Antisymmetric paramodular forms of weight 3

Haowu Wang; Valery Gritsenko

arxiv: 1906.09869 · v1 · pith:GQMUW333new · submitted 2019-06-24 · 🧮 math.NT · math.AG

Antisymmetric paramodular forms of weight 3

Valery Gritsenko , Haowu Wang This is my paper

Pith reviewed 2026-05-25 17:12 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords paramodular formsBorcherds productstheta blocksKummer surfacesmoduli spacescusp formsweight 3antisymmetric forms

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The pith

An infinite family of antisymmetric paramodular forms of weight 3 is built as Borcherds products starting from theta blocks.

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

The problem of constructing antisymmetric paramodular forms of weight 3 has remained open since 1998. Any cusp form of this type produces a canonical differential form on a smooth compactification of the moduli space of Kummer surfaces attached to (1,t)-polarized abelian surfaces. The paper supplies the first infinite family of such forms by realizing them as Borcherds products whose leading Fourier-Jacobi coefficient is a theta block. These products are shown to be cusp forms on the paramodular group and to satisfy the required antisymmetry and weight conditions.

Core claim

We construct the first infinite family of antisymmetric paramodular forms of weight 3 as Borcherds products whose first Fourier-Jacobi coefficient is a theta block.

What carries the argument

Borcherds products whose first Fourier-Jacobi coefficient is a theta block

Load-bearing premise

The Borcherds products built from the chosen theta blocks are antisymmetric, have weight exactly 3, and are cusp forms for the paramodular group.

What would settle it

Explicit computation of the Fourier expansion or the action of the paramodular group on the first few members of the family that shows the weight is not 3 or that the form is not antisymmetric.

read the original abstract

The problem on the construction of antisymmetric paramodular forms of canonical weight 3 was open since 1998. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to $(1,t)$-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight 3 as Borcherds products whose first Fourier-Jacobi coefficient is a theta block.

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

They give the first infinite family of antisymmetric paramodular forms of weight 3 by taking Borcherds products whose leading Fourier-Jacobi coefficient is a theta block.

read the letter

The central result is a concrete construction that closes the 1998 problem on antisymmetric paramodular forms of weight 3. The authors produce an infinite family as Borcherds products whose first Fourier-Jacobi coefficient is a theta block, and these forms are claimed to be cusp forms for the paramodular group. That supplies explicit objects that determine canonical differentials on the moduli spaces of Kummer surfaces coming from (1,t)-polarized abelian surfaces. The method is direct: choose suitable theta blocks so that the resulting Borcherds product lands in weight 3 and satisfies the antisymmetry condition under the relevant involution. Borcherds products are a standard tool in this area, so the novelty sits in identifying the right input blocks that work for every t in an arithmetic progression or similar infinite set. The abstract states the construction succeeds, and the stress-test note finds no internal inconsistency in the claim. The main limitation visible from the given material is that the abstract itself contains no sample Fourier expansions, no explicit list of the theta blocks used, and no verification that the weight and antisymmetry conditions hold after the product is formed. A reader therefore has to trust that the full paper carries out those checks in detail. If those checks are present and correct, the construction is usable. This paper is aimed at people who work with Siegel modular forms of genus 2, paramodular groups, or the geometry of moduli spaces of abelian surfaces. It will be cited by anyone who needs explicit examples of weight-3 forms with the antisymmetry property or who wants to compute the corresponding differentials. It deserves a serious referee because it resolves a stated open construction problem with a reproducible method rather than an existence argument.

Referee Report

0 major / 2 minor

Summary. The paper constructs the first infinite family of antisymmetric paramodular forms of weight 3 as Borcherds products whose first Fourier-Jacobi coefficient is a theta block. This resolves an open problem from 1998; such forms determine canonical differential forms on smooth compactifications of the moduli space of Kummer surfaces associated to (1,t)-polarised abelian surfaces.

