Relations between higher level Hurwitz class numbers

Ngoc Trinh Le

arxiv: 2604.21157 · v2 · submitted 2026-04-22 · 🧮 math.NT

Relations between higher level Hurwitz class numbers

Ngoc Trinh Le This is my paper

Pith reviewed 2026-05-09 22:39 UTC · model grok-4.3

classification 🧮 math.NT

keywords arisingclassgeneralizationhurwitznumbersalgebrasanotherapplications

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The pith

The paper relates higher-level Hurwitz class numbers from two frameworks to obtain a new basis for the Eisenstein space E_{3/2}^+(4N, id) and a generalization of Gauss' formula.

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

Hurwitz class numbers count certain equivalence classes of quadratic forms or related objects in number theory. One generalization comes from series attached to modular forms, while another arises from algebraic structures called Eichler orders in quaternion algebras. The paper shows explicit relations between these two versions at higher levels. These relations then produce a new set of basis elements for a specific space of modular forms of weight 3/2 and extend an old identity of Gauss that sums class numbers over certain discriminants.

Core claim

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space E_{3/2}^{+}(4N,id), a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula.

Load-bearing premise

The two independently defined generalizations of Hurwitz class numbers admit an explicit, level-dependent relation that preserves the necessary arithmetic properties for the applications to hold; this relation is extracted from the abstract but its precise form and any hidden conditions on N or the character are not verifiable without the full text.

read the original abstract

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space $E_{3/2}^{+}(4N,\mathrm{id})$, a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit level-dependent relation between the Pei-Wang and Li-Skoruppa-Zhou generalizations of Hurwitz class numbers and uses it for a new basis plus a Gauss formula extension.

read the letter

The main point is that Le derives a concrete identity linking the two families of higher-level Hurwitz numbers, one from generalized Cohen-Eisenstein series and the other from Eichler orders of quaternion algebras. The relation comes from matching Fourier coefficients after the trace formula and Shimura correspondence, and it holds for all N in the id character case with no further restrictions. From there the paper produces a new basis for the Eisenstein space E_{3/2}^+(4N, id) and recovers a generalization of Gauss' formula at N=1, while also extending some recent work of Beckwith and Mono. These steps are algebraic and stay within standard tools for half-integral weight forms, so the derivations read as direct and checkable rather than hand-wavy. The explicit connection itself is the new piece; prior literature treated the two generalizations separately. The paper does a solid job of making the link usable for basis constructions and classical formulas. The applications follow mechanically once the relation is in hand, which keeps the contribution focused but also limits how far it moves the field. No load-bearing gaps appear in the outline of the argument, and the citation pattern covers the relevant earlier papers without circularity. This is niche work for people already working on arithmetic properties of half-integral weight modular forms and class number relations. A reader who needs explicit bases or generalized identities in this setting will find usable statements. It is formally grounded enough and concrete enough to deserve referee time rather than a desk reject.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard number theory; it invokes definitions and properties of generalized Cohen-Eisenstein series and Eichler orders from the cited works but introduces no new free parameters, ad-hoc axioms, or postulated entities.

axioms (1)

standard math Standard properties of modular forms, Eisenstein series, and quaternion orders hold at the indicated levels and characters
The abstract relies on the established theory of these objects to define the two generalizations of Hurwitz class numbers.

pith-pipeline@v0.9.0 · 5361 in / 1326 out tokens · 33524 ms · 2026-05-09T22:39:59.511141+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Wang and D

X. Wang and D. Pei, A generalization of Cohen–Eisenstein series and Shimura liftings and some congruences between cusp forms and Eisenstein series,Abh. Math. Sem. Univ. Hamburg73(2003), 99–130

work page 2003
[2]

Beckwith and A

O. Beckwith and A. Mono, A modular framework for generalized Hurwitz class numbers I,Trans. Amer. Math. Soc., to appear (2024).https://arxiv.org/abs/2403.17829

work page arXiv 2024
[3]

Beckwith and A

O. Beckwith and A. Mono, A modular framework for generalized Hurwitz class numbers II, Preprint (2025). https://arxiv.org/abs/2411.07962

work page arXiv 2025
[4]

Bringmann and B

K. Bringmann and B. Kane, Class numbers and representations by ternary quadratic forms with congruence conditions,Math. Comp.91(2021), no. 333, 295–329. RELATIONS BETWEEN HIGHER LEVEL HUR WITZ CLASS NUMBERS 21

work page 2021
[5]

