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Non-abelian amplification and bilinear forms with Kloosterman sums

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Pith's one-line read By replacing abelian characters with a non-abelian amplifier on SL2(Z/cZ), this paper proves a power saving over the trivial bound for bilinear Kloosterman sums at square-root length for every composite modulus.

desk verdict A dense but genuine advance: new power savings for bilinear Kloosterman sums at composite moduli, with the counting lemma surviving close scrutiny; worth a serious referee. read the letter →

arxiv 2511.08445 v2 pith:SMLZ4DYO submitted 2025-11-11 math.NT

Alexandru Pascadi This is my paper · ORCID
classification math.NT MSC 11L0511F6611F72
keywords Kloostermansumsbilinearformsnon-abelianFourieranalysisamplificationSL2(Z/cZ)momentsofL-functionslargesieveexceptionalcusp
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a new upper bound for bilinear sums of Kloosterman sums modulo composite integers c: for sequences supported on intervals of length about sqrt(c), the sum is at most ||alpha|| ||beta|| c^{1-1/700+o(1)}, improving on the trivial bound c. Previous power savings of this type were available only for prime moduli or moduli with special factorizations, and the missing cases—squares of primes and products of two comparable primes—are exactly where the new method works best, saving c^{-1/12}. The argument identifies bilinear Kloosterman matrices with Fourier coefficients of functions on SL2(Z/cZ), then amplifies with a normal-subgroup-supported character sum to reduce the problem to counting solutions of a matrix equation. As applications, the paper proves an asymptotic for averaged second moments of twisted cuspidal L-functions for every modulus, and a large-sieve inequality for exceptional cusp forms.

What carries the argument

The central objects are the permutation representation rho_c of SL2(Z/cZ) on the projective line P1(Z/cZ), given by Möbius transformations, and its 'sifted' subrepresentation rho^o_c, obtained by removing all subrepresentations that factor through reduction mod d for d|c. A Kloosterman matrix is shown to have operator norm bounded by c times the spectral norm of a Fourier coefficient of a short-word function at rho^o_c. The new amplifier, Proposition 5.1, weights the sixth moment of singular values by a character sum over a normal congruence subgroup Gamma_c(d); this turns the problem into counting 6-tuples h1,...,h6 with |hi| at most H satisfying T^{h1} S ... T^{h6} S = I in PSL2(Z/dZ). The

What would settle it

Compute, for c=p^2 and H1=H2=p, the exact number of 6-tuples (h1,...,h6) in [-p,p]^6 satisfying T^{h1}S...T^{h6}S ≡ ±I mod p^2 or ≡ ±I mod p. Proposition 6.4 predicts at most p^{2+o(1)} and p^{3+o(1)} solutions respectively; finding p^{2+δ} or p^{3+δ} for a fixed δ>0 would invalidate the saving.

Watch

Extended reading notes

Core claim

Let c be any positive integer and let M,N not exceed c^{1/2+o(1)}. The paper proves that the bilinear sum of Kloosterman sums S(am,n;c) over m≤M, n≤N with (m,n,c)=1 is bounded by ||alpha|| ||beta|| c^{1-1/700+o(1)} for arbitrary complex sequences, and by sqrt(M) ||beta|| c^{1-1/276+o(1)} when the alpha coefficients are bounded by 1. This beats the trivial bound c^{1+o(1)} in the hardest balanced regime where MN is about c. The engine is Theorem 1.2: after writing c=dd'e with d'|d and (d,e)=1, the saving is governed by (f/min(c,d^2))^{1/6}, where f is the largest integer with f^2|cd. For c a square of a prime or a product of two primes of comparable size, this yields a c^{-1/12} saving, and c

Load-bearing premise

The power saving enters only through the elementary count of six-fold products T^{h1}S...T^{h6}S equal to the identity in PSL2(Z/cZ) (Proposition 6.4); if that count were larger than c^{o(1)}(H2^2 + (H1H2)^2/c), the main theorem's saving would fail.