Significance. If verified, the explicit infinite family is a substantial advance in the theory of paramodular forms and their geometric applications. The construction via Borcherds products with theta-block input supplies the required objects directly and is a strength of the work.

minor comments (2)

The abstract states the main result but the introduction or §1 should include a brief outline of the choice of theta blocks that produce weight exactly 3 and the verification that the resulting Borcherds product is antisymmetric under the paramodular group action.
Add a short table or list in §3 or §4 enumerating the first few members of the infinite family with their level t and the explicit theta block used, to make the construction immediately usable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The referee summary correctly identifies the resolution of the 1998 open problem via the explicit infinite family of antisymmetric paramodular forms of weight 3 constructed as Borcherds products.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is an explicit construction of an infinite family of antisymmetric paramodular forms of weight 3 realized as Borcherds products whose leading Fourier-Jacobi coefficient is a theta block. No step in the provided abstract or description reduces a prediction or uniqueness claim to a fitted parameter, self-definition, or load-bearing self-citation chain; the construction itself supplies the objects once the input theta blocks are selected to yield weight 3. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the construction is presented as relying on standard properties of Borcherds products and theta blocks.

pith-pipeline@v0.9.0 · 5598 in / 1029 out tokens · 27983 ms · 2026-05-25T17:12:58.340938+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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R. E. Borcherds, Automorphic forms on Os+2,2(R) and inﬁnite products, Invent. Math. 120:1 (1995), 161–213

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J. Breeding II, C. Poor, D. S. Yuen, Computations of spaces of paramodular forms of general level. J. Korean Math. Soc. 53 (2016), 645–689

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A. Brumer, K. Kramer, Paramodular abelian varieties of odd conductor. Trans. Amer. Math. Soc. 366 (2014), 2463–2516

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Cl´ ery, V

F. Cl´ ery, V. Gritsenko,Modular forms of orthogonal type and Jacobi theta-series. Abh. Math. Semin. Univ. Hambg 83 (2013), 187–217

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Eichler, D

M. Eichler, D. Zagier, The Theory of Jacobi Forms. Progress in Mathematics 55. Birkh¨ auser, Boston, Mass., 1985

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Freitag, Siegelsche Modulfunktionen

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Gritsenko, Irrationality of the moduli spaces of polarized abelian sur faces

V. Gritsenko, Irrationality of the moduli spaces of polarized abelian sur faces. Int. Math. Res. Not. IMRN 6 (1994), 235–243

work page 1994
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Gritsenko, Reﬂective modular forms and their applications

V. Gritsenko, Reﬂective modular forms and their applications. Russian Math. Surveys 73:5 (2018), 797–864

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Gritsenko, K

V. Gritsenko, K. Hulek, Minimal Siegel modular threefolds. Math. Proc. Cambridge Philos. Soc. 123 (1998) 461–485

work page 1998
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Gritsenko, K

V. Gritsenko, K. Hulek, Uniruledness of orthogonal modular varieties. J. of Algebraic Geom. 23 (2014), 711–725

work page 2014
[14]

Gritsenko, K

V. Gritsenko, K. Hulek, G. K. Sankaran, Abelianisation of orthogonal groups and the funda- mental group of modular varieties . J. Algebra 322:2 (2009), 463–478

work page 2009
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Gritsenko, K

V. Gritsenko, K. Hulek, G. K. Sankaran, The Kodaira dimension of the moduli of K3 surfaces. Invent. Math. 169:3 (2007), 519–567

work page 2007
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Gritsenko, V

V. Gritsenko, V. V. Nikulin, Igusa modular forms and the simplest Lorentzian KacMoody algebras, Sb. Math. 187:11 (1996), 1601-1641

work page 1996
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Gritsenko, V

V. Gritsenko, V. Nikulin, Automorphic forms and Lorentzian Kac–Moody algebras. Part II. Internat. J. Math. 9 (1998), 201–275. 20 V ALERY GRITSENKO AND HAOWU W ANG

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[18]

Gritsenko, V

V. Gritsenko, V. V. Nikulin, Lorentzian Kac-Moody algebras with Weyl groups of 2-reﬂections. Proc. Lond. Math. Soc. 116:3 (2018), 485–533

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Gritsenko, C

V. Gritsenko, C. Poor, D. S. Yuen, Antisymmetric paramodular forms of weights 2 and 3. Int. Math. Res. Not. IMRN, https://doi.org/10.1093/imrn/ rnz011

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[20]

Gritsenko, N

V. Gritsenko, N. P. Skoruppa, D. Zagier, Theta blocks , preprint 2018, 56 pp. https://math.univ-lille1.fr/d7/sites/ default/ﬁles/T HETA%20BLOCKS22.09.18 1.pdf

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Gritsenko, H

V. Gritsenko, H. W ang, Conjecture on theta-blocks of order 1. Russian Math. Surveys 72:5 (2017), 968–970

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[22]

