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arxiv: 2603.18511 · v2 · submitted 2026-03-19 · 🧮 math.NT

Norm-trace and Kloosterman sums in finite semi-simple algebras

Daqing Wan This is my paper

Pith reviewed 2026-05-15 09:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords norm-trace sumsKloosterman sumsfinite semisimple algebrasGauss sumsHasse-Davenportasymptotic formulascharacter sums
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The pith

An asymptotic formula with square root error counts elements by trace and norm in finite semisimple algebras over finite fields.

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

The paper establishes an asymptotic formula with a square root error term for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from the commutative etale case to the non-commutative semisimple case. The main technique is to use the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the counting problem to the classical geometric case where the result is already known. The same reduction gives a square root estimate for Kloosterman sums over these algebras. The work also discusses the product-trace version and poses a new conjecture for that counting problem.

Core claim

An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field by applying the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the problem to the classical geometric case.

What carries the argument

The Eichler formula for Gauss sums over the general linear group combined with the Hasse-Davenport relation, which reduces the non-commutative semisimple case to the known etale geometric case.

If this is right

  • The asymptotic count with square root error holds for all finite semisimple algebras.
  • Kloosterman sums over finite semisimple algebras satisfy a square root estimate.
  • A conjecture is formulated for the analogous counting problem using product and trace instead of norm and trace.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction method may extend to other types of character sums in non-commutative settings.
  • The uniformity of trace and norm distribution in semisimple algebras matches that in fields up to square root terms.
  • Direct computation in low-dimensional examples like matrix rings could test the error bound quickly.

Load-bearing premise

The Eichler formula for Gauss sums and the Hasse-Davenport relation extend from the commutative etale case to the non-commutative semisimple case without obstruction.

What would settle it

Explicit enumeration of all elements in a small semisimple algebra such as M_2 over a small finite field to check if the error in the trace-norm count stays within the square root bound.

read the original abstract

An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from finite etale algebras (commutative case) to finite semi-simple algebras (non-commutative case). The main idea is to apply the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the problem to the classical geometric case where the result is known to be true. As an application of this reduction, we also obtain a square root estimate for Kloosterman sums over semi-simple algebras. Similar square root estimates are discussed when norm-trace is replaced by product-trace, leading to a new conjecture on product-trace counting over finite semi-simple algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims an asymptotic formula with square-root error term for the number of elements with prescribed trace and norm in a finite semisimple algebra over a finite field. The argument reduces the count to the known geometric result for étale algebras by applying the Eichler formula for Gauss sums over GL_n and the Hasse-Davenport relation. Applications include square-root bounds for Kloosterman sums over semisimple algebras, and a conjecture is proposed for analogous product-trace counts.

Significance. If the reduction is valid, the result extends classical square-root bounds on norm-trace distributions from commutative étale algebras to the non-commutative semisimple setting, with direct consequences for character-sum estimates in matrix algebras over finite fields. The approach is economical and the error term is of optimal quality when it holds.

major comments (2)
  1. [central reduction argument] The reduction in the proof of the main asymptotic (stated after the abstract and detailed in the central section) asserts that the Eichler formula for Gauss sums over GL and the Hasse-Davenport relation extend verbatim to A ≅ ∏ M_{n_i}(F_{q^{d_i}}). No explicit verification is given that the pairing of the additive character on the trace with the multiplicative character on the norm factors without introducing extra error terms that would spoil the square-root bound.
  2. [Kloosterman-sum application] Theorem on the Kloosterman-sum estimate (the application section) inherits the same reduction; any failure of exact product decomposition for the character sums in the non-commutative case would invalidate the claimed square-root bound, yet no separate error analysis or small-case check (e.g., for M_2(F_q)) is supplied.
minor comments (2)
  1. [preliminaries] Notation for the trace and norm maps on the product algebra should be introduced with an explicit formula before the reduction is invoked.
  2. [final section] The conjecture on product-trace counts is stated without any numerical checks or partial results for small algebras.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of our results. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [central reduction argument] The reduction in the proof of the main asymptotic (stated after the abstract and detailed in the central section) asserts that the Eichler formula for Gauss sums over GL and the Hasse-Davenport relation extend verbatim to A ≅ ∏ M_{n_i}(F_{q^{d_i}}). No explicit verification is given that the pairing of the additive character on the trace with the multiplicative character on the norm factors without introducing extra error terms that would spoil the square-root bound.

