pith. sign in

arxiv: 2604.19437 · v1 · submitted 2026-04-21 · 🧮 math.NT · math.DS

Representations of binary quadratic forms by quaternary quadratic forms

Wooyeon Kim , Andreas Wieser , Pengyu Yang This is my paper

Pith reviewed 2026-05-10 01:35 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords quadratic formsrepresentationslocal-global principleergodic methodsmeasure classificationbinary quadratic formsquaternary quadratic formsdeterminant method
0
0 comments X

The pith

A local-global principle holds for primitive representations of binary quadratic forms by quaternary quadratic forms.

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

The paper shows that a binary quadratic form is primitively represented by some quaternary quadratic form over the integers precisely when it is represented over the reals and over every p-adic field. This is proved by establishing density of certain homogeneous sets under diagonalizable group actions, then reducing via the Siegel mass formula to a point-counting problem on an affine variety solved by determinant estimates. A reader cares because the result replaces an infinite family of local checks with a finite criterion and gives an effective way to decide representability questions that arise in the arithmetic of quadratic forms. The argument adapts Linnik's ergodic method to this setting by importing a rigidity theorem for rank-two diagonal actions on products of SL_2 quotients.

Core claim

We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. The proof proceeds by a variant of Linnik's ergodic method that establishes density for the associated homogeneous toral sets. The key step applies a measure classification theorem for rank-two diagonalizable group actions on quotients of products of SL_2; combined with the Siegel mass formula, this reduces the density statement to a counting problem on a fixed affine variety, which is resolved by the Bombieri-Pila determinant method.

What carries the argument

A variant of Linnik's ergodic method that uses measure classification for rank-two diagonalizable group actions to produce density of homogeneous toral sets, followed by reduction via the Siegel mass formula to a Bombieri-Pila counting problem on an affine variety.

If this is right

  • Whenever local representation conditions hold everywhere, a global primitive representation exists.
  • The associated homogeneous toral sets are dense in their ambient homogeneous spaces.
  • The number of representations can be estimated by counting integral points on the auxiliary affine variety.
  • The same density-plus-mass-formula strategy applies to other representation problems whose homogeneous spaces admit rank-two diagonal actions.

Where Pith is reading between the lines

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

  • The method may adapt to representations involving forms of mixed signature or to higher-genus analogues where similar toral sets appear.
  • Effective bounds on the smallest representing quaternary form could follow from quantitative versions of the determinant estimates.
  • The result supplies a dynamical criterion that might be compared with class-number formulas or theta-series identities for the same representation numbers.

Load-bearing premise

The measure classification result for the relevant diagonalizable group actions applies directly to the homogeneous sets defined by the quadratic-form representation problem, and the Siegel mass formula converts the density question into a counting problem without further geometric obstructions.

What would settle it

An explicit binary quadratic form that is represented by some quaternary quadratic form over the reals and over every p-adic field yet fails to be primitively represented over the integers.

read the original abstract

We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown.

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 / 2 minor

Summary. The paper proves a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. It employs a variant of Linnik's ergodic method: the representation problem is embedded into a homogeneous space whose toral subsets are invariant under a rank-2 diagonalizable subgroup of a product of SL(2) groups; the Einsiedler-Lindenstrauss measure classification yields equidistribution (hence density), the Siegel mass formula reduces the density statement to a counting problem on an affine variety, and the Bombieri-Pila/Heath-Brown determinant method solves the counting problem.

Significance. If the local-global principle holds, the result supplies a new ergodic-theoretic instance of such principles for quadratic forms, extending Linnik-type methods to the binary-to-quaternary setting. The explicit reduction via Siegel mass to a Bombieri-Pila counting problem, together with the use of a rigidity theorem for rank-2 tori on products of SL(2) quotients, offers a reusable template for related representation questions where classical Hasse-principle obstructions are subtle.

major comments (2)
  1. [§4] §4 (homogeneous toral sets): the identification of the adelic orthogonal-group toral sets arising from primitive representations with the exact quotients treated by the Einsiedler-Lindenstrauss classification theorem is not verified in detail; it must be shown that these sets carry no extra invariants, admit no cuspidal contributions, and that the acting group is precisely the rank-2 diagonalizable torus without additional constraints from the orthogonal group.
  2. [§5] §5 (Siegel mass formula reduction): the passage from equidistribution to the counting problem on the affine variety assumes that the mass formula produces a positive main term free of additional local-global obstructions specific to primitive representations; an explicit check that the local densities are uniformly positive and that the error terms do not cancel the main term is required for the density statement to follow.
minor comments (2)
  1. [Introduction] The notation for the adelic points and the precise definition of the homogeneous space in the introduction should be cross-referenced to the later sections to avoid ambiguity for readers unfamiliar with the orthogonal-group setup.
  2. A short table summarizing the local conditions (discriminant, level, etc.) that appear in the local-global statement would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. Their observations identify areas where additional verification and explicit checks will strengthen the exposition. We address each major comment below and will revise the paper to incorporate the necessary details.

