Pith. sign in

REVIEW 3 minor 35 references

Independence of CM points in Elliptic Curves

T0 review · 0 major / 3 minor · reviewed 2026-05-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read All linear dependencies of CM points on elliptic curves are described

desk verdict Pila-Tsimerman unify earlier results into one statement describing linear dependencies among images of CM points in elliptic curves under fixed maps from Shimura data. read the letter →

arxiv 1907.02737 v1 pith:C5YWEURK submitted 2019-07-05 math.NT math.AGmath.LO

Jonathan Pila , Jacob Tsimerman This is my paper · ORCID
classification math.NTmath.AGmath.LO
keywords CMpointsellipticcurveslineardependenciesmodularShimuraspecialcorrespondences
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a complete description of the linear relations that hold among any number of points on elliptic curves obtained as images of CM points from modular or Shimura curves via fixed parameterizations. A reader would care because this controls the possible additive structures involving these arithmetic special points and provides a uniform framework that covers previous partial results. The description applies for each n separately and accounts for all such dependencies.

What carries the argument

The fixed parameterizations or correspondences mapping CM points from modular or Shimura curves to points on elliptic curves.

What would settle it

An explicit set of n CM points whose images satisfy an unexpected linear relation not included in the described list would disprove the result.

Watch

Extended reading notes

Core claim

The central claim is that for each n at least 1, there is an explicit description of all linear dependencies among n images in elliptic curves of special CM points coming from modular or Shimura curves under given parameterizations or correspondences. This unifies and improves upon earlier results in certain aspects.

Load-bearing premise

The parameterizations from the modular or Shimura curves to the elliptic curves are fixed independently of the CM points chosen.

Editorial extensions

If this is right

  • The description applies uniformly to any finite number n of such points.
  • Previous partial classifications are subsumed and extended in some cases.
  • Only the dependencies that arise from the geometry of the source curves occur.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This classification could be applied to determine independence in concrete instances of maps and points.
  • It may connect to broader questions about the distribution of special points in arithmetic geometry.
  • Testable by checking small n cases against known examples from prior work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a result that, for each integer n ≥ 1, describes all linear dependencies among the n images (in elliptic curves) of special points lying on modular or Shimura curves, where the images are obtained via fixed parameterizations or correspondences. The result is presented as unifying and improving upon the earlier theorems of Rosen–Silverman–Kühne and Buium–Poonen.

Significance. If correct, the theorem supplies a uniform description of the linear relations satisfied by images of CM points under the indicated maps. This would consolidate two previously independent lines of work into a single statement and could serve as a reference point for further questions on heights or ranks of CM points in elliptic curves.

minor comments (3)
  1. The abstract states the result for 'special points' but the introduction should explicitly recall the precise definition of CM points on the source Shimura varieties that is used throughout the paper.
  2. Notation for the target elliptic curves and the parameterizations should be introduced once in §1 and then used consistently; several ad-hoc symbols appear in the statements of the main theorems.
  3. The comparison with the cited works of Rosen–Silverman–Kühne and Buium–Poonen would be clearer if a short table or paragraph listed the precise improvements (e.g., removal of a height bound, extension to higher-dimensional Shimura varieties).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in derivation chain

full rationale

The paper states a theorem unifying and improving prior independent results by Rosen-Silverman--Kühne and Buium-Poonen on linear dependencies of CM point images under fixed parameterizations. No self-citations appear in the abstract or described claim, no parameters are fitted and relabeled as predictions, and no ansatz or uniqueness result is imported from the authors' own prior work. The derivation is presented as a self-contained mathematical proof against external benchmarks in the theory of Shimura varieties and elliptic curves.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Independence of CM points in Elliptic Curves." pith.science (2026). https://pith.science/paper/C5YWEURK

@misc{pith2026190702737,
  author       = {Pith},
  title        = {Pith review of: Independence of CM points in Elliptic Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C5YWEURK}},
  note         = {Machine review of arXiv:1907.02737}
}
abstract

We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--K\"uhne and Buium-Poonen.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Baldi, On a conjecture of Buium and Poonen, arXiv:1803 .04946

    G. Baldi, On a conjecture of Buium and Poonen, arXiv:1803 .04946

    work page

  2. [2]

    Barroero, CM relations in fibred powers of elliptic fam ilies, J

    F. Barroero, CM relations in fibred powers of elliptic fam ilies, J. Inst. Math. Jussieu (2017) 1–16 (and corrigendum), and arXiv:1611.01955v5

    work page arXiv 2017

  3. [3]