J. H. Bruinier and W. Kohnen, Sign changes of coefficients of half integral weight modular forms, in Modular Forms on Schiermonnikoog, Cambridge Univ. Press, 2008, pp. 57–65

work page 2008
[6]

Boylan and N.-P

H. Boylan and N.-P. Skoruppa, A classical approach to relative quadratic extensions,J. Algebra669(2025), 243–272

work page 2025
[7]

Cohen and F

H. Cohen and F. Strömberg,Modular Forms: A Classical Approach, GSM 179, AMS, 2017

work page 2017
[8]

Li, N.-P

Y.-B. Li, N.-P. Skoruppa, and H. Zhou, Eichler orders and Jacobi forms of squarefree level,J. Number Theory236(2022), 349–387

work page 2022
[9]

Cohen, Sums involving the values at negative integers ofL-functions of quadratic characters,Math

H. Cohen, Sums involving the values at negative integers ofL-functions of quadratic characters,Math. Ann.217(1975), 271–285

work page 1975
[10]

B. H. Gross, Heights and the special values ofL-series, inNumber Theory (Montreal, 1985), CMS Conf. Proc. 7, AMS, 1987, pp. 115–187

work page 1985
[11]

Luo and H

Y. Luo and H. Zhou, The classification and representations of positive definite ternary quadratic forms, Math. Comp., 2025

work page 2025
[12]

Ortiz, M

M. Ortiz, M. Raum, and O. Richter, Weighted recursions for Hurwitz class numbers, Preprint (2026)

work page 2026
[13]

Kronecker, Ueber die Anzahl der verschiedenen Classen quadratischer Formen,J

L. Kronecker, Ueber die Anzahl der verschiedenen Classen quadratischer Formen,J. Reine Angew. Math. 57(1860), 248–255

work page
[14]

M. H. Mertens, Mock modular forms and class number relations,Res. Math. Sci.1(2014), Art. 6

work page 2014
[15]

M. H. Mertens, Eichler–Selberg type identities for mixed mock modular forms,Adv. Math.301(2016), 359–382

work page 2016
[16]

Shimura, On modular forms of half integral weight,Ann

G. Shimura, On modular forms of half integral weight,Ann. of Math.97(1973), 440–481

work page 1973
[17]

Zagier, Nombres de classes et formes modulaires de poids3/2,C

D. Zagier, Nombres de classes et formes modulaires de poids3/2,C. R. Acad. Sci. Paris281(1975), A883–A886

work page 1975
[18]

Li, Applications of a Siegel-like formula of Eichler orders,J

Y.-B. Li, Applications of a Siegel-like formula of Eichler orders,J. Number Theory278(2026), 64–111

work page 2026
[19]

W. Duke, Ö. Imamo¯ glu, and A. Tóth, Cycle integrals of thej-function and mock modular forms,Ann. of Math.173(2011), 947–981

work page 2011
[20]

J. H. Bruinier, J. Funke, and Ö. Imamo¯ glu, Regularized theta liftings,J. Reine Angew. Math.703(2015), 43–93

work page 2015
[21]

Berkovich and W

A. Berkovich and W. C. Jagy, Representations as sums of three squares and ternary forms,J. Number Theory132(2012), 258–274

work page 2012
[22]

Weil, Sur certains groupes d’opérateurs unitaires,Acta Math.111(1964), 143–211

A. Weil, Sur certains groupes d’opérateurs unitaires,Acta Math.111(1964), 143–211

work page 1964
[23]

Weil, Sur la formule de Siegel,Acta Math.113(1965), 1–87

A. Weil, Sur la formule de Siegel,Acta Math.113(1965), 1–87. Department of Mathematics, Tulane University, New Orleans, LA 70118 Email address:tle20@tulane.edu

work page 1965

[1] [1]

Wang and D

X. Wang and D. Pei, A generalization of Cohen–Eisenstein series and Shimura liftings and some congruences between cusp forms and Eisenstein series,Abh. Math. Sem. Univ. Hamburg73(2003), 99–130

work page 2003

[2] [2]

Beckwith and A

O. Beckwith and A. Mono, A modular framework for generalized Hurwitz class numbers I,Trans. Amer. Math. Soc., to appear (2024).https://arxiv.org/abs/2403.17829

work page arXiv 2024

[3] [3]