Editorial extensions

If this is right

  • For every composite modulus c, bilinear Kloosterman sums at length about sqrt(c) now admit a uniform power saving over the trivial bound, including the previously resistant cases p^2 and pq with comparable primes.
  • For moduli such as p^2 or pq with comparable primes, the saving reaches c^{-1/12} in the balanced range, and the method beats the trivial bound for M,N between c^{5/12+o(1)} and c^{5/8-o(1)}.
  • The averaged second moment of two twisted cuspidal L-functions over primitive characters mod q satisfies the predicted asymptotic with an error q^{1-1/674+o(1)} for every modulus q.
  • For composite levels q with a divisor d of size sqrt(q) such that q/d is square-free, the exceptional-spectrum large sieve inequality holds with a factor q^{6θ/5}, improving on the standard q^θ loss when N is about sqrt(q).
  • The non-abelian amplification lemma applies to arbitrary finite groups and normal subgroups, making the structure reusable for other exponential sums with SL2 or GL2 geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test is the q=8 analogue of the counting bound: if the number of 8-tuples obeys Conjecture 6.2 in the range H roughly sqrt(c), the same framework would give nontrivial prime-modulus bounds for lengths beyond p^{3/8+o(1)}, a barrier that also appears in known results; the paper explicitly leaves this open.
  • The exponents 1/700 and 1/276 are small because a sixth moment is used to control the top singular value; using a higher even moment or refining the character-size lower bound should improve the exponents without changing the mechanism.
  • The same pattern—choose a large normal subgroup, amplify, then count solutions in PSL2(Z/dZ)—should also bound bilinear sums of other exponential sums with projective-line geometry, such as additive characters of Möbius transformations; the author notes this possibility but does not carry it out.
  • The moment application is stated for holomorphic cusp forms; extending it to Maass cusp forms would require additional care about the Ramanujan bound, as the paper indicates, so a fully unconditional Maass-level version is not immediate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper develops a new method, based on non-abelian Fourier analysis and a non-abelian amplification argument, for bounding bilinear (Type II) sums of Kloosterman sums with composite moduli. The central result, Theorem 1.1, gives a power saving over the trivial bound c for sequences of length at most c^{1/2+o(1)} for every modulus, with exponents 1/700 in general and 1/276 when one sequence is bounded. The argument proceeds by realizing Kloosterman matrices as Fourier coefficients of a representation ρ_c^∘ of SL_2(Z/cZ), applying an amplifier supported on a congruence subgroup Γ_c(d), and reducing to an elementary count (Proposition 6.4) of solutions in PSL_2(Z/\tilde d Z). The paper also proves an asymptotic for twisted second moments of modular L-functions (Theorem 1.5) and an improved large sieve for exceptional Maass forms for composite levels (Theorem 9.4 / Corollary 1.6).

Significance. If the main results are correct, this is a substantial advance: it supplies the first nontrivial Type II Kloosterman bound at square-root length for the composite moduli (such as p^2 or pq) that were the remaining barriers for the moment application, and it does so with an explicit, self-contained method that avoids the algebraic-geometric machinery used in the prime-modulus results. The non-abelian amplifier construction is new and likely to be useful independently. A particularly valuable feature is that the power saving is produced by a purely elementary counting argument, with no fitted parameters or circular benchmarks; the savings are measured directly against the trivial bound (1.2), and external results are used only for the near-prime and prime-modulus cases. The paper also honestly acknowledges the simultaneous independent work [29]. The main risk is the intricacy of Proposition 6.4, the sole source of the saving; I checked the main danger points and found no counterexample, but the proof is delicate enough that a fully expanded verification of the endpoint maximization is desirable.

