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arxiv: 2208.08737 · v4 · pith:DNRQ4MFHnew · submitted 2022-08-18 · 🧮 math.AG

Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions

Dimitri Markushevich , Anne Moreau This is my paper

Pith reviewed 2026-05-24 11:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords Bernstein-Schwarzman conjecturecrystallographic reflection groupsKlein's quartic curveinvariant theta functionsweighted projective spaceautomorphism groupJacobian variety
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The pith

The quotient by the 3D crystallographic reflection group for Klein's order-168 group is the weighted projective space with weights 1, 2, 4, 7.

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

The paper proves the Bernstein-Schwarzman conjecture in the case of the irreducible complex crystallographic reflection group in dimension 3 whose associated collineation group is Klein's simple group of order 168. It shows that the quotient of complex affine 3-space by this group is the weighted projective space with weights 1, 2, 4, 7. The proof rests on an explicit computation of the algebra of invariant theta functions, which is not freely generated by polynomials. A sympathetic reader cares because this settles the conjecture for a non-Coxeter example in dimension 3 and supplies a concrete description of the quotient.

Core claim

We prove that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient is the computation of the algebra of invariant theta functions.

What carries the argument

The algebra of invariant theta functions, whose explicit structure determines the quotient to be the weighted projective space with weights 1, 2, 4, 7.

If this is right

  • The Bernstein-Schwarzman conjecture holds for this non-Coxeter crystallographic reflection group in dimension 3.
  • The invariant algebra is not a free polynomial algebra, unlike the Coxeter cases treated earlier.
  • The quotient admits an explicit description as weighted projective space with weights 1, 2, 4, 7.
  • The method via invariant theta functions succeeds where direct invariant theory was blocked.

Where Pith is reading between the lines

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

  • The same theta-function technique might resolve the conjecture for other non-Coxeter groups in dimension 3 or higher.
  • The weights 1, 2, 4, 7 likely encode the degrees of the basic invariants tied to the action of Klein's group on the Jacobian.
  • This example supplies a test case for whether the weighted projective space description persists when the reflection group is not of Coxeter type.

Load-bearing premise

The algebra of invariant theta functions can be computed explicitly and its structure determines that the quotient is the weighted projective space with the stated weights.

What would settle it

An explicit basis computation for the algebra of invariant theta functions showing generators in degrees other than those corresponding to weights 1, 2, 4, 7, or a geometric check that the quotient has a different singularity structure, would falsify the claim.

read the original abstract

Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling 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.

Referee Report

1 major / 0 minor

Summary. The manuscript proves the Bernstein-Schwarzman conjecture for the irreducible complex crystallographic reflection group in dimension 3 whose associated collineation group is Klein's simple group of order 168. The quotient of affine 3-space by this group is shown to be the weighted projective space P(1,2,4,7). The proof proceeds by explicit computation of the algebra of invariant theta functions on the Jacobian of Klein's quartic curve; unlike the Coxeter cases, this algebra is not a free polynomial algebra.

Significance. If the identification of the quotient holds, the result verifies the conjecture in the first non-Coxeter case in dimension 3 and supplies an explicit example in which the invariant algebra has relations. The computation of the invariant theta functions is a concrete, verifiable ingredient that strengthens the claim beyond abstract existence arguments.

major comments (1)
  1. [computation of the algebra of invariant theta functions] The central claim that the quotient is exactly P(1,2,4,7) rests on the structure of the computed invariant algebra. The homogeneous coordinate ring of P(1,2,4,7) is the free graded polynomial ring on four generators of degrees 1, 2, 4 and 7. The abstract states that the invariant algebra is not free polynomial. The manuscript must therefore show explicitly (in the section containing the computation of the invariant theta functions) how the relations in this algebra nevertheless produce a quotient isomorphic to the full weighted projective space rather than a proper closed subvariety.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. We address the major comment below.

read point-by-point responses

  1. Referee: [computation of the algebra of invariant theta functions] The central claim that the quotient is exactly P(1,2,4,7) rests on the structure of the computed invariant algebra. The homogeneous coordinate ring of P(1,2,4,7) is the free graded polynomial ring on four generators of degrees 1, 2, 4 and 7. The abstract states that the invariant algebra is not free polynomial. The manuscript must therefore show explicitly (in the section containing the computation of the invariant theta functions) how the relations in this algebra nevertheless produce a quotient isomorphic to the full weighted projective space rather than a proper closed subvariety.

