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Bilinear forms with Kloosterman sums via quadratic characters

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Pith's one-line read Bilinear Kloosterman forms save c to the minus 1/32 in the square-root range for every modulus, via quadratic characters.

desk verdict Uniform c^{-1/32} bilinear Kloosterman bound via a clean SL_2-to-Jacobi link; hybrid with [29] checks out and the applications are real. read the letter →

arxiv 2607.24311 v1 pith:XAT6PGTB submitted 2026-07-27 math.NT

Valentin Blomer , Alexandru Pascadi This is my paper · ORCID
classification math.NT MSC 11L0511L4011M4111F30
keywords KloostermansumsbilinearformsquadraticcharacterstwistedL-functionsexceptionalMaasslargesieveSL(2)representations
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

Kloosterman sums appear throughout analytic number theory. When they are paired against two arbitrary sequences of length about the square root of the modulus, the usual Weil bound and a crude Cauchy estimate give the same size, so any genuine cancellation is hard-won. This paper proves a uniform saving of c to the power minus 1/32 over that trivial size, valid for every positive integer modulus, not merely primes. The gain comes from rewriting a fourth-moment matrix of Kloosterman sums in terms of characters of SL(2) and then recognizing those characters, after a short amplification that peels off the square-full part of the modulus, as ordinary Jacobi symbols. The resulting complete character sums are estimated by Weil’s bound and elementary counting. Two concrete payoffs follow at once: a second-moment asymptotic for twisted L-functions with error q to the minus 1/90, and a large-sieve inequality for exceptional Maass forms that improves the classical Deshouillers–Iwaniec range without any factorization hypothesis on the level.

What carries the argument

The fourth-moment spectral norm of the Kloosterman matrix is rewritten, via the special representations of SL(2,Z/cZ), as a sum of non-abelian characters; after amplification that isolates the square-free part of c those characters become Jacobi symbols of a quadratic polynomial in four variables, which are then bounded by Hölder, elementary counting, and Weil’s estimate for character sums.

What would settle it

Take a prime power modulus p^k with k large and N equal to p^{k/2}; compute the bilinear form for random unit-coefficient sequences and check whether the observed size is smaller than the trivial bound by a factor p^{k/32} or better. If the ratio stays larger than p^{k/32+ε} for large k, the uniform claim fails.

Watch

Extended reading notes

Core claim

For any modulus c and any intervals of length at most N less than or equal to c, the bilinear form of Kloosterman sums against arbitrary complex coefficients is bounded by the product of the l2-norms times c to the 1+o(1) times (N^{1/8}c^{-3/32}+N^{5/16}c^{-3/16}+N^{2/3}c^{-7/18}). In the critical range N equal to the square root of c this is a saving of c to the minus 1/32 over the trivial bound, and the result holds for every c.

Load-bearing premise

The clean reduction from non-abelian characters to Jacobi symbols works only after the square-full part of the modulus has been amplified away; the remaining square-full contribution is controlled by a coarser sixth-moment bound that must not lose more than claimed when that part is large.

Editorial extensions

If this is right

  • The second moment of L(1/2,f1×χ)L(1/2,f2×χ) over primitive characters modulo q equals the expected main term plus an error O(q^{-1/90+ε}).
  • The exceptional-spectrum large sieve for Maass forms of level q gains an extra factor roughly q^{2 max θ_j /29} when the Fourier coefficients are supported near length √q, with no factorization hypothesis on q.
  • The same bilinear bound improves the range of non-trivial estimates for shifted convolution problems and for the greatest prime factor of n^{2}+1 that rely on Kloosterman sums of general modulus.
  • For square-free moduli the hybrid step can be omitted and the pure fourth-moment argument already yields the full c^{-1/32} saving.

Reading between the lines

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

  • Higher even moments or different Hölder exponents are unlikely to improve the critical exponent 1/32, because the diagonal contribution in the character-sum estimate already saturates the available cancellation.
  • The same SL(2)-to-Jacobi dictionary should apply, with only minor changes, to bilinear forms involving other algebraic trace functions that arise from the same permutation representation.
  • Once the square-full obstruction is removed by a better local estimate, the method would give a pure power-saving large sieve for exceptional eigenvalues at every level.
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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

