pith. sign in

arxiv: 2604.11301 · v1 · submitted 2026-04-13 · 🧮 math.NT · math.AG

Branched covers of mathbb{P}¹ and divisibility in class group

Kalyan Banerjee , Kalyan Chakraborty , Azizul Hoque This is my paper

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords branched coversclass groupsJacobian torsionm-gonal curvesnumber fieldsdivisibilityprojective line
0
0 comments X

The pith

Branched covers of the projective line map n-torsion from m-gonal curve Jacobians into class groups of associated number fields.

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

The paper constructs a link starting from points of order n on the Jacobian of an m-gonal curve and using branched covers of the projective line to produce elements of order n in the class group of a related number field K. A sympathetic reader cares because the construction supplies explicit geometric sources for arithmetic torsion, offering concrete ways to realize divisibility by n in class groups. If the mapping preserves order, it gives a systematic source of number fields whose ideal class groups contain prescribed torsion subgroups.

Core claim

Starting with n-torsion points in the Jacobian of an m-gonal curve, branched covers of the projective line are employed to produce n-torsion elements inside the class group of a certain number field K.

What carries the argument

The branched cover of the projective line, which supplies a correspondence carrying Jacobian torsion to ideal classes while preserving exact order n.

If this is right

  • Certain number fields K possess class groups divisible by n for chosen n determined by the geometry of the cover.
  • The construction yields explicit examples of class-group torsion coming from curve Jacobians.
  • Divisibility properties in class groups become controllable through choices of m-gonal curves and their branched covers over the line.

Where Pith is reading between the lines

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

  • The same geometric input might be varied to produce fields whose class groups contain torsion of arbitrarily high order.
  • The method could be tested on low-genus m-gonal curves to generate concrete numerical examples of class groups with known torsion.

Load-bearing premise

A suitable branched cover or correspondence exists that sends the n-torsion point on the Jacobian to an element of the class group whose order remains exactly n.

What would settle it

An explicit m-gonal curve together with an n-torsion point on its Jacobian for which the associated number field K has no element of order n in its class group, or the order of the image drops below n.

read the original abstract

We start with $n$-torsions in the Jacobian of an $m$-gonal curve and produce $n$-torsions in the class group of certain number field $K$.

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 paper claims that n-torsion points in the Jacobian of an m-gonal curve can be mapped, via a branched cover of the projective line, to n-torsion elements in the class group of a suitable number field K, with the order exactly preserved.

Significance. A successful construction would link geometric torsion on curves to arithmetic torsion in class groups, offering a potential method for producing explicit high-order class group elements. This is relevant to questions about the distribution of class groups and the existence of fields with prescribed torsion. However, the absence of any explicit construction, equations, or verification steps makes the significance impossible to evaluate at present.

major comments (1)
  1. Abstract: the central claim is stated but the manuscript supplies no proof, no definition of the branched cover or correspondence, no equations describing the map from Jacobian torsion to class group elements, and no verification that the order n is preserved. This renders the result unverifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential significance of linking Jacobian torsion to class group torsion. We address the major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: Abstract: the central claim is stated but the manuscript supplies no proof, no definition of the branched cover or correspondence, no equations describing the map from Jacobian torsion to class group elements, and no verification that the order n is preserved. This renders the result unverifiable.

    Authors: We agree that the submitted manuscript is concise and states the main result without supplying the full construction, definitions, equations, or proof. The branched cover arises from an m-gonal curve C over P^1, and the correspondence is induced by the associated function field extension that produces the number field K; the map sends n-torsion points on Jac(C) to ideal classes in Cl(K). In the revised version we will add the explicit equations defining the cover, the precise definition of the map from Jacobian torsion to class group elements, and the argument showing that the order is exactly preserved. This will make the result verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation chain not inspectable

full rationale

The visible paper text is limited to a one-sentence abstract stating that n-torsions in the Jacobian of an m-gonal curve are mapped to n-torsions in the class group of K. No equations, explicit constructions, self-citations, or derivation steps are provided, so no load-bearing claim can be shown to reduce to its own inputs by construction. The central claim therefore cannot be analyzed for self-definitional, fitted-input, or self-citation circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5316 in / 966 out tokens · 42131 ms · 2026-05-10T15:29:04.393896+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    Agboola and G