    Authors: We agree that an explicit demonstration is required to reconcile the non-free nature of the invariant algebra with the claim that the quotient is the full weighted projective space P(1,2,4,7). The current manuscript presents the generators and relations obtained from the theta-function computation but does not include a dedicated verification that these relations define an ideal whose Proj recovers the entire space (rather than a proper subvariety). In the revised version we will add, in the section on the computation of the invariant theta functions, a direct argument establishing the isomorphism: we will compare the Hilbert series of the computed algebra with that of the polynomial ring on generators of degrees 1,2,4,7 and confirm that the relations do not alter the Proj beyond the weighted projective space itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit computation of invariant algebra is independent of the claimed quotient identification

full rationale

The paper's central step is an explicit computation of the algebra of invariant theta functions for the specific crystallographic reflection group associated to Klein's group of order 168. This computation is presented as direct and is used to determine the structure of the quotient, without any reduction of the result to a fitted parameter, self-definition, or load-bearing self-citation. The abstract and description emphasize that the algebra is computed explicitly (unlike prior Coxeter cases), and the conclusion that the quotient is the weighted projective space follows from the structure of this computed algebra. No equations or steps are shown to be equivalent to their inputs by construction, and external results (Bernstein-Schwarzman conjecture, prior proofs in dim 2 or Coxeter cases) are cited as background rather than as the sole justification for the present case. The derivation is therefore self-contained against the computation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof uses standard mathematical axioms from algebraic geometry and invariant theory; no new entities or free parameters are introduced in the abstract description.

axioms (2)
  • standard math Standard properties of theta functions on abelian varieties and their transformation under group actions
    The computation of the invariant algebra relies on known transformation properties of theta functions.
  • domain assumption The given group is the crystallographic reflection group associated with Klein's simple group of order 168
    The paper assumes this identification to apply the conjecture to this case.

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Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Bernstein and O

    J. Bernstein and O. Schwarzman, Chevalley's theorem for the complex crystallographic groups , J. Nonlinear Math.\ Phys.\ 13 (2006), no. 3, 323--351

    work page 2006

  2. [2]

    Berndt, R.\,J

    B.\,C. Berndt, R.\,J. Evans and K.\,S. Williams, Gauss and Jacobi sums , Canad.\ Math.\ Soc.\ Ser.\ Monogr.\ Adv.\ Texts, John Wiley & Sons, Inc., New York, 1998

    work page 1998

  3. [3]

    Birkenhake and H

    C. Birkenhake and H. Lange, Complex abelian varieties , 2nd ed., Grundlehren math.\ Wiss., vol 302, Springer-Verlag, Berlin, 2004

    work page 2004

  4. [4]

    Bourbaki, \' E l\' e ments de math\' e matique, Fasc

    N. Bourbaki, \' E l\' e ments de math\' e matique, Fasc. XXXIV, Groupes et alg\` e bres de Lie, Chapitres IV--VI , Actualit\'es Sci.\ Industr., No. 1337, Hermann, Paris, 1968

    work page 1968

  5. [5]

    Cohen, Finite complex reflection groups , Ann.\ Sci.\ \' E cole Norm.\ Sup

    A.\,M. Cohen, Finite complex reflection groups , Ann.\ Sci.\ \' E cole Norm.\ Sup. (4) 9 (1976), no. 3, 379--436

    work page 1976

  6. [6]

    Dedieu and E

    T. Dedieu and E. Sernesi, Deformations and Extensions of Gorenstein Weighted Projective Spaces , in: The Art of Doing Algebraic Geometry , pp. 119--143, Trends Math., Birkh\"auser/Springer, Cham, 2023

    work page 2023

  7. [7]

    Fulton, Introduction to toric varieties , Ann.\ of Math.\ Stud., vol

    W. Fulton, Introduction to toric varieties , Ann.\ of Math.\ Stud., vol. 131, William Roever Lectures Geom., Princeton Univ.\ Press, Princeton, NJ, 1993

    work page 1993

  8. [8]