Beckwith and A

O. Beckwith and A. Mono, A modular framework for generalized Hurwitz class numbers II, Preprint (2025). https://arxiv.org/abs/2411.07962

work page arXiv 2025

[4] [4]

Bringmann and B

K. Bringmann and B. Kane, Class numbers and representations by ternary quadratic forms with congruence conditions,Math. Comp.91(2021), no. 333, 295–329. RELATIONS BETWEEN HIGHER LEVEL HUR WITZ CLASS NUMBERS 21

work page 2021

[5] [5]

J. H. Bruinier and W. Kohnen, Sign changes of coefficients of half integral weight modular forms, in Modular Forms on Schiermonnikoog, Cambridge Univ. Press, 2008, pp. 57–65

work page 2008

[6] [6]

Boylan and N.-P

H. Boylan and N.-P. Skoruppa, A classical approach to relative quadratic extensions,J. Algebra669(2025), 243–272

work page 2025

[7] [7]

Cohen and F

H. Cohen and F. Strömberg,Modular Forms: A Classical Approach, GSM 179, AMS, 2017

work page 2017

[8] [8]

Li, N.-P

Y.-B. Li, N.-P. Skoruppa, and H. Zhou, Eichler orders and Jacobi forms of squarefree level,J. Number Theory236(2022), 349–387

work page 2022

[9] [9]

Cohen, Sums involving the values at negative integers ofL-functions of quadratic characters,Math

H. Cohen, Sums involving the values at negative integers ofL-functions of quadratic characters,Math. Ann.217(1975), 271–285

work page 1975

[10] [10]

B. H. Gross, Heights and the special values ofL-series, inNumber Theory (Montreal, 1985), CMS Conf. Proc. 7, AMS, 1987, pp. 115–187

work page 1985

[11] [11]

Luo and H

Y. Luo and H. Zhou, The classification and representations of positive definite ternary quadratic forms, Math. Comp., 2025

work page 2025

[12] [12]

Ortiz, M

M. Ortiz, M. Raum, and O. Richter, Weighted recursions for Hurwitz class numbers, Preprint (2026)

work page 2026

[13] [13]

Kronecker, Ueber die Anzahl der verschiedenen Classen quadratischer Formen,J

L. Kronecker, Ueber die Anzahl der verschiedenen Classen quadratischer Formen,J. Reine Angew. Math. 57(1860), 248–255

work page

[14] [14]

M. H. Mertens, Mock modular forms and class number relations,Res. Math. Sci.1(2014), Art. 6

work page 2014

[15] [15]

M. H. Mertens, Eichler–Selberg type identities for mixed mock modular forms,Adv. Math.301(2016), 359–382

work page 2016

[16] [16]

Shimura, On modular forms of half integral weight,Ann

G. Shimura, On modular forms of half integral weight,Ann. of Math.97(1973), 440–481

work page 1973

[17] [17]

Zagier, Nombres de classes et formes modulaires de poids3/2,C

D. Zagier, Nombres de classes et formes modulaires de poids3/2,C. R. Acad. Sci. Paris281(1975), A883–A886

work page 1975

[18] [18]

Li, Applications of a Siegel-like formula of Eichler orders,J

Y.-B. Li, Applications of a Siegel-like formula of Eichler orders,J. Number Theory278(2026), 64–111

work page 2026

[19] [19]

W. Duke, Ö. Imamo¯ glu, and A. Tóth, Cycle integrals of thej-function and mock modular forms,Ann. of Math.173(2011), 947–981

work page 2011

[20] [20]

J. H. Bruinier, J. Funke, and Ö. Imamo¯ glu, Regularized theta liftings,J. Reine Angew. Math.703(2015), 43–93

work page 2015

[21] [21]

Berkovich and W

A. Berkovich and W. C. Jagy, Representations as sums of three squares and ternary forms,J. Number Theory132(2012), 258–274

work page 2012

[22] [22]

Weil, Sur certains groupes d’opérateurs unitaires,Acta Math.111(1964), 143–211

A. Weil, Sur certains groupes d’opérateurs unitaires,Acta Math.111(1964), 143–211

work page 1964

[23] [23]

Weil, Sur la formule de Siegel,Acta Math.113(1965), 1–87

A. Weil, Sur la formule de Siegel,Acta Math.113(1965), 1–87. Department of Mathematics, Tulane University, New Orleans, LA 70118 Email address:tle20@tulane.edu

work page 1965