Bounds for Kloosterman Sums for $\mathrm{GL}_n$

Johannes Linn

REVIEW 3 major objections 5 minor 1 cited by

Bounds for Kloosterman Sums for $\mathrm{GL}_n$

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that every admissible Weyl-element Kloosterman sum on GL_n admits a power saving of size 1/(4l(w)) over the trivial bound.

desk verdict A serious, credible preprint that plausibly proves power-saving bounds for all admissible Weyl elements on GL_n, but the pivotal combinatorial inequality (8) is under-verified and needs a real referee. read the letter →

arxiv 2412.04976 v2 pith:EZMW5NY6 submitted 2024-12-06 math.NT

Johannes Linn This is my paper · ORCID

classification math.NT MSC 11L05

keywords KloostermansumsGL_nWeylelementsexponentialWeilboundBruhatdecompositionp-adicgroupsrelativetraceformula

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 power-saving upper bound for the local Kloosterman sums of GL_{N+1} attached to any admissible Weyl element, not just the special cases treated before. For a Weyl element w of length l(w), the sum is shown to be at most $C^{l(w)/2}(\prod_{k=1}^N p^{r_k})^{1-1/(4l(w))+\varepsilon}$, where the $r_k$ encode the diagonal modulus and $C$ measures the conductors of the characters. This improves the trivial bound by a power saving of $1/(4l(w))$; for all but three Weyl elements it is the first non-trivial bound, and for the long Weyl element it quadruples the previous saving. The same bound holds for the congruence subgroup $\Gamma_0(q)$ with $q$ a power of $p$. The reason to care is that these sums appear on the geometric side of relative trace formulae and in Fourier coefficients of automorphic forms, where bounds control the size of the terms.

What carries the argument

The load-bearing objects are, first, the explicit parametrization of the Kloosterman sum by the sets $C_w(m)$ of exponent coordinates and, second, a directed diagram whose vertices are the roots in $R(w^{-1})$ and whose edges carry the coordinates; the diagram is augmented with dotted edges for the second character and is used both to write the exponential sum (Corollary 2) and to define the grouping of summation variables. The grouping into two families of non-adjacent variables is what allows several Weil bounds to be applied at once, and an induction over the number of blocks in the Weyl element reduces every admissible $w$ to two-block elements.

What would settle it

Take the five-block Weyl element $w_2$ in Section 4.2, enumerate all admissible $m$ for a small prime $p$, compute the sets $B_k$ explicitly, and check inequality (8): any tuple with $l(w)\sum b^*_{i,j} > -\sum (j-i+1)m_{i,j}$ would destroy the $p^{1/(4l(w))}$ saving and falsify Theorem 1.

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Extended reading notes

Core claim

The central claim, Theorem 1, is that for an admissible block-diagonal Weyl element $w$ in $\mathrm{GL}_{N+1}$ and a modulus with exponent vector $r$, the Kloosterman sum satisfies $\mathrm{Kl}_p(\psi,\psi',n)\ll_\varepsilon C^{l(w)/2}(\prod_{k=1}^N p^{r_k})^{1-1/(4l(w))+\varepsilon}$ with $C=\max_j \min(|\psi_j|_p^{-1/2}, p^{r_j/2})$, and the identical bound holds for $\Gamma_0(q)$ with $q$ a power of $p$. The proof is constructive: the sum is parametrized explicitly as an exponential sum over a product of congruence classes (Theorem 3), using the Bruhat decomposition of the group elements $b_\alpha(a)$. A diagram attached to the Weyl element encodes the $p$-adic powers in the summands, and its vertices are two-coloured so that disjoint groups of variables can be summed one after another; each single-variable sum is a $\mathrm{GL}_2$ Kloosterman sum to which the Weil bound applies. A lemma controlling the average dependence on neighbouring variables gives the stated saving.

Load-bearing premise

The bound collapses if the combinatorial grouping of exponent coordinates in Section 4.2 fails to satisfy inequality (8), a step that is asserted with only a sketch and no worked induction.

Editorial extensions

If this is right

For every admissible Weyl element other than the identity, $w^*$, and the long element, Theorem 1 is the first non-trivial bound; applications to trace formulae no longer have to single out these cases.
The bound applies uniformly to $\mathrm{GL}_{N+1}(\mathbb{Z}_p)$ and to $\Gamma_0(q)$, $q$ a power of $p$, so the same saving is available in congruence-subgroup settings used in density theorems.
Theorem 2 gives an alternative with a much lighter dependence on the characters, replacing $C^{l(w)/2}$ by $C$ at the cost of a saving of $1/(2Nl(w))$ instead of $1/(4l(w))$.
Since the proof uses only the stratification of the Kloosterman set and the Weil bound, the same statements hold over any non-archimedean local field, not only $\mathbb{Q}_p$.