Gritsenko, H

V. Gritsenko, H. W ang, Theta block conjecture for paramodular forms of weight 2. Preprint 2018, 15 pp, arXiv:1812.08698

work page arXiv 2018
[23]

Gross, S

M. Gross, S. Popescu, Calabi–Yau three-folds and moduli of abelian surfaces II. Trans. Amer. Math. Soc. 363 (2011), 3573–3599

work page 2011
[24]

Nikulin, Integer symmetric bilinear forms and some of their geometri c applications

V. Nikulin, Integer symmetric bilinear forms and some of their geometri c applications. Math. USSR Izv. 14 (1980), 103–167

work page 1980
[25]

C. Poor, J. Shurman, D. S. Yuen, Siegel paramodular forms of weight 2 and squarefree level. Int. J. Number Theory 13 (2017), 2627–2652

work page 2017
[26]

C. Poor, D. S. Yuen, Paramodular Cusp Forms, Math. Comp. 84 (no. 293) (2015), 1401–1438

work page 2015
[27]

N. R. Scheithauer, On the classiﬁcation of automorphic products and generaliz ed Kac-Moody algebras. Invent. Math. 164 (2006), 641–678

work page 2006
[28]

N. R. Scheithauer, The Weil representation of SL2(Z) and some applications. Int. Math. Res. Not. IMRN (2009) no.8, 1488–1545

work page 2009
[29]

N. R. Scheithauer, Some constructions of modular forms for the Weil representa tion of SL(2, Z). Nagoya Math. J. 220 (2015), 1–43

work page 2015
[30]

N. R. Scheithauer, Automorphic products of singular weight. Compos. Math. 153 (2017), 1855–1892. Laboratoire Paul Painlev ´e, Universit ´e de Lille and IUF, 59655 Villeneuve d’Ascq Cedex, France and National Research University Higher School o f Economics, Russian Federation E-mail address : valery.gritsenko@univ-lille.fr Laboratoire Paul Painlev ´e, Univ...

work page 2017

[1] [1]

A. Ash, P. E. Gunnells, M. McConnell, Cohomology of congruence subgroups of SL(4; Z). III. Math. Comp. 79 (2010), 1811–1831

work page 2010

[2] [2]

R. E. Borcherds, Automorphic forms on Os+2,2(R) and inﬁnite products, Invent. Math. 120:1 (1995), 161–213

work page 1995

[3] [3]

R. E. Borcherds, Automorphic forms with singularities on Grassmannians. Invent. Math., 123 (1998), no. 3, 491–562

work page 1998

[4] [4]

Bourbaki, Groupes et alg` ebres de Lie

N. Bourbaki, Groupes et alg` ebres de Lie. Chapter 4,5 et 6

work page

[5] [5]

Breeding II, C

J. Breeding II, C. Poor, D. S. Yuen, Computations of spaces of paramodular forms of general level. J. Korean Math. Soc. 53 (2016), 645–689

work page 2016

[6] [6]

Brumer, K

A. Brumer, K. Kramer, Paramodular abelian varieties of odd conductor. Trans. Amer. Math. Soc. 366 (2014), 2463–2516

work page 2014

[7] [7]

Cl´ ery, V

F. Cl´ ery, V. Gritsenko,Modular forms of orthogonal type and Jacobi theta-series. Abh. Math. Semin. Univ. Hambg 83 (2013), 187–217

work page 2013

[8] [8]

Eichler, D

M. Eichler, D. Zagier, The Theory of Jacobi Forms. Progress in Mathematics 55. Birkh¨ auser, Boston, Mass., 1985

work page 1985

[9] [9]

Freitag, Siegelsche Modulfunktionen

E. Freitag, Siegelsche Modulfunktionen. Grundlehren der mathematischen Wissenschaften 254, Springer-Verlag (1983)

work page 1983

[10] [10]

Gritsenko, Irrationality of the moduli spaces of polarized abelian sur faces

V. Gritsenko, Irrationality of the moduli spaces of polarized abelian sur faces. Int. Math. Res. Not. IMRN 6 (1994), 235–243

work page 1994

[11] [11]

Gritsenko, Reﬂective modular forms and their applications

V. Gritsenko, Reﬂective modular forms and their applications. Russian Math. Surveys 73:5 (2018), 797–864

work page 2018

[12] [12]

Gritsenko, K

V. Gritsenko, K. Hulek, Minimal Siegel modular threefolds. Math. Proc. Cambridge Philos. Soc. 123 (1998) 461–485

work page 1998

[13] [13]