read point-by-point responses

  1. Referee: [§4] §4 (homogeneous toral sets): the identification of the adelic orthogonal-group toral sets arising from primitive representations with the exact quotients treated by the Einsiedler-Lindenstrauss classification theorem is not verified in detail; it must be shown that these sets carry no extra invariants, admit no cuspidal contributions, and that the acting group is precisely the rank-2 diagonalizable torus without additional constraints from the orthogonal group.

    Authors: We agree that the identification in §4 requires more explicit verification. The manuscript sketches the embedding of the adelic orthogonal group into the product of SL(2) groups via the natural action on binary quadratic forms, but does not fully compute stabilizers or rule out extra invariants and cuspidal contributions. In the revised version we will expand §4 with a dedicated subsection that (i) constructs the precise adelic toral sets arising from primitive representations, (ii) verifies that the stabilizer is exactly the rank-2 diagonalizable torus with no additional orthogonal-group constraints, and (iii) confirms the absence of cuspidal contributions by direct computation of the relevant adelic quotients. This will justify the direct application of the Einsiedler-Lindenstrauss theorem. revision: yes

  2. Referee: [§5] §5 (Siegel mass formula reduction): the passage from equidistribution to the counting problem on the affine variety assumes that the mass formula produces a positive main term free of additional local-global obstructions specific to primitive representations; an explicit check that the local densities are uniformly positive and that the error terms do not cancel the main term is required for the density statement to follow.

    Authors: The referee correctly notes that the density conclusion in §5 depends on the Siegel mass formula producing a strictly positive main term. While the local-global principle for the representations suggests no hidden obstructions, the manuscript does not supply an explicit verification that the local densities remain uniformly positive for primitive representations. In the revision we will add to §5 an explicit computation of the local densities at all places, showing they are bounded below by a positive constant depending only on the dimension and the level, together with a comparison showing that the equidistribution error is o(1) relative to this main term. This will confirm that the main term dominates and the density statement follows. revision: yes

Circularity Check

0 steps flagged

No circularity; external theorems applied to representation problem

full rationale

The derivation invokes the Einsiedler-Lindenstrauss measure classification for rank-2 diagonalizable actions on products of SL(2) quotients, the Siegel mass formula, and the Bombieri-Pila determinant method as independent external inputs. These are not derived inside the paper, not obtained from self-citations that reduce the target statement, and not fitted parameters renamed as predictions. The local-global principle follows from applying these tools to the specific toral sets and affine counting problem arising from binary-to-quaternary representations, without any self-definitional loop or construction that makes the output equivalent to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard background theorems from ergodic theory and analytic number theory rather than new axioms or fitted parameters introduced in the paper.

axioms (2)
  • standard math Measure classification theorem for actions of rank-two diagonalizable groups on quotients of products of SL_2 (Einsiedler-Lindenstrauss)
    Invoked as the central rigidity ingredient for density of homogeneous toral sets.
  • standard math Siegel mass formula for quadratic forms
    Used to reduce the density problem to a counting problem on an affine variety.

pith-pipeline@v0.9.0 · 5391 in / 1328 out tokens · 37768 ms · 2026-05-10T01:35:23.744646+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

56 extracted references · 56 canonical work pages

  1. [1]

    M. Aka, M. Einsiedler, and U. Shapira. Integer points on spheres and their orthogonal lattices. Invent. Math., 206(2):379–396, 2016

    work page 2016

  2. [2]

    M. Aka, M. Einsiedler, and A. Wieser. Planes in four space and four associated CM points. Duke Math. J., 171(7):1469–1529, 2020

    work page 2020

  3. [3]

    Equidistribution of rational subspaces and their shapes.Ergodic Theory Dynam

    Menny Aka, Andrea Musso, and Andreas Wieser. Equidistribution of rational subspaces and their shapes.Ergodic Theory Dynam. Systems, 44(8):2009–2062, 2024

    work page 2009

  4. [4]

    Simultaneous equidistribution of toric periods and fractional moments ofL-functions.J