3 major / 6 minor

Summary. The paper proves new bounds for bilinear forms ∑_{m∈I,n∈J} α_m β_n S(am,n;c) with (m,n,c)=1, valid for all moduli c. Theorem 1.1 gives a saving factor with three terms (N^{1/8}c^{-3/32} + N^{5/16}c^{-3/16} + N^{2/3}c^{-7/18}), which in the critical range N=√c yields a saving of c^{-1/32} over the trivial bound ‖α‖‖β‖Nc^{1/2+o(1)} — doubling the previous p^{-1/64} saving of Kowalski–Michel–Sawin for prime moduli and improving on Milićević–Qin–Wu for general moduli. The method reduces the bilinear form, via the spectral norm and a fourth-moment trace bound, to a character sum of SL₂(ℤ/cℤ) (Proposition 3.1, with non-abelian amplification to strip the square-full part), identifies the resulting character χ°_c with a Jacobi symbol in the discriminant Tr(g)²−4 for odd square-free c (Lemma 3.3), and treats the resulting quadratic character sum by a Hölder reduction to two counting problems and one complete character sum (Propositions 3.6, 4.1–4.5, using Weil's bound). For moduli with large square-full part the result is combined with the sixth-moment bound from the second author's prior work [29] (Theorems 5.2–5.5). Applications give an error term q^{-1/90+ε} in the second moment of twisted L-functions (Theorem 1.3) and a uniform large sieve inequality for exceptional Maass forms (Theorem 1.6).

Significance. If the estimates hold — and I found no gap — this is a strong result in the analytic theory of exponential sums: it doubles the best previous saving in the critical range even for prime moduli, is the first bound beating the trivial estimate in the range c^{13/28+ε} < N < c^{7/12−ε} for arbitrary c, and yields the best known error terms in the second moment of twisted L-functions (q^{-1/90+ε}) and a level-uniform improvement of the Deshouillers–Iwaniec exceptional-spectrum large sieve. Particular strengths: the argument is self-contained modulo clearly cited theorems (Weil bounds, [29]); it introduces a genuinely new mechanism (the trace χ°_c equals a Jacobi symbol of Tr(g)²−4); the exponents are explicit and hence falsifiable; and the proof uses no unproved hypotheses. The auxiliary bound of Lemma 5.6/Theorem 5.7 for arbitrary moduli is of independent interest.

major comments (3)
  1. [§5, proof of Theorem 5.2, (5.7)-(5.9)] The passage from (5.7) to (5.9) is the place where the exponents of Theorem 1.1 are actually fixed, and it rests on two monotonicity assertions that are not fully substantiated: (i) that every term in the expansion of (5.7) grows at most like d^{-1}, so the maximum over d in [c_2, c] is attained at d = c_2; and (ii) that at d = c_2 the expression is non-increasing in c_2, justifying the replacement c_2, d \mapsto 1. Because (5.7) contains a minimum of two expressions with different d-dependence (the elementary bound of Prop. 4.2 and the Fourier bound of Prop. 4.3), the d^{-1} claim is not immediate from the displayed formula; a term-by-term expansion or a small lemma tabulating the d-exponents would make this checkable. As written, the reader must redo a nontrivial optimization to verify the load-bearing step of the paper.
  2. [§6, Theorem 1.3 (end of section, In[1]/Out[1])] The error term q^{-1/90} in Theorem 1.3 is one of the headline results, but its derivation is outsourced to a Mathematica Maximize computation whose printed output is the only verification. The combination of (6.2), (6.3), (6.4) under (6.1) is a finite piecewise-linear optimization, so a human-checkable certificate should be easy to give: please identify the active constraints at the optimum (m, n) = (43/90, 43/30), state which of the three bounds is attained there, and confirm the value -1/90 by exact rational arithmetic. Also, in the derivation of (6.4), the claim that 'the smallest power of r is positive' (so that r = q may be substituted) is asserted after substituting N^* = q^2/N in all but one factor; one line identifying the two extremal r-exponents would remove any ambiguity.
  3. [§3.1, Proposition 3.1] Proposition 3.1 is imported almost entirely from [29] (Corollary 4.11, and the amplification of Proposition 5.1), and the manuscript's uniformity over all moduli depends on the amplified estimate (3.3) holding with the indicated error O_epsilon(c^{-100}) and on the nonnegativity of S used in the amplification step. The argument given is convincing, but since this is the only place where the square-full part c_2 enters before the hybrid combination of Theorem 5.5, I ask the authors to state explicitly which hypotheses of [29, Corollary 4.11] are being invoked (in particular the role of the weight nu_{(m,n,c_1)} and the indicator 1_{(m,n,c_2)=1}) and to confirm that the error term survives the 2k-th root in (3.1) unchanged. A short remark would suffice; no new mathematics is needed.
minor comments (6)
  1. [§1.2, paragraph before Theorem 1.3] 'Here prove the following result.' — missing 'we'.
  2. [§4.2, proof of Proposition 4.3] In the displayed estimate there appears to be a stray bracket: 'H_1^3 H^{3/2}]_2 / (cd^{5/4})'. Please check the source; the same display also has heavy exponent bookkeeping that would benefit from one more intermediate line.
  3. [§2.1, Eq. (2.5)] For complex matrices the variational characterization should read |w^* A v| (conjugate transpose), not w^T A v.
  4. [§4.3, Remark 4.6] The claim that E_{c/d} is 'essentially optimal' is plausible (diagonal vs. generic pairing), but for composite c/d the interpolating terms depend on the factorization; a half-sentence clarifying in what sense optimality is meant (e.g., matching the two displayed extremes G=1 and G=c/d) would help.
  5. [§5, Remark 1.2 / Theorem 5.5] The third member of the minimum in Theorem 5.5 is used silently in the proof of Theorem 1.1 when N is small (the N^{1/15}/c^{1/15} branch); a forward reference from Theorem 1.1 to the case analysis H(N,N,c) would make the range N <= c^{13/28} easier to follow.
  6. [References / acknowledgements] Two references central to the comparison are currently preprints ([27], [13]); if they have appeared or been accepted by revision time, please update. The acknowledgements' mention of AI assistance could be moved to a footnote describing its scope, per emerging journal conventions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the c^{-1/32} saving is derived from Weil bounds, SL_2 representation theory, and elementary counting, not from a tautology or fitted input.