Reading between the lines

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

The genuine bottleneck is the combinatorial inequality (8); if it fails, the clean exponent $1/(4l(w))$ would need to be replaced by a smaller saving, but the parametrization and two-colouring scheme would still give some power saving.
The paper's example with all $m_{i,j}$ equal suggests that the method cannot reach savings better than $1/(2(N-1))$ for the long element; a plausible next step is to compute the exact saving for small $N$ to see whether $1/O(N)$ is the right order.
The diagram language is group-independent in structure, so a parallel treatment for other reductive groups with the same kind of stratification and a Weil bound would be a natural test of the method's scope.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline is simple: this paper claims the first power-saving bound for every admissible Weyl element of GL_n, and the claim is plausible. The parametrization via Dabrowski-Reeder stratification is explicit, the Weil-bound application is structured, and the main theorem's exponent 1/(4l(w)) is derived, not fitted. Prior work covered only special elements (long, w*, GL3, GL4, Sp4, orthogonal), so the breadth here is genuinely new. The induction over blocks in Section 3 is the right idea, and Theorem 4 gives real explicit Bruhat-decomposition formulas that look like they hold.

Where it is soft, in proportion: the proof of Lemma 4.1 is only sketched and carries a product-of-e_i factor that the author says 'can maybe be removed' but then absorbs into epsilon. That is a minor gap, since the absorption is legitimate even without removal. The bigger soft spot is Section 4.2. Inequality (8) is the load-bearing combinatorial step: it converts the Weil-bound savings into the final power saving. The proof there is a heuristic paragraph. The sets B_k are defined by a recursive 'furthest to the right' procedure with three replacement rules for m_{i,j}=0, and the termination/uniqueness of that procedure is not demonstrated. The claimed bound 'each b_{i,j} appears at most l(w) times' is asserted with a sketch, not a proof. The m=0 replacement rules on p.34 are under-specified; it is not clear that they preserve inequalities (6) and (7). If (8) fails, Theorem 1's exponent breaks. Nothing in my reading suggests it actually fails—the example and the one-block computation are consistent—but the proof as written is not checkable without extra work.

The reader's stress-test note flagged exactly this, and I agree with the diagnosis. I do not agree with any suggestion that the whole paper is hollow: the parametrization theorem and the two-block theorem are substantial, and the combinatorial step is isolated. I also note the paper is formally self-contained against external benchmarks: no circularity, no fitted exponents, no invented entities. The citation pattern is appropriate, including use of [BM24] and [DR98].

My bottom line: this deserves a serious referee and likely a conditional acceptance after the combinatorial section is expanded. The paper is for analytic number theorists working on relative trace formulae and density theorems. I would bring it to a reading group, and I would cite it if the combinatorial lemma survives scrutiny. Recommend sending it to peer review; do not desk reject.

Best, [You]

Referee Report

3 major / 5 minor

Summary. The paper develops an explicit parametrization of local Kloosterman sums for GL_{N+1} for all admissible Weyl elements, building on the Dąbrowski–Reeder stratification. Two-block Weyl elements are treated by an explicit Bruhat decomposition encoded in path diagrams, and general Weyl elements are handled by an induction that splits off the top block. The resulting exponential sums are bounded by repeated applications of the Weil bound together with a combinatorial inequality (8), yielding Theorem 1 (power saving 1/(4l(w))) and Theorem 2 (a variant with a milder character dependency), including a version for the congruence subgroup Γ0(q).

Significance. If the combinatorial inequality (8) is fully established, Theorem 1 supplies the first non-trivial bounds for many admissible Weyl elements and improves the long-element bound of Blomer–Man by a factor of 4. The parametrization itself (Theorem 3 and Corollary 2) is a concrete and potentially reusable tool. The paper is largely self-contained relative to the cited stratification and Weil-bound benchmarks, and the main derivation is not circular. However, the advertised power saving rests on an unproved and under-specified combinatorial claim, and the uniform character dependence in Theorem 1 also needs qualification. No machine-checked proofs or exhaustive verifications are provided.

major comments (3)