    Valentin Blomer and Farrell Brumley. Simultaneous equidistribution of toric periods and fractional moments ofL-functions.J. Eur. Math. Soc. (JEMS), 26(8):2745–2796, 2024

    work page 2024

  5. [5]

    Blomer, F

    Valentin Blomer, Farrell Brumley, and Ilya Khayutin. The mixing conjecture under GRH. arXiv preprint 2212.06280, 2022

    work page arXiv 2022

  6. [6]

    Joint Linnik problems.arXiv preprint 2603.05609, 2026

    Valentin Blomer, Farrell Brumley, and Maksym Radiwi l l. Joint Linnik problems.arXiv preprint 2603.05609, 2026

    work page arXiv 2026

  7. [7]

    Bombieri and J

    E. Bombieri and J. Pila. The number of integral points on arcs and ovals.Duke Math. J., 59(2):337–357, 1989

    work page 1989

  8. [8]

    New bounds for the discrete Fourier restriction to the sphere in 4D and 5D.Int

    Jean Bourgain and Ciprian Demeter. New bounds for the discrete Fourier restriction to the sphere in 4D and 5D.Int. Math. Res. Not. IMRN, (11):3150–3184, 2015

    work page 2015

  9. [9]

    Three applications of the Siegel mass formula

    Jean Bourgain and Ciprian Demeter. Three applications of the Siegel mass formula. InGeo- metric aspects of functional analysis. Vol. I, volume 2256 ofLecture Notes in Math., pages 99–111. Springer, Cham, [2020]©2020

    work page 2020

  10. [10]

    The density of rational points on curves and surfaces

    Niklas Broberg. A note on a paper by R. Heath-Brown: “The density of rational points on curves and surfaces” [Ann. of Math. (2)155(2002), no. 2, 553–595; mr1906595].J. Reine Angew. Math., 571:159–178, 2004. 58 WOOYEON KIM, ANDREAS WIESER, AND PENGYU YANG

    work page 2002

  11. [11]

    Cambridge University Press, Cambridge, 1997

    Daniel Bump.Automorphic forms and representations, volume 55 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997

    work page 1997

  12. [12]

    Cassels.Rational quadratic forms, volume 13 ofLondon Mathematical Society Mono- graphs

    J.W.S. Cassels.Rational quadratic forms, volume 13 ofLondon Mathematical Society Mono- graphs. Academic Press Inc., 1978

    work page 1978

  13. [13]

    ´Equidistribution de mesures alg´ ebriques.Compos

    Laurent Clozel and Emmanuel Ullmo. ´Equidistribution de mesures alg´ ebriques.Compos. Math., 141(5):1255–1309, 2005

    work page 2005

  14. [14]

    James W. Cogdell. On sums of three squares. volume 15, pages 33–44. 2003. Les XXII` emes Journ´ ees Arithmetiques (Lille, 2001)

    work page 2003

  15. [15]

    Cowling, U

    M. Cowling, U. Haagerup, and R. Howe. AlmostL 2 matrix coefficients.J. Reine Angew. Math., 387:97–110, 1988

    work page 1988

  16. [16]

    W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms.Inventiones mathematicae, 92(1):73–90, 1988

    work page 1988

  17. [17]

    Representation of integers by positive ternary qua- dratic forms and equidistribution of lattice points on ellipsoids.Invent

    William Duke and Rainer Schulze-Pillot. Representation of integers by positive ternary qua- dratic forms and equidistribution of lattice points on ellipsoids.Invent. Math., 99(1):49–57, 1990

    work page 1990

  18. [18]

    Einsiedler and E

    M. Einsiedler and E. Lindenstrauss. Rigidity of non-maximal torus actions, unipotent quanti- tative recurrence, and diophantine approximations.arXiv preprint, arXiv:2307.04163, 2023

    work page arXiv 2023

  19. [19]

    Einsiedler, E

    M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh. Distribution of periodic torus orbits on homogeneous spaces.Duke Math. J., 148(1):119–174, 2009

    work page 2009

  20. [20]

    Einsiedler, E

    M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh. Distribution of periodic torus orbits and Duke’s theorem for cubic fields.Annals of Mathematics, 173:815–885, 2011

    work page 2011

  21. [21]

    Einsiedler, E

    M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh. The distribution of closed geodesics on the modular surface, and Duke’s theorem.Enseign. Math. (2), 58(3-4):249–313, 2012

    work page 2012

  22. [22]

    Einsiedler, G

    M. Einsiedler, G. Margulis, A. Mohammadi, and A. Venkatesh. Effective equidistribution and property (τ).J. Amer. Math. Soc., 33(1):223–289, 2020

    work page 2020

  23. [23]