    Authors: Because A is semisimple it decomposes as a direct product of matrix algebras M_{n_i}(F_{q^{d_i}}). The global trace is the sum of the component traces and the global norm is the product of the component norms; consequently the additive character attached to the trace and the multiplicative character attached to the norm factor exactly as a product over the components. The Eichler formula therefore applies independently to each GL_{n_i} factor, and the Hasse-Davenport relation lifts the characters componentwise. The resulting Gauss sum is therefore the product of the local sums, yielding an exact reduction to the known étale case with no cross terms or additional error. We will add a short paragraph in the central section spelling out this factorization. revision: yes

  2. Referee: [Kloosterman-sum application] Theorem on the Kloosterman-sum estimate (the application section) inherits the same reduction; any failure of exact product decomposition for the character sums in the non-commutative case would invalidate the claimed square-root bound, yet no separate error analysis or small-case check (e.g., for M_2(F_q)) is supplied.

    Authors: The Kloosterman sum is built from the same trace and norm characters, so the exact product decomposition already established for the main count carries over verbatim and preserves the square-root error term. To address the request for an explicit check we will insert a short remark in the application section verifying the bound directly for A = M_2(F_q) when q is small (e.g., q = 2, 3). revision: yes

Circularity Check

0 steps flagged

No circularity; reduction to independently known etale geometric case

full rationale

The paper derives the square-root-error asymptotic for norm-trace counts in finite semisimple algebras by invoking the Eichler formula for Gauss sums over GL and the Hasse-Davenport relation to reduce directly to the classical geometric result already established for the commutative etale case. No equations, definitions, or steps in the abstract or described method equate the target count to its own inputs by construction, rename a fitted parameter as a prediction, or rely on a self-citation chain whose validity is internal to the present work. The argument is therefore self-contained against external benchmarks in the literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on standard properties of finite fields, semisimple algebras, and two named classical formulas whose validity is taken from prior literature.

axioms (1)
  • standard math Finite semisimple algebras over finite fields admit the Eichler formula for Gauss sums over GL(n) and satisfy the Hasse-Davenport relation.
    Invoked in the abstract as the mechanism that reduces the problem to the classical geometric case.

pith-pipeline@v0.9.0 · 5418 in / 1275 out tokens · 47774 ms · 2026-05-15T09:10:20.814681+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Adolphson and S

    A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: cohomology and estimates,Ann. Math., 130(1989), 367-406

    work page 1989

  2. [2]

    Eichler, Allgemeine Kongruenz-Klasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkör- pern und ihre L-Reihen.J

    M. Eichler, Allgemeine Kongruenz-Klasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkör- pern und ihre L-Reihen.J. Reine Angew. Math.,179(1937), 227 - 251

    work page 1937

  3. [3]

    G. L. Matthews, T. Morrison, and A. W. Murphy, Curve-lifted codes for local recovery using lines,Des. Codes Cryptogr.,92(2024), 3645-3664

    work page 2024

  4. [4]

    Cohomologie étale (SGA 4 1 2 )

    P. Deligne, “Cohomologie étale (SGA 4 1 2 )",Lecture Notes in Mathematics, V ol. 569, Springer-Verlag, Berlin/Heidelberg/New York, 1977

    work page 1977

  5. [5]

    Denef and F

    J. Denef and F. Loeser, Weights of exponential sums, intersection cohomology, and Newton polyhedra,Invent. Math., V ol.106(1991), 275-294

    work page 1991

  6. [6]

    Katz, Gauss Sums,Kloosterman Sums, and Monodromy Groups, Princeton University Press, 1988