[§4.2, Eq. (8)] Inequality (8) is the load-bearing step that converts the Weil-bound savings into the exponent 1/(4l(w)) in Theorem 1. The proof given for the first inequality in (8) is the assertion that each b_{i,j} appears at most l(w) times, followed by 'The maximal sum over these multiplicities can be computed inductively over the number of blocks and is exactly l(w)'. The induction is not carried out, and the sets B_k are only described informally. Since any weakening or hidden multiplicity in this step would change the final exponent, a complete definition of B_k for arbitrary multi-block diagrams and a full proof of (8) are required.
[§4.2, m_{i,j}=0 replacement rules] The three replacement rules on p. 34 for handling m_{i,j}=0 are recursive and contain instructions such as 'continue as if b_{i,j} was added'. It is not demonstrated that this procedure terminates for all diagrams, nor is it proved that inequalities (6) and (7) survive the replacement. The final claim that the new sets B_k still satisfy both inequalities in (8) is asserted rather than proved. This is an essential part of the argument because zero exponents occur generically when the exponent vector r has zero coordinates.
[Theorem 1 and Remark 2] The statement 'improves the trivial bound by a power saving of 1/(4l(w))' is not uniform in the characters. For a character with |ψ_j|^{-1/2}_p = p^{r_j/2} (for example a character of conductor p^{r_j}), one has C = p^{r_j/2}; then C^{l(w)/2} already contributes p^{l(w)r_j/4}. When only one r_j is nonzero and l(w)>4, the right-hand side of Theorem 1 exceeds the trivial size p^{Σ r_k}. The authors should either state the character regime in which the displayed bound actually beats the Dąbrowski–Reeder trivial bound or explicitly compare the full expression (including C) with the trivial bound.

minor comments (5)

[Remark 15] In Lemma 4.1, the sentence 'It can maybe be removed' about the factor ∏ e_i is not a mathematical statement; because the factor is later absorbed into ε, this is harmless, but the remark should be phrased more precisely or deleted.
[§4.2, paragraph after Eq. (5)] The text says 'Theorem 3 claims C = 1/l(w)', but Theorem 3 is the parametrization; the relevant claim is inequality (8) in Section 4.2, not Theorem 3. This misreference should be corrected.
[Remark 16] The expression '1/2(N−1)' is ambiguous; it should read '1/(2(N−1))' or be parenthesized accordingly.
[Section 3.2, example after Theorem 4] In the displayed formula for L_{2,5}, the exponent '−m22,−m2,3−m2,4' appears to be a typo for '−m_{2,2}−m_{2,3}−m_{2,4}'.
[Notation] The symbol C_w(m) is defined once just before Theorem 3 and again in Theorem 5, with slightly different conventions for the range of c_{i,j}. The two definitions should be explicitly reconciled.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the power-saving bound is derived from external Weil-bound input plus combinatorial bookkeeping; the weak point is an unproved combinatorial inequality, not circular reasoning.

full rationale

The derivation chain is self-contained against external benchmarks. The parametrization in Theorem 3 is proved from the Bruhat decomposition using the Dąbrowski–Reeder stratification, which is cited from [DR98] as an external input and is not derived from the target bound. The power-saving exponent 1/(4l(w)) is obtained by applying the Weil bound to the explicitly parametrized exponential sum and then performing combinatorial bookkeeping over the diagram; it is not fitted to data, nor is any parameter defined in terms of the final bound. The only significant weakness is internal rather than circular: the proof of inequality (8) in §4.2 is heuristic and under-specified. The paper explicitly says the first inequality holds because each b_{i,j} appears at most l(w) times and that the maximal sum 'can be computed inductively over the number of blocks and is exactly l(w)', but no induction details are given, and the m_{i,j}=0 replacement rules are recursive. That is a rigor gap in the proof of the claimed saving, not a reduction of the conclusion to the hypotheses. There are no load-bearing self-citations: the author cites [BM24] for prior results and for code, but the cited results are independently stated and are not used to define away the main theorem. In particular, the paper improves rather than assumes the [BM24] saving for the long Weyl element. Overall, no step exhibits the pattern of a fitted input being renamed a prediction, an ansatz smuggled in via citation, or a uniqueness theorem imported from the author's own prior work. The combinatorial gap should be flagged as a correctness risk, but it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the constants in the bounds are explicit. The axioms are: the Dąbrowski-Reeder parametrization and size formula (Lemma 2.2), the Weil bound for GL2, the admissibility condition for Weyl elements, and standard p-adic additive character theory. No new entities are introduced beyond the diagram, which is a bookkeeping device.