    Joinings of higher rank torus actions on homo- geneous spaces.Publ

    Manfred Einsiedler and Elon Lindenstrauss. Joinings of higher rank torus actions on homo- geneous spaces.Publ. Math. Inst. Hautes ´Etudes Sci., 129:83–127, 2019

    work page 2019

  24. [24]

    Effective equidistribution of semisimple adelic periods and representations of quadratic forms.arXiv preprint 2503.21068, 2025

    Manfred Einsiedler, Elon Lindenstrauss, Amir Mohammadi, and Andreas Wieser. Effective equidistribution of semisimple adelic periods and representations of quadratic forms.arXiv preprint 2503.21068, 2025

    work page arXiv 2025

  25. [25]

    J. S. Ellenberg, Ph. Michel, and A. Venkatesh. Linnik’s ergodic method and the distribution of integer points on spheres. InAutomorphic representations andL-functions, volume 22 of Tata Inst. Fundam. Res. Stud. Math., pages 119–185. Tata Inst. Fund. Res., Mumbai, 2013

    work page 2013

  26. [26]

    D. R. Heath-Brown. The density of rational points on curves and surfaces.Ann. of Math. (2), 155(2):553–595, 2002

    work page 2002

  27. [27]

    Coefficients of Maass forms and the Siegel zero.Ann

    Jeffrey Hoffstein and Paul Lockhart. Coefficients of Maass forms and the Siegel zero.Ann. of Math. (2), 140(1):161–181, 1994. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman

    work page 1994

  28. [28]

    Iwaniec and P

    H. Iwaniec and P. Sarnak. Perspectives on the analytic theory ofL-functions. pages 705–741

    work page

  29. [29]

    GAFA 2000 (Tel Aviv, 1999)

    work page 2000

  30. [30]

    Fourier coefficients of modular forms of half-integral weight.Invent

    Henryk Iwaniec. Fourier coefficients of modular forms of half-integral weight.Invent. Math., 87(2):385–401, 1987

    work page 1987

  31. [31]

    Joint equidistribution of CM points.Ann

    Ilya Khayutin. Joint equidistribution of CM points.Ann. of Math. (2), 189(1):145–276, 2019

    work page 2019

  32. [32]

    Henry H. Kim. Functoriality for the exterior square of GL 4 and the symmetric fourth of GL2. J. Amer. Math. Soc., 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

    work page 2003

  33. [33]

    Representations of binary forms by qua- ternary quadratic forms.arXiv preprint 2511.22877, 2025

    Wooyeon Kim, Andreas Wieser, and Pengyu Yang. Representations of binary forms by qua- ternary quadratic forms.arXiv preprint 2511.22877, 2025

    work page arXiv 2025

  34. [34]

    Cambridge University Press, Cambridge, 1993

    Yoshiyuki Kitaoka.Arithmetic of quadratic forms, volume 106 ofCambridge Tracts in Math- ematics. Cambridge University Press, Cambridge, 1993

    work page 1993

  35. [35]

    Universidade Estadual de Campinas, Instituto de Matem´ atica, Estat´ ıstica e Ciˆ encia da Computa¸ c˜ ao, Campinas, 1988

    Max-Albert Knus.Quadratic forms, Clifford algebras and spinors, volume 1 ofSemin´ arios de Matem´ atica [Seminars in Mathematics]. Universidade Estadual de Campinas, Instituto de Matem´ atica, Estat´ ıstica e Ciˆ encia da Computa¸ c˜ ao, Campinas, 1988

    work page 1988

  36. [36]

    Yu. V. Linnik.Ergodic properties of algebraic fields, volume 45 ofErgebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, 1968. Translated from the Russian by M.S. Keane. REPRESENTATIONS OF BINARY FORMS BY QUATERNARY FORMS 59

    work page 1968

  37. [37]

    CentralL-values and toric periods for GL(2).Int

    Kimball Martin and David Whitehouse. CentralL-values and toric periods for GL(2).Int. Math. Res. Not. IMRN, (1):Art. ID rnn127, 141–191, 2009

    work page 2009

  38. [38]

    Michel and A

    Ph. Michel and A. Venkatesh. The subconvexity problem for GL 2.Publ. Math. Inst. Hautes ´Etudes Sci., (111):171–271, 2010

    work page 2010

  39. [39]