    N. Katz, Gauss Sums,Kloosterman Sums, and Monodromy Groups, Princeton University Press, 1988

    work page 1988

  7. [7]

    Katz, Estimates for Soto-Andrade sums,J

    N. Katz, Estimates for Soto-Andrade sums,J. Reine Angew. Math.,438(1993), 143-161

    work page 1993

  8. [8]

    Kim, Gauss sums for general and special linear groups over a finite field,Arch

    D. Kim, Gauss sums for general and special linear groups over a finite field,Arch. Math.,69(1997), 297-304

    work page 1997

  9. [9]

    Kim, Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums, Annali di Matematica,190(2011), 61-76

    D. Kim, Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums, Annali di Matematica,190(2011), 61-76

    work page 2011

  10. [10]

    Lamprecht, Struktur und Relationen allgemeiner Gaußcher Summen in endlichen Ringen I, II,J

    E. Lamprecht, Struktur und Relationen allgemeiner Gaußcher Summen in endlichen Ringen I, II,J. Reine Angew. Math.,197(1957), 1-48

    work page 1957

  11. [11]

    Li and S

    Y . Li and S. Hu, Gauss sums over some matrix groups,J. Number Theory,132(2012), 2967-2976

    work page 2012

  12. [12]

    Lin and D

    X. Lin and D. Wan, Counting elements with given trace and norm in étale algebras,International J. Number Theory, 21(2025), 1955-1965

    work page 2025

  13. [13]

    Moisio, Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm,Acta Arith.,132(2008), 329-350

    M. Moisio, Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm,Acta Arith.,132(2008), 329-350

    work page 2008

  14. [14]

    Moisio and D

    M. Moisio and D. Wan, On Katz’s bound for the number of elements with given trace and norm,J. Reine Angew. Math.,638(2010), 69-74

    work page 2010

  15. [15]

    Rojas-Leon, Rationality of trace and normL-functions,Duke Math

    A. Rojas-Leon, Rationality of trace and normL-functions,Duke Math. J.,161(2012), 1751-1795

    work page 2012

  16. [16]

    Rojas-Leon, On the number of rational points on curves over finite fields with many automorphisms,Finite Fields & Appl.,19(2013), 1-15

    A. Rojas-Leon, On the number of rational points on curves over finite fields with many automorphisms,Finite Fields & Appl.,19(2013), 1-15

    work page 2013

  17. [17]

    Rojas-Leon, Local convolution ofℓ-adic sheaves on the torus,Math Z.,274(2013), 1211-1230

    A. Rojas-Leon, Local convolution ofℓ-adic sheaves on the torus,Math Z.,274(2013), 1211-1230

    work page 2013

  18. [18]

    Rojas-Leon and D

    A. Rojas-Leon and D. Wan, Moment zeta functions for toric Calabi-Yau hypersurfaces,Commun. Number Th. Phys.,1(2007), 539-578

    work page 2007

  19. [19]

    Rojas-Leon and D

    A. Rojas-Leon and D. Wan, Improvements of the Weil bound for Artin-Schreier curves,Math. Ann.,351(2011), 417–442

    work page 2011

  20. [20]

    Wan, Lectures on zeta functions over finite fields,Higher Dimensional Geometry over Finite Fields, D

    D. Wan, Lectures on zeta functions over finite fields,Higher Dimensional Geometry over Finite Fields, D. Kaledin and Y . Tschinkel, eds., ISO Press (2008), 244-268

    work page 2008

  21. [21]

    Zelingher, On matrix Kloosterman sums and Hall-Littlewood polynomials,Trans

    E. Zelingher, On matrix Kloosterman sums and Hall-Littlewood polynomials,Trans. Amer. Math. Soc.,378(2025), 3597–3623. CENTER FORDISCRETEMATHEMATICS, COLLEGE OFMATHEMATICS ANDSTATISTICS, CHONGQINGUNIVER- SITY, CHONGQING401331, CHINA. Email address:dwan@math.uci.edu

    work page 2025