p^2, pq$ (with $p \asymp q$, $d = \sqrt{c}$, $f = d$), this gives a saving of $c^{-1/12}$ for $M=N=\sqrt{c}$. For general modulus $c$, employing hybrid arguments (gluing results at prime factors), the bound improves over Pólya–Vinogradov for all $c$ and all $M, N \leq c$, specifically with

$$\sum_{m, n} \alpha_m \beta_n \mathrm{Kl}(a m, n; c) \ll \| \alpha \| \|\beta\|\, \sqrt{MN c} \left( c^{(11+53\delta)/64}(MN)^{-3/16} + c^{(1-\delta)/6} N^{-1/3} + \cdots \right)$$

for any $\delta \in [0, 1/24]$.

6. Applications: $L$-Function Moments and Large Sieve Bounds

Non-abelian amplification yields new asymptotics:

• Twisted moments of cuspidal $L$-functions: For fixed holomorphic newforms $f_1, f_2$ and any modulus $q$,

$$\sum^{\ast}_{\chi \bmod q} L\left(\tfrac{1}{2}, f_1 \otimes \chi \right) L\left(\tfrac{1}{2}, f_2 \otimes \chi \right) = \text{MainTerm} + O\left( q^{1 - 1/674 + o(1)} \right).$$

• Exceptional-spectrum large sieve for Maass forms: For $\Gamma_0(q)$ Maass forms $f_j$ with eigenvalues $\frac{1}{4} - \theta_j^2$, coefficients $\rho_j(n)$, any $N \ll q^{1/2 + o(1)}$, and coprime sequence $\alpha_n$,

$$\sum_{\lambda_j < 1/4} q^{6\theta_j/5} \left| \sum_{n \sim N} \alpha_n \rho_j(n) \right|^2 \ll (qN)^{o(1)} \| \alpha \|^2,$$

substantially improving prior $(q/N)^{2\theta}$ savings when $N \asymp \sqrt{q}$ and $q$ is composite.

7. Connections to Infinite Characters and Traceable Representations

Amplification arguments with non-abelian characters exploit deep representation-theoretic input, analogous to the infinite character constructions in non-amenable groups acting on trees and higher-rank arithmetic groups (Bekka, 2018). While the Kloosterman amplification operates in the setting of $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$, the existence of large irreducibles with strong orthogonality and traceability echoes the method of inducing representations from non-amenable subgroups and deforming permutation representations (Julg–Valette trick), which produce uncountably many infinite-dimensional, traceable irreducible representations for non-abelian groups. This highlights the overarching theme: leveraging non-abelian harmonic analysis to both extend the analytic toolkit of number theory—e.g., bounding exponential sums—and to construct new classes of representations with rich properties.

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In summary, the amplification argument with non-abelian characters merges non-abelian Fourier techniques, explicit character sum bounds, combinatorial group theory, and analytic number theory to achieve unprecedented savings in bilinear Kloosterman sum estimates for composite moduli. This methodology directly translates to sharper asymptotics in $L$-function moments and exceptional large sieve bounds, while also elucidating representation-theoretic phenomena such as infinite character construction for non-abelian groups (Pascadi, 11 Nov 2025, Bekka, 2018).