    Bombieri, D

    E. Bombieri, D. Masser and U. Zannier, Anomalous subvari eties – structure theorems and applications, IMRN 19 (2007), 33 pages

    work page 2007

  4. [4]

    Bruin, Bornes optimales pour la diff´ erence entre la ha uteur de Weil et la hauteur de N´ eron-Tate sur les courbes elliptiques sur Q, Acta Arith

    P. Bruin, Bornes optimales pour la diff´ erence entre la ha uteur de Weil et la hauteur de N´ eron-Tate sur les courbes elliptiques sur Q, Acta Arith. 160 (2013), 385–397

    work page 2013

  5. [5]

    Buium and B

    A. Buium and B. Poonen, Independence of points on ellipti c curves arising from special points on modular and Shimura curves, I: Global resu lts, Duke Math. J. 147 (2009), 181–191

    work page 2009

  6. [6]

    Buium and B

    A. Buium and B. Poonen, Independence of points on ellipti c curves arising from special points on modular and Shimura curves, II: Local resu lts, Compositio 145 (2009), 566–602

    work page 2009

  7. [7]

    Daw and J

    C. Daw and J. Ren, Some applications of the hyperbolic Ax- Schanuel conjecture, Compositio Math. 154 (2018), 1843–1888

    work page 2018

  8. [8]

    Dill, Unlikely intersections between isogeny orbits and curves, arXiv:1801.05701

    G. Dill, Unlikely intersections between isogeny orbits and curves, arXiv:1801.05701

    work page arXiv

Show all 35 references
  1. [9]

    Dill, Unlikely intersections with isogeny orbits in a product of elliptic schemes, arXiv:1902.01323

    G. Dill, Unlikely intersections with isogeny orbits in a product of elliptic schemes, arXiv:1902.01323

    1902

  2. [10]

    Gao, Mixed Ax-Schanuel for the universal abelian var ieties and some applications, 2018 preprint, arXiv:1806.01408

    Z. Gao, Mixed Ax-Schanuel for the universal abelian var ieties and some applications, 2018 preprint, arXiv:1806.01408

    2018

  3. [11]

    Habegger, Singular moduli that are algebraic units, Algebra and Number Theory 9 (2015), 1515–1524

    P. Habegger, Singular moduli that are algebraic units, Algebra and Number Theory 9 (2015), 1515–1524

    2015

  4. [12]

    Habegger and J

    P. Habegger and J. Pila, Some unlikely intersections be yond Andr´ e-Oort,Compositio 148 (2012), 1–27

    2012

  5. [13]

    Habegger and J

    P. Habegger and J. Pila, O-minimality and certain atypi cal intersections, Annales Sci. Ecole Norm. Sup. (4) 49 (2016), 813–858

    2016

  6. [14]

    Khare and C

    C. Khare and C. S. Rajan, On Heegner points of large condu ctors, Math. Res. Lett. 8 (2001), 275–278

    2001

  7. [15]

    K¨ uhne, Intersections of class fields, 2017 preprint , arXiv:1709.00998v2

    L. K¨ uhne, Intersections of class fields, 2017 preprint , arXiv:1709.00998v2

    2017 arXiv

  8. [16]

    Masser, Linear relation on algebraic groups, New advances in transcendence theory, Baker (ed.), 248–262, CUP, 1988

    D. Masser, Linear relation on algebraic groups, New advances in transcendence theory, Baker (ed.), 248–262, CUP, 1988

    1988

  9. [17]

    Masser, Specializations of finitely generated subgr oups of Abelian varieties, Trans- actions of the AMS , Vol 311, Number 1 (1989), 413–424

    D. Masser, Specializations of finitely generated subgr oups of Abelian varieties, Trans- actions of the AMS , Vol 311, Number 1 (1989), 413–424

    1989

  10. [18]

    Masser, G.W¨ ustholz, Some effective estimates for el liptic curves, Arithmetic of Complex Manifolds , Lecture notes in mathematics, Vol 1399, Springer, Berlin, Hei- delberg, 1989

    D. Masser, G.W¨ ustholz, Some effective estimates for el liptic curves, Arithmetic of Complex Manifolds , Lecture notes in mathematics, Vol 1399, Springer, Berlin, Hei- delberg, 1989. 18 JONATHAN PILA AND JACOB TSIMERMAN

    1989

  11. [19]