    A. Agboola and G. Pappas, Line bundles, rational points and ideal classes , Math. Res. Lett. 7 (2000), no. 5-6, 709--717

    work page 2000

  2. [2]

    N. C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields , Pacific J. Math. 5 (1955), 321--324

    work page 1955

  3. [3]

    Banerjee and A

    K. Banerjee and A. Hoque, Chow groups, pull back and class groups , Monatsh. Math. 205 (2024), no. 3, 433--454

    work page 2024

  4. [4]

    \'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

    K. Banerjee and V. Guletskii, Rational equivalence for line configurations on cubic hypersurfaces in P^5 , 2014. arXiv:1405.6430v1

    work page Pith review arXiv 2014

  5. [5]

    Bugeaud and T

    Y. Bugeaud and T. N. Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation , J. Reine Angew. Math. 539 (2001), 55--74

    work page 2001

  6. [6]

    Chakraborty, A

    K. Chakraborty, A. Hoque, Y. Kishi and P. P. Pandey, Divisibility of the class numbers of imaginary quadratic fields , J. Number Theory 185 (2018), 339--348

    work page 2018

  7. [7]

    Chakraborty, A

    K. Chakraborty, A. Hoque and R. Sharma, Divisibility of class numbers of quadratic fields: Qualitative aspects , Advances in Mathematical inequalities and Application, 247--264, Trends Math., Birkh\" a user/Springer, Singapore, 2018

    work page 2018

  8. [8]

    Chakraborty and A

    K. Chakraborty and A. Hoque, Exponents of class groups of certain imaginary quadratic fields , To appear in Czechoslovak Math. J. (2020). arXiv:1801.00392

    work page arXiv 2020

  9. [9]

    Cohen and H

    H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields , in Number theory (Noordwijkerhout 1983), Lecture Notes in Math. 1068 , Springer, Berlin (1984), 33--62

    work page 1983

  10. [10]

    J. H. E. Cohn, Square Fibonacci numbers, etc. , Fibonacci Quart. 2 (1964), 109--113

    work page 1964

  11. [11]

    J. H. E. Cohn, On the Diophantine equation x^n=Dy^2+1 , Acta Arith. 106 (2003), 73--83

    work page 2003

  12. [12]

    J. -H. Evertse and J. H. Silverman, Uniform bounds for the number of solutions to Y^n=f(X) , Math. Proc. Camb. Phil. Soc. 100 (1986), 237--248

    work page 1986

  13. [13]

    Fantechi, L

    B. Fantechi, L. Gottsche, L. Illusie, S. Kleiman, N. Nitsure and A. Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained , Mathematical surveys and monographs, AMS, Volume 123, 2005

    work page 2005

  14. [14]

    H.Gillet and C.Soule, Arithmetic intersection theory , Publications Mathématiques de l'IH\'ES, Volume 72 (1990), p. 93--174

    work page 1990

  15. [15]

    Gillibert and A

    J. Gillibert and A. Levin, Pulling back torsion line bundles to ideal classes , Math. Res. Lett. 19 (2012), no. 6, 1171--1184

    work page 2012

  16. [16]

    Gillibert and A

    J. Gillibert and A. Levin, A geometric approach to large class groups: A survey , Class Groups of Number Fields and Related Topics, 1-- 15, Number Theory & Discrete Math. Springer, Singapore, 2020

    work page 2020

  17. [17]

    Gillibert, From Picard groups of hyperelliptic curves to class groups of quadratic fields , Trans

    J. Gillibert, From Picard groups of hyperelliptic curves to class groups of quadratic fields , Trans. Amer. Math. Soc. 374 (2021), no. 6, 3919--3946

    work page 2021

  18. [18]

    B. H. Gross and D. E. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve , Invent. Math. 44 (1978), 201--224

    work page 1978

  19. [19]

    Hoque and K

    A. Hoque and K. Chakraborty, Divisibility of class numbers of certain families of quadratic fields , J. Ramanujan Math. Soc. 34 (2019), no. 3, 281--289

    work page 2019

  20. [20]