full rationale

The central claim (Theorem 1.1) is obtained by reducing bilinear Kloosterman forms, via Fourier analysis on SL_2(Z/cZ) and non-abelian amplification (Proposition 3.1), to a fourth-moment character sum that is rewritten, for the square-free part of the modulus, as an explicit Jacobi-symbol sum (Lemma 3.3 / Proposition 3.4). The resulting incomplete character sum is estimated by Hölder (Proposition 3.6) together with elementary counting (Propositions 4.1–4.3) and the classical Weil bound for quadratic characters (Lemma 4.4 / Proposition 4.5). The square-full contribution is controlled by an independent sixth-moment bound of Pascadi [29, Thm 7.1] (Theorem 5.4) and by the diagonal term already present in the fourth-moment analysis; the hybrid combination (Theorem 5.5) is an ordinary min of independently derived expressions and does not force the target exponent by construction. Prior work of one co-author is used only as a black-box input whose hypotheses do not include the claimed saving. No parameter is fitted to data and then re-predicted; no uniqueness theorem is imported to forbid alternatives; the derivation is self-contained against external benchmarks (Weil, fixed-point counts on P^1). Score 0 is therefore appropriate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard analytic-number-theory toolkit (Weil bound, Poisson summation, representation theory of SL_2(Z/cZ), Kim–Sarnak θ≤7/64) plus the authors’ prior non-abelian amplification framework [29]. No free parameters are fitted to data; Hölder exponents and amplification cutoffs are chosen for optimization but are explicit and not data-dependent. No new physical or arithmetic entities are postulated.

assumptions (5)
  • standard math Weil bound |S(m,n;c)| ≤ τ(c)√((m,n,c)c) and Weil bound for non-square quadratic character sums over F_p
    Used throughout for pointwise Kloosterman estimates and for C_ℓ (Lemma 4.4, Prop 4.5).
  • domain assumption Properties of the special representations ρ_c, ρ°_c of SL_2(Z/cZ) (multiplicativity, dimension ≍c, large irreducible constituents)
    Taken from Pascadi [29, §4]; Lemma 2.3 records what is used.
  • domain assumption Kim–Sarnak bound θ_j ≤ 7/64 toward Ramanujan–Petersson
    Invoked in the twisted-moment application (§6) and in the exceptional large sieve setup.
  • domain assumption Kuznetsov formula and Deshouillers–Iwaniec spectral large sieve framework
    Used to convert exceptional-spectrum sums into bilinear Kloosterman forms (§7).
  • domain assumption Sixth-moment bilinear bound of Pascadi [29, Thm 7.1] for the square-full part
    Hybridized with the new fourth-moment/quadratic-character bound in Thm 5.5 to remove factorization dependence.

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Pith. "Pith review of Bilinear forms with Kloosterman sums via quadratic characters." pith.science (2026). https://pith.science/paper/XAT6PGTB

@misc{pith2026260724311,
  author       = {Pith},
  title        = {Pith review of: Bilinear forms with Kloosterman sums via quadratic characters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XAT6PGTB}},
  note         = {Machine review of arXiv:2607.24311}
}
read the original abstract

We prove new bounds for bilinear forms with Kloosterman sums, valid for all moduli c. In the critical range where the summation length is the square root of the modulus, the saving over the trivial bound is c^{-1/32}, improving on all previous approaches even for prime moduli. This is based on a new connection to quadratic character sums. Applications to moments of twisted L-functions and to the large sieve for exceptional Maass forms are given.

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