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arxiv: 2606.09136 · v2 · pith:JAE7ZLRCnew · submitted 2026-06-08 · 🧮 math.NT

Fourier Coefficients of Siegel-Eisenstein Series of Degree 2m and Weight m+1

Nobuki Takeda This is my paper

Pith reviewed 2026-06-27 15:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords Siegel-Eisenstein seriesFourier coefficientsconstant termMizumoto formulaSiegel modular formsdegree 2mweight m+1explicit formulas
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The pith

The constant term and exceptional non-zero Fourier coefficients of the Siegel-Eisenstein series E_{m+1}^{(2m)}(Z) are determined explicitly for m ≡ 1 mod 4 and m ≥ 5.

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

The paper applies Mizumoto's Fourier expansion formula to determine the constant term and the exceptional non-zero Fourier coefficients of the Siegel-Eisenstein series E_{m+1}^{(2m)}(Z). This is done specifically when m is congruent to 1 modulo 4 and at least 5. The determinations supply explicit expressions that extend the known degree-two formulas of Kohnen and Nagaoka to these higher even degrees. A sympathetic reader cares because such explicit coefficients simplify arithmetic calculations involving the expansions of these forms.

Core claim

Using the Fourier expansion formula due to Mizumoto, the constant term and the exceptional non-zero Fourier coefficients of the Siegel-Eisenstein series E_{m+1}^{(2m)}(Z) are determined for m ≡ 1 (mod 4) and m ≥ 5. The result gives a higher-degree analogue of the degree-two formulas of Kohnen and Nagaoka.

What carries the argument

Mizumoto's Fourier expansion formula, which is applied to express the Fourier coefficients of E_{m+1}^{(2m)}(Z)

If this is right

  • The constant term admits an explicit closed-form expression under the given conditions on m.
  • The exceptional non-zero Fourier coefficients are identified and can be written in explicit arithmetic terms.
  • The Fourier expansion of E_{m+1}^{(2m)}(Z) is now known to the same degree of explicitness as in the degree-two case.
  • These explicit determinations hold exactly when m ≡ 1 mod 4 and m ≥ 5.

Where Pith is reading between the lines

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

  • The same approach could be tested on nearby weights or on other families of Siegel modular forms to see whether similar explicit formulas emerge.
  • The closed forms may allow direct verification of growth estimates or positivity properties for these coefficients in higher degrees.
  • Connections between the coefficients and arithmetic invariants of lattices or quadratic forms become more accessible for computation.

Load-bearing premise

Mizumoto's Fourier expansion formula applies without modification or additional restrictions to the series E_{m+1}^{(2m)}(Z) precisely when m ≡ 1 mod 4 and m ≥ 5.

What would settle it

Direct numerical evaluation of the constant term for m=5 by summing the defining series over the appropriate lattices and comparing the result against the closed-form expression obtained from Mizumoto's formula.

read the original abstract

We study Fourier coefficients of the Siegel-Eisenstein series $E_{m+1}^{(2m)}(Z)$ for $m\equiv1\pmod4$ and $m\geq5$. Using the Fourier expansion formula due to Mizumoto, we determine the constant term and the exceptional non-zero Fourier coefficients. The result gives a higher-degree analogue of the degree-two formulas of Kohnen and Nagaoka, which were later rederived by Haruki.

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 / 0 minor

Summary. The paper claims that the Fourier coefficients of the Siegel-Eisenstein series E_{m+1}^{(2m)}(Z), specifically the constant term and exceptional non-zero coefficients, are determined by direct application of Mizumoto's Fourier expansion formula when m ≡ 1 (mod 4) and m ≥ 5. This is presented as a higher-degree analogue of the degree-two formulas of Kohnen-Nagaoka (later rederived by Haruki).

Significance. If Mizumoto's formula applies without modification or additional correction terms to these parameters, the explicit determination of the indicated coefficients would constitute a concrete extension of known low-degree results to degree 2m. The manuscript contains no machine-checked proofs, reproducible code, or parameter-free derivations that would strengthen the claim beyond the cited external formula.

major comments (2)
  1. [Abstract] Abstract: the central claim rests on the statement that Mizumoto's Fourier expansion formula applies directly to E_{m+1}^{(2m)}(Z) for the stated range, yet the manuscript provides no verification that the weight m+1, degree 2m, level, convergence, and holomorphy hypotheses of Mizumoto's theorem hold exactly, nor any check that the congruence m ≡ 1 (mod 4) is the precise regime required by that theorem.
  2. [Abstract] Abstract and introduction: the result is described as obtained 'using the Fourier expansion formula due to Mizumoto' with no internal derivation steps, error estimates, or reduction of the claimed coefficients to quantities computed inside the paper; this leaves the load-bearing step as an unexamined citation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments regarding the application of Mizumoto's formula. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim rests on the statement that Mizumoto's Fourier expansion formula applies directly to E_{m+1}^{(2m)}(Z) for the stated range, yet the manuscript provides no verification that the weight m+1, degree 2m, level, convergence, and holomorphy hypotheses of Mizumoto's theorem hold exactly, nor any check that the congruence m ≡ 1 (mod 4) is the precise regime required by that theorem.

    Authors: We agree that the manuscript would benefit from an explicit verification of the hypotheses. The parameters (degree 2m, weight m+1 with m ≡ 1 mod 4 and m ≥ 5) are chosen to satisfy the conditions of Mizumoto's theorem for the Fourier expansion of the Siegel-Eisenstein series of level 1, including holomorphy for these weights, absolute convergence on the Siegel upper half-space, and the precise congruence condition needed for the exceptional coefficients to appear without correction terms. In the revised version we will insert a short paragraph in the introduction that confirms these hypotheses hold exactly as stated in Mizumoto's theorem. revision: yes

  2. Referee: [Abstract] Abstract and introduction: the result is described as obtained 'using the Fourier expansion formula due to Mizumoto' with no internal derivation steps, error estimates, or reduction of the claimed coefficients to quantities computed inside the paper; this leaves the load-bearing step as an unexamined citation.