minor comments (6)
  1. [§6, Proposition 6.4, Case 2] In the sentence 'Since h5h6 ≡ 1 (mod d), there are O(1 + H1H2/c) ways to pick the nonzero integer h5h6', the denominator should be d, not c. The factor (1+H1H2/d) appears correctly in (6.8), but the displayed sentence is internally inconsistent and, read literally, would undercount the h5,h6 possibilities. Please correct and clarify.
  2. [§6, Eq. (6.8)] The transition from the bound after (6.8) to the final maximum is compressed. It is asserted that after expansion each term is monotone in d, so the maximum occurs at d=1 or d=c; however the displayed algebra also contains an inequality that should be ≤ rather than = when min(H1/(cd),1) is replaced by H1/(cd). Since Proposition 6.4 is the load-bearing combinatorial step, please spell out this maximization and the inequalities in full.
  3. [§6, proof of Proposition 6.4] In the sum over g in the estimation of the h2,h3 contribution, the congruence for h2' should be h2' ≡ g^{-1} r (mod d), not h2' ≡ gr (mod d). The resulting count is the same, but the displayed congruence is a typo that should be corrected.
  4. [§5, Proposition 5.1] The right-hand side of the amplifier inequality is written as a complex-valued sum involving χ(g1···gq). The proof shows that this quantity arises from a nonnegative expression, but the statement itself would benefit from a remark that the displayed sum is real (or that one takes its real part), to avoid an apparent type mismatch with the real left-hand side.
  5. [§1.2, Remark on [29]] The acknowledgment of simultaneous independent work is appropriate and should remain. It may be worth noting explicitly which results overlap and which remain unique to this paper, since the current remark is brief.
  6. [§A, Lemma 5.4] The remark after Lemma 5.4 states that the bound is expected to hold for all 0≤j≤k. As written, Proposition 5.3 uses only the proved range, so this is not a gap, but the reader would benefit from a sentence clarifying that the unproved extension is not needed for the main theorems.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the power saving comes from an independent counting lemma (Prop 6.4), not from the target bound.

full rationale

The central claim (Theorem 1.1) is not circular. The new saving is produced by Theorem 7.1, whose chain is: Proposition 4.10/Corollary 4.11 (exact reduction of a Kloosterman matrix to a Fourier coefficient on SL_2(Z/cZ) via Poisson summation and Möbius inversion), Proposition 5.5 (amplification passes to a weighted count of solutions in PSL_2(Z/dZ), using character bounds from Lemmas 5.2 and 5.4, whose proofs are local matrix/Clifford-theory computations), and Proposition 6.4 (an elementary count for q=6 solutions in PSL_2(Z/cZ), independent of the bilinear-form statement). None of these steps is a fitted parameter renamed as a prediction; the counting target (6.1) is structurally distinct from the desired bilinear bound, and no equation in the proof is equivalent by construction to the statement being proved. Section 9 cites the author's [32, Corollary I], but that is a prior large-sieve result derived from the Kuznetsov trace formula and Deshouillers--Iwaniec [11], not the theorem being established, and the core composite-modulus bilinear theorem does not rely on it. Near-prime cases use external results of Kowalski--Michel--Sawin [24] and Blomer--Milićević [3]. The remaining open Conjecture 6.2 is explicitly not used for q=6. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard external theorems (Weil, KMS, Blomer–Milićević, Deligne, DI) and explicit representation theory; no ad hoc parameters are fitted, and no new entities are postulated.

assumptions (6)
  • standard math Weil bound for Kloosterman sums: |S(m,n;c)| ≪ c^{o(1)} sqrt((m,n,c)c) (Lemma 3.3)
    Used for the trivial bound (1.2) and in the near-prime Corollary 7.3.
  • standard math Kowalski–Michel–Sawin bilinear-form bound for prime moduli (Theorem 3.4)
    Used to handle near-prime moduli in Theorem 7.4.
  • standard math Blomer–Milićević bound for moduli with a large divisor (Theorem 3.5)
    Used in Theorem 7.4 for prime-power moduli.
  • standard math Deligne's bound on Hecke eigenvalues: λ_f(n) ≪ n^{o(1)} (8.2)
    Used in the moment application Theorem 1.5 for holomorphic cusp forms.
  • standard math Bourgain–Gamburd dimension bound for primitive representations of SL2(Z/p^kZ) (Lemma 3.16)
    Gives dim ρ ≫ p^k; used in Proposition 4.6 and Lemma 5.4.
  • standard math Deshouillers–Iwaniec large sieve inequality (Theorem 9.3)
    Baseline for the Section 9 application; the paper improves it for composite levels.

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Pith. "Pith review of Non-abelian amplification and bilinear forms with Kloosterman sums." pith.science (2026). https://pith.science/paper/SMLZ4DYO

@misc{pith2026251108445,
  author       = {Pith},
  title        = {Pith review of: Non-abelian amplification and bilinear forms with Kloosterman sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SMLZ4DYO}},
  note         = {Machine review of arXiv:2511.08445}
}
abstract

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$, our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the P\'olya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal $L$-functions, and to large sieve inequalities for exceptional cusp forms with composite levels.

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Cited by 1 Pith paper

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