    Hoque, On the exponents of class groups of some families of imaginary quadratic fields , Preprint

    A. Hoque, On the exponents of class groups of some families of imaginary quadratic fields , Preprint

    work page

  21. [21]

    Kishi, Note on the divisibility of the class number of certain imaginary quadratic fields , Glasgow Math

    Y. Kishi, Note on the divisibility of the class number of certain imaginary quadratic fields , Glasgow Math. J. 51 (2009), 187--191; corrigendum, ibid. 52 (2010), 207--208

    work page 2009

  22. [22]

    Kollar, Rational curves on algebraic varieties , Springer, New York, 1996

    J. Kollar, Rational curves on algebraic varieties , Springer, New York, 1996

    work page 1996

  23. [23]

    Ljunggren, Some theorems on indeterminate equations of the form x^n-1 x-1 =y^q , Norsk Mat

    W. Ljunggren, Some theorems on indeterminate equations of the form x^n-1 x-1 =y^q , Norsk Mat. Tidsskr. 25 (1943), 17--20

    work page 1943

  24. [24]

    S. R. Louboutin, On the divisibility of the class number of imaginary quadratic number fields , Proc. Amer. Math. Soc. 137 (2009), 4025--4028

    work page 2009

  25. [25]

    Mumford, Rational equivalence for 0 -cycles on surfaces , J

    D. Mumford, Rational equivalence for 0 -cycles on surfaces , J. Math Kyoto Univ. 9 (1968), 195--204

    work page 1968

  26. [26]

    M. R. Murty, Exponents of class groups of quadratic fields , Topics in number theory (University Park, PA, 1997), Math. Appl. 467 , Kluwer Acad. Publ., Dordrecht (1999), 229--239

    work page 1997

  27. [27]

    Ribet, A modular construction of unramified p -extensions of ( _p) , Invent

    K. Ribet, A modular construction of unramified p -extensions of ( _p) , Invent. Math. 34 (1976), 151--162

    work page 1976

  28. [28]

    A. A. Ro i tman, -equivalence of zero-dimensional cycles , Mat. Sb. (N. S.) 86 (1971), 557--570

    work page 1971

  29. [29]

    Rydh, Families of cycles and Chow schemes , PhD Thesis, KTH, Stockholm Sweden 2008

    D. Rydh, Families of cycles and Chow schemes , PhD Thesis, KTH, Stockholm Sweden 2008

    work page 2008

  30. [30]

    C. L. Siegel, Uber einige Anwendungen Diophantischer Approximationen , Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1 (1929), 1-70; Ges. Abh., Band 1 , 209--266

    work page 1929

  31. [31]

    W. M. Schmidt, Equations over finite fields, An elementary approach, Lecture Notes in Mathematics, 536, Springer-Verlag, Berlin-New York, 1976

    work page 1976

  32. [32]

    R.Soleng Homomorphisms From the Group of Rational Points On Elliptic Curves to Class Groups of Quadratic Number Fields , Journal of Number Theory, Volume 46, issue 2, February 1994, 214-229

    work page 1994

  33. [33]

    Soundararajan, Divisibility of class numbers of imaginary quadratic fields , J

    K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields , J. London Math. Soc. 61 (2000), 681--690

    work page 2000

  34. [34]

    Stack Exchange, Smooth morphisms , section 29.32 and 10.136

    work page

  35. [35]

    II , Cambridge Studies in Advanced Mathematics 77 , Cambridge University Press, Cambridge, 2003

    C.Voisin, Hodge theory and complex algebraic geometry. II , Cambridge Studies in Advanced Mathematics 77 , Cambridge University Press, Cambridge, 2003

    work page 2003

  36. [36]

    Voisin, Unirational threefolds with no universal codimension 2 cycle , Invent

    C. Voisin, Unirational threefolds with no universal codimension 2 cycle , Invent. Math. 201 (2015), no. 1, 207--237

    work page 2015

  37. [37]

    Yamamoto, On unramified Galois extensions of quadratic number fields , Osaka J

    Y. Yamamoto, On unramified Galois extensions of quadratic number fields , Osaka J. Math. 7 (1970), 57--76

    work page 1970