    N. Mok, J. Pila, and J. Tsimerman, Ax-Schanuel for Shimu ra varieties, Annals Math. 189 (2019), 945–978, and arXiv:1711.02189

    2019 arXiv

  12. [20]

    Nekov´ aˇ r and N

    J. Nekov´ aˇ r and N. Schappacher, On the asymptotic beha viour of Heegner points Turkish Math. J. 23 (1999), 549–556

    1999

  13. [21]

    Paulin, An explicit Andr´ e-Oort type theorem for P1(C) × Gm(C) based on loga- rithmic forms, Publ

    R. Paulin, An explicit Andr´ e-Oort type theorem for P1(C) × Gm(C) based on loga- rithmic forms, Publ. Math. Debrecen 88 (2016), 21–33

    2016

  14. [22]

    Pazuki, Hauteur Thˆ eta et hauteur de Faltings, Bull

    F. Pazuki, Hauteur Thˆ eta et hauteur de Faltings, Bull. Soc. Math. France 140 (2012), 19–49

    2012

  15. [23]

    Peterzil and S

    Y. Peterzil and S. Starchenko, Definability of restrict ed theta functions and families of abelian varieties, Duke Math. J. 162 (2013), 731–765

    2013

  16. [24]

    Pila, O-minimality and the Andr´ e-Oort conjecture f or Cn, Annals Math

    J. Pila, O-minimality and the Andr´ e-Oort conjecture f or Cn, Annals Math. 173 (2011), 1779–1840

    2011

  17. [25]

    Pila and J

    J. Pila and J. Tsimerman, The Andr´ e-Oort conjecture fo r the moduli space of abelian surfaces, Compositio 149 (2013), 204–214

    2013

  18. [26]

    Pila and J

    J. Pila and J. Tsimerman, Multiplicative relations amo ng singular moduli, Ann. Sc. Norm. Super. Pisa (5) XVII (2017), 1357–1382

    2017

  19. [27]

    Pila and A

    J. Pila and A. J. Wilkie, The rational points of a definabl e set, Duke Math. J. 133 (2006), 591–616

    2006

  20. [28]

    R. Pink, A combination of the conjectures of Mordell-La ng and Andr´ e-Oort, Geo- metric methods in algebra and geometry, 251–282, Progress in Mathematics 235, Birkh¨ auser, Boston, 2005

    2005

  21. [29]

    Pink, A common generalization of the conjectures of A ndr´ e-Oort, Manin- Mumford, and Mordell-Lang, manuscript dated 17 April 2005 a vailable from http://www.math.ethz.ch/∼pink/

    R. Pink, A common generalization of the conjectures of A ndr´ e-Oort, Manin- Mumford, and Mordell-Lang, manuscript dated 17 April 2005 a vailable from http://www.math.ethz.ch/∼pink/

    2005

  22. [30]

    Poizat, L’egalit´ e au cube, J

    B. Poizat, L’egalit´ e au cube, J. Symbolic Logic 66 (2001), 1647–1676

    2001

  23. [31]

    Rosen and J

    M. Rosen and J. H. Silverman, On the independence of Heeg ner points associated to distinct imaginary fields, J. Number Th. 127 (2007), 10–36

    2007

  24. [32]

    S ¸ahino˘ glu, On the independence of Heegner points o n CM curves associated to distinct quadratic imaginary fields, Proc

    H. S ¸ahino˘ glu, On the independence of Heegner points o n CM curves associated to distinct quadratic imaginary fields, Proc. AMS 141 (20012), 813–826

  25. [33]

    Tsimerman, The Andr´ e-Oort conjecture forAg, Annals Math

    J. Tsimerman, The Andr´ e-Oort conjecture forAg, Annals Math. 187 (2018), 379–390

    2018

  26. [34]

    Zhang, Equidistribution of CM points in Shimura Vari eites, Int.Math.Res.Not

    S. Zhang, Equidistribution of CM points in Shimura Vari eites, Int.Math.Res.Not. No. 59 (2005), 3657–3689

    2005

  27. [35]

    Zilber, Exponential sums equations and the Schanuel conjecture, J

    B. Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), 27–44. JP: Mathematical Institute, University of Oxford, Oxford, UK. pila@maths.ox.ac.uk JT: Department of Mathematics, University of Toronto, Toro nto, Canada. jacobt@math.t...

    2002

Pith tools

Reviewed May 25, 2026 · model on record in the stance chip above.