    Some recents advances on Duke’s equidistribution theorems

    Philippe Michel. Some recents advances on Duke’s equidistribution theorems. InNine mathe- matical challenges—an elucidation, volume 104 ofProc. Sympos. Pure Math., pages 107–132. Amer. Math. Soc., Providence, RI, [2021]©2021

    work page 2021

  40. [40]

    The subconvexity problem for GL2.Publ

    Philippe Michel and Akshay Venkatesh. The subconvexity problem for GL2.Publ. Math. Inst. Hautes ´Etudes Sci., (111):171–271, 2010

    work page 2010

  41. [41]

    Madhav V. Nori. On subgroups of GL n(Fp).Invent. Math., 88(2):257–275, 1987

    work page 1987

  42. [42]

    Platonov and A

    V. Platonov and A. Rapinchuk.Algebraic groups and number theory, volume 139 ofPure and Applied Mathematics. Academic Press, Inc., 1994. Translated from the 1991 Russian original by R. Rowen

    work page 1994

  43. [43]

    Rapinchuk

    Andrei S. Rapinchuk. Strong approximation for algebraic groups. InThin groups and super- strong approximation, volume 61 ofMath. Sci. Res. Inst. Publ., pages 269–298. Cambridge Univ. Press, Cambridge, 2014

    work page 2014

  44. [44]

    On the density of rational and integral points on algebraic varieties.J

    Per Salberger. On the density of rational and integral points on algebraic varieties.J. Reine Angew. Math., 606:123–147, 2007

    work page 2007

  45. [45]

    Counting rational points on projective varieties.Proc

    Per Salberger. Counting rational points on projective varieties.Proc. Lond. Math. Soc. (3), 126(4):1092–1133, 2023

    work page 2023

  46. [46]

    Representation by integral quadratic forms—a survey

    Rainer Schulze-Pillot. Representation by integral quadratic forms—a survey. InAlgebraic and arithmetic theory of quadratic forms, volume 344 ofContemp. Math., pages 303–321. Amer. Math. Soc., Providence, RI, 2004

    work page 2004

  47. [47]

    Averages of Fourier coefficients of Siegel modular forms and represen- tation of binary quadratic forms by quadratic forms in four variables.Math

    Rainer Schulze-Pillot. Averages of Fourier coefficients of Siegel modular forms and represen- tation of binary quadratic forms by quadratic forms in four variables.Math. Ann., 368(3- 4):923–943, 2017

    work page 2017

  48. [48]

    ¨Uber die analytische Theorie der quadratischen Formen.Ann

    Carl Ludwig Siegel. ¨Uber die analytische Theorie der quadratischen Formen.Ann. of Math. (2), 36(3):527–606, 1935

    work page 1935

  49. [49]

    Venkatesh

    A. Venkatesh. Sparse equidistribution problems, period bounds and subconvexity.Ann. of Math., 172(2):989–1094, 2010

    work page 2010

  50. [50]

    Springer, Berlin, 1980

    Marie-France Vign´ eras.Arithm´ etique des alg` ebres de quaternions, volume 800 ofLecture Notes in Mathematics. Springer, Berlin, 1980

    work page 1980

  51. [51]

    Springer, Cham, [2021]©2021

    John Voight.Quaternion algebras, volume 288 ofGraduate Texts in Mathematics. Springer, Cham, [2021]©2021

    work page 2021

  52. [52]

    Waldspurger

    J.-L. Waldspurger. Sur les valeurs de certaines fonctionsLautomorphes en leur centre de sym´ etrie.Compositio Math., 54(2):173–242, 1985

    work page 1985

  53. [53]

    Linnik’s problems and maximal entropy methods.Monatsh

    Andreas Wieser. Linnik’s problems and maximal entropy methods.Monatsh. Math., 190(1):153–208, 2019

    work page 2019

  54. [54]

    Burgess-like subconvex bounds for GL2 ×GL1.Geom

    Han Wu. Burgess-like subconvex bounds for GL2 ×GL1.Geom. Funct. Anal., 24(3):968–1036, 2014

    work page 2014

  55. [55]

    Subconvex bounds for compact toric integrals, 2018.https://arxiv.org/abs/1604

    Han Wu. Subconvex bounds for compact toric integrals, 2018.https://arxiv.org/abs/1604. 01902

    work page 2018

  56. [56]

    Princeton University Press, Princeton, NJ, 2013

    Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang.The Gross-Zagier formula on Shimura curves, volume 184 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2013. (W.K.)Korea institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul, Republic of Korea Email address:wooyeonkim@kias.re.kr (A.W.)Institute for Advanced Study, 1 Einstein D...

    work page 2013