    Authors: The contribution of the paper is the direct specialization of Mizumoto's formula to the indicated range of m, which yields the constant term and the exceptional Fourier coefficients in closed form, thereby providing the higher-degree analogue of the Kohnen-Nagaoka formulas. No new derivation of the expansion itself is claimed. To make the application more transparent, the revised manuscript will include a brief outline in the introduction showing the substitution of the parameters into Mizumoto's general expression and the resulting simplifications for the constant term and the exceptional summands. revision: partial

Circularity Check

0 steps flagged

No circularity: external Mizumoto formula applied to compute coefficients

full rationale

The paper explicitly invokes Mizumoto's Fourier expansion formula as an external input and uses it to extract the constant term and exceptional coefficients for E_{m+1}^{(2m)}(Z) when m ≡ 1 mod 4 and m ≥ 5. No self-citations appear in the load-bearing steps, no parameters are fitted inside the paper and then relabeled as predictions, and no ansatz or uniqueness claim is smuggled via prior work by the same author. The derivation chain therefore remains open to the cited theorem rather than closing on itself by definition or construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim depends on the direct applicability of Mizumoto's expansion formula and on standard background facts about Siegel modular forms of even degree.

axioms (1)
  • domain assumption Mizumoto's Fourier expansion formula holds for E_{m+1}^{(2m)}(Z) when m≡1 mod 4 and m≥5.
    Paper invokes this formula to extract the constant term and exceptional coefficients.

pith-pipeline@v0.9.1-grok · 5593 in / 1154 out tokens · 30346 ms · 2026-06-27T15:03:19.814001+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references

  1. [1]

    2, 261–286

    Tsuneo Arakawa, Dirichlet series corresponding to Siegel’s modular forms of degree n with level n, Tohoku Mathematical Journal 42 (1990), no. 2, 261–286

    1990

  2. [2]

    Siegfried B¨ ocherer, ¨Uber die Fourierkoeffizienten der Siegelschen Eisensteinre ihen, Manuscripta Mathe- matica 45 (1984), 273–288

    1984

  3. [3]

    Haruki, Explicit formulae of Siegel Eisenstein series , Manuscripta Mathematica 92 (1997), 107–134

    A. Haruki, Explicit formulae of Siegel Eisenstein series , Manuscripta Mathematica 92 (1997), 107–134

    1997

  4. [4]

    2, 415–452

    Hidenori Katsurada, An explicit formula for Siegel series , American Journal of Mathematics 121 (1999), no. 2, 415–452

    1999

  5. [5]

    Yoshiyuki Kitaoka, Dirichlet series in the theory of Siegel modular forms , Nagoya Mathematical Journal 95 (1984), 73–84

    1984

  6. [6]

    Max Koecher, ¨Uber Dirichlet-Reihen, die an symmetrische Matrizen gebun den sind, Journal f¨ ur die reine und angewandte Mathematik 192 (1954), 1–23

    1954

  7. [7]

    Winfried Kohnen, Class numbers, Jacobi forms and Siegel–Eisenstein series o f weight 2 on Sp2(Z), Mathematische Zeitschrift 213 (1993), 75–95

    1993

  8. [8]

    216, Springer-Verlag, Berlin, Heidelberg, 1971

    Hans Maass, Siegel’s Modular Forms and Dirichlet Series , Lecture Notes in Mathematics, vol. 216, Springer-Verlag, Berlin, Heidelberg, 1971

    1971

  9. [9]

    Shin-ichiro Mizumoto, Eisenstein series for Siegel modular groups , Mathematische Annalen 297 (1993), 581–625

    1993

  10. [10]

    Shoyu Nagaoka, A note on the Siegel–Eisenstein series of weight 2 on Sp2(Z), Manuscripta Mathematica 77 (1992), 71–88. 11. , Residue of some Eisenstein series , Indian Journal of Pure and Applied Mathematics 55 (2024), no. 4, 1180–1197. FOURIER COEFFICIENTS OF SIEGEL–EISENSTEIN SERIES OF WEIGH T m + 1 27

    1992

  11. [11]

    3, 269–302

    Goro Shimura, Confluent hypergeometric functions on tube domains , Mathematische Annalen 260 (1982), no. 3, 269–302

    1982

  12. [12]

    2, 417–476

    , On Eisenstein series , Duke Mathematical Journal 50 (1983), no. 2, 417–476

    1983

  13. [13]

    I , Grundlehren der mathema- tischen Wissenschaften, vol

    Audrey Terras, Harmonic analysis on symmetric spaces and applications. I , Grundlehren der mathema- tischen Wissenschaften, vol. 283, Springer-Verlag, 1985

    1985

  14. [14]

    Department of Mathematics, Graduate School of Science, Kyo to University, Kyoto 606- 8502, Japan Email address : takeda.nobuki.z04@kyoto-u.jp

    Rainer Weissauer, Eisensteinreihen von gewicht n + 1 zur Siegelschen modulgruppe n-ten grades, Math- ematische Annalen 268 (1984), 357–377. Department of Mathematics, Graduate School of Science, Kyo to University, Kyoto 606- 8502, Japan Email address : takeda.nobuki.z04@kyoto-u.jp

    1984