assumptions (4)

domain assumption Dąbrowski-Reeder parametrization and size formula (Lemma 2.2 and the injectivity from DR98, Props 3.1-3.4).
The whole parametrization of the Kloosterman sum rests on the bijection between Y_λ/U_w(Z_p) and the Kloosterman set, quoted from DR98.
standard math Weil bound for GL2 Kloosterman sums: |S(m,n;c)| ≤ τ(c) c^{1/2} gcd(m,n,c)^{1/2}.
Applied repeatedly in Section 4.1 to bound the one-variable exponential sums after grouping the variables.
domain assumption Only Weyl elements with identity blocks (admissible) give well-defined Kloosterman sums (Friedberg, p.175).
The paper restricts to these without proof, citing Friedberg.
standard math Standard p-adic additive character Ψ with kernel Z_p and norm properties used in Lemma 4.1.
Background on p-adic characters and valuations.

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Pith. "Pith review of Bounds for Kloosterman Sums for $\mathrm{GL}_n$." pith.science (2026). https://pith.science/paper/EZMW5NY6

@misc{pith2026241204976,
  author       = {Pith},
  title        = {Pith review of: Bounds for Kloosterman Sums for $\mathrmGL_n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZMW5NY6}},
  note         = {Machine review of arXiv:2412.04976}
}

abstract

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by D\k{a}browski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the spectral aspect density hypothesis and application
math.NT 2025-04 accept novelty 8.0 of 10

For n >= 4, the paper establishes Sarnak's density hypothesis in the spectral aspect for GL_n(Z) cuspidal representations and uses it to prove the Diophantine exponent of the SL_n(Z[1/p])-action is optimal (kappa = 1).

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

[1]

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[AB24] Edgar Assing and Valentin Blomer,The density conjecture for principal congruence subgroups, Duke Math. J.173(2024), no. 7, 1359 –

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[Man22] Siu Man,Symplectic Kloosterman sums and Poincaré series, Ramanujan J.57(2022), 707–753. [Mia24] Xinchen Miao,Bessel functions and Kloosterman integrals on GL(n), J. Funct. Anal.286(2024), no. 4, 110267. [Muj24] Catinca Mujdei,Kloosterman sums on orthogonal groups(2024), arXiv 2410.20569, primary class math.AT [Poi11] H. Poincaré,Fonctions modulair...

work page Pith review arXiv 2022
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[BFG88] Daniel Bump, Solomon Friedberg, and Dorian Goldfeld,Poincaré series and Kloosterman sums forSL(3,Z), Acta Arith.50(1988), no. 1, 31-89. [Blo23] Valentin Blomer,Density theorems forGL(n), Invent. math.232(2023), 683-711. [BM24] Valentin Blomer and Siu Hang Man,Bounds for Kloosterman sums onGL(n), Math. Ann.390(2024), no. 1, 1171–1200. [Dą93] Romual...

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[Klo26] Hendrik Kloosterman,On the representation of numbers in the formax2 +by 2 +cz 2 +dt 2, Acta Math.49 (1926), no. 3-4, 407 –

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

J.173(2024), no

[AB24] Edgar Assing and Valentin Blomer,The density conjecture for principal congruence subgroups, Duke Math. J.173(2024), no. 7, 1359 –

work page 2024

[2] [464]

Kloosterman sums on orthogonal groups

[Man22] Siu Man,Symplectic Kloosterman sums and Poincaré series, Ramanujan J.57(2022), 707–753. [Mia24] Xinchen Miao,Bessel functions and Kloosterman integrals on GL(n), J. Funct. Anal.286(2024), no. 4, 110267. [Muj24] Catinca Mujdei,Kloosterman sums on orthogonal groups(2024), arXiv 2410.20569, primary class math.AT [Poi11] H. Poincaré,Fonctions modulair...

work page Pith review arXiv 2022

[3] [1426]

1, 31-89

[BFG88] Daniel Bump, Solomon Friedberg, and Dorian Goldfeld,Poincaré series and Kloosterman sums forSL(3,Z), Acta Arith.50(1988), no. 1, 31-89. [Blo23] Valentin Blomer,Density theorems forGL(n), Invent. math.232(2023), 683-711. [BM24] Valentin Blomer and Siu Hang Man,Bounds for Kloosterman sums onGL(n), Math. Ann.390(2024), no. 1, 1171–1200. [Dą93] Romual...

work page 1988

[4] [2006]

3-4, 407 –

[Klo26] Hendrik Kloosterman,On the representation of numbers in the formax2 +by 2 +cz 2 +dt 2, Acta Math.49 (1926), no. 3-4, 407 –

work page 1926