    Gunning, Lectures on modular forms (Notes by A

    R.\,C. Gunning, Lectures on modular forms (Notes by A. Brumer), Ann.\ of Math.\ Stud., vol. 48, Princeton Univ.\ Press, Princeton, NJ, 1962

    work page 1962

  9. [9]

    Igusa, Theta functions , Grundlehren math.\ Wiss., vol

    J.-i. Igusa, Theta functions , Grundlehren math.\ Wiss., vol. 194, Springer-Verlag, New York-Heidel\-berg, 1972

    work page 1972

  10. [10]

    Kasprzyk, Bounds on fake weighted projective space , Kodai Math

    A.\,M. Kasprzyk, Bounds on fake weighted projective space , Kodai Math. J.\ 32 (2009), no. 2, 197--208

    work page 2009

  11. [11]

    Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen , Math.\ Ann.\ 14 (1878) , no

    F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen , Math.\ Ann.\ 14 (1878) , no. 3, 428--471 (1878)

    work page

  12. [12]

    Looijenga, Root systems and elliptic curves , Invent.\ Math.\ 38 (1976/77), no

    E. Looijenga, Root systems and elliptic curves , Invent.\ Math.\ 38 (1976/77), no. 1, 17--32

    work page 1976

  13. [13]

    Grayson and M.\,E

    D.\,R. Grayson and M.\,E. Stillman, Macaulay2, a software system for research in algebraic geometry , http://www.math.uiuc.edu/Macaulay2/

    work page

  14. [14]

    Markushevich and A

    D. Markushevich and A. Moreau, Action of the Automorphism Group on the Jacobian of Klein's Quartic Curve , in: Birational Geometry, K\" a hler--Einstein Metrics and Degenerations , 591--607, Springer Proc.\ Math.\ Stat., vol. 409, Springer, Cham, 2023

    work page 2023

  15. [15]

    Mazur, Arithmetic on curves , Bull.\ Amer.\ Math.\ Soc.\ (N.S.) 14 (1986), no

    B. Mazur, Arithmetic on curves , Bull.\ Amer.\ Math.\ Soc.\ (N.S.) 14 (1986), no. 2, 207--259

    work page 1986

  16. [16]

    Popov, Discrete complex reflection groups , Commun.\ Math.\ Inst.\ Rijksuniv.\ Utrecht, vol

    V.\,L. Popov, Discrete complex reflection groups , Commun.\ Math.\ Inst.\ Rijksuniv.\ Utrecht, vol. 15, Rijksuniv.\ Utrecht, Math.\ Inst., Utrecht, 1982

    work page 1982

  17. [17]

    Rains, Quotients of abelian varieties by reflection groups , preprint arXiv:2303.04786 (2023)

    E.\,M. Rains, Quotients of abelian varieties by reflection groups , preprint arXiv:2303.04786 (2023)

    work page arXiv 2023

  18. [18]

    Reid, Canonical 3-folds , in: Journ\'ees de G\'eom\'etrie Alg\'ebrique d'Angers, Juillet 1979 , pp

    M. Reid, Canonical 3-folds , in: Journ\'ees de G\'eom\'etrie Alg\'ebrique d'Angers, Juillet 1979 , pp. 273--310, Sijthoff &\ Noordhoff, Alphen aan den Rijn---Germantown, MD, 1980

    work page 1979

  19. [19]

    Schlessinger, Rigidity of quotient singularities , Invent.\ Math.\ 14 (1971), 17--26

    M. Schlessinger, Rigidity of quotient singularities , Invent.\ Math.\ 14 (1971), 17--26

    work page 1971

  20. [20]

    Shephard and J.\,A

    G.\,C. Shephard and J.\,A. Todd, Finite unitary reflection groups , Canad. J.\ Math.\ 6 (1954), 274--304

    work page 1954

  21. [21]

    Stanley, Enumerative combinatorics, Vol

    R.\,P. Stanley, Enumerative combinatorics, Vol. I , Wadsworth & Brooks/Cole Math.\ Ser., Wadsworth & Brooks/Cole Advanced Books & Software, Montereye , CA, 1986

    work page 1986