Gritsenko, K

V. Gritsenko, K. Hulek, Uniruledness of orthogonal modular varieties. J. of Algebraic Geom. 23 (2014), 711–725

work page 2014

[14] [14]

Gritsenko, K

V. Gritsenko, K. Hulek, G. K. Sankaran, Abelianisation of orthogonal groups and the funda- mental group of modular varieties . J. Algebra 322:2 (2009), 463–478

work page 2009

[15] [15]

Gritsenko, K

V. Gritsenko, K. Hulek, G. K. Sankaran, The Kodaira dimension of the moduli of K3 surfaces. Invent. Math. 169:3 (2007), 519–567

work page 2007

[16] [16]

Gritsenko, V

V. Gritsenko, V. V. Nikulin, Igusa modular forms and the simplest Lorentzian KacMoody algebras, Sb. Math. 187:11 (1996), 1601-1641

work page 1996

[17] [17]

Gritsenko, V

V. Gritsenko, V. Nikulin, Automorphic forms and Lorentzian Kac–Moody algebras. Part II. Internat. J. Math. 9 (1998), 201–275. 20 V ALERY GRITSENKO AND HAOWU W ANG

work page 1998

[18] [18]

Gritsenko, V

V. Gritsenko, V. V. Nikulin, Lorentzian Kac-Moody algebras with Weyl groups of 2-reﬂections. Proc. Lond. Math. Soc. 116:3 (2018), 485–533

work page 2018

[19] [19]

Gritsenko, C

V. Gritsenko, C. Poor, D. S. Yuen, Antisymmetric paramodular forms of weights 2 and 3. Int. Math. Res. Not. IMRN, https://doi.org/10.1093/imrn/ rnz011

work page doi:10.1093/imrn/

[20] [20]

Gritsenko, N

V. Gritsenko, N. P. Skoruppa, D. Zagier, Theta blocks , preprint 2018, 56 pp. https://math.univ-lille1.fr/d7/sites/ default/ﬁles/T HETA%20BLOCKS22.09.18 1.pdf

work page 2018

[21] [21]

Gritsenko, H

V. Gritsenko, H. W ang, Conjecture on theta-blocks of order 1. Russian Math. Surveys 72:5 (2017), 968–970

work page 2017

[22] [22]

Gritsenko, H

V. Gritsenko, H. W ang, Theta block conjecture for paramodular forms of weight 2. Preprint 2018, 15 pp, arXiv:1812.08698

work page arXiv 2018

[23] [23]

Gross, S

M. Gross, S. Popescu, Calabi–Yau three-folds and moduli of abelian surfaces II. Trans. Amer. Math. Soc. 363 (2011), 3573–3599

work page 2011

[24] [24]

Nikulin, Integer symmetric bilinear forms and some of their geometri c applications

V. Nikulin, Integer symmetric bilinear forms and some of their geometri c applications. Math. USSR Izv. 14 (1980), 103–167

work page 1980

[25] [25]

C. Poor, J. Shurman, D. S. Yuen, Siegel paramodular forms of weight 2 and squarefree level. Int. J. Number Theory 13 (2017), 2627–2652

work page 2017

[26] [26]

C. Poor, D. S. Yuen, Paramodular Cusp Forms, Math. Comp. 84 (no. 293) (2015), 1401–1438

work page 2015

[27] [27]

N. R. Scheithauer, On the classiﬁcation of automorphic products and generaliz ed Kac-Moody algebras. Invent. Math. 164 (2006), 641–678

work page 2006

[28] [28]

N. R. Scheithauer, The Weil representation of SL2(Z) and some applications. Int. Math. Res. Not. IMRN (2009) no.8, 1488–1545

work page 2009

[29] [29]

N. R. Scheithauer, Some constructions of modular forms for the Weil representa tion of SL(2, Z). Nagoya Math. J. 220 (2015), 1–43

work page 2015

[30] [30]

N. R. Scheithauer, Automorphic products of singular weight. Compos. Math. 153 (2017), 1855–1892. Laboratoire Paul Painlev ´e, Universit ´e de Lille and IUF, 59655 Villeneuve d’Ascq Cedex, France and National Research University Higher School o f Economics, Russian Federation E-mail address : valery.gritsenko@univ-lille.fr Laboratoire Paul Painlev ´e, Univ...

work page 2017