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Isogeny relations in products of families of elliptic curves

  • Luca Ferrigno ORCID logo EMAIL logo
Veröffentlicht/Copyright: 23. Oktober 2025
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 38 Heft 3

Abstract

Let E λ be the Legendre family of elliptic curves with equation Y 2 = X ⁢ ( X − 1 ) ⁢ ( X − λ ) . Given a curve 𝒞, satisfying a condition on the degrees of some of its coordinates and parametrizing 𝑚 points P 1 , … , P m ∈ E λ and 𝑛 points Q 1 , … , Q n ∈ E μ and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points c 0 on 𝒞 such that there exists an isogeny ϕ : E μ ⁢ ( c 0 ) → E λ ⁢ ( c 0 ) and the m + n points P 1 ⁢ ( c 0 ) , … , P m ⁢ ( c 0 ) , ϕ ⁢ ( Q 1 ⁢ ( c 0 ) ) , … , ϕ ⁢ ( Q n ⁢ ( c 0 ) ) ∈ E λ ⁢ ( c 0 ) are linearly dependent over End ⁡ ( E λ ⁢ ( c 0 ) ) .

Keywords: Unlikely intersections; Zilber–Pink conjecture; elliptic curves; isogenies
MSC 2020: 11G05; 11G50; 11U09; 14K05

Funding source: Ministero dell’Università e della Ricerca

Award Identifier / Grant number: 2022HPSNCR

Funding statement: The author was supported by the PRIN 2022 project 2022HPSNCR: Semiabelian varieties, Galois representations and related Diophantine problems and the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA INdAM).

Acknowledgements

We would like to thank Fabrizio Barroero and Laura Capuano for many useful discussions and comments, and Gabriel Dill for his comments, for pointing out an alternative proof of cases (ii) and (iii), and for his hospitality in Bonn. We are also grateful to Francesco Veneziano and to the anonymous referee for their many helpful comments, which greatly improved the exposition.

  1. Communicated by: Jan Bruinier

References

[1] Y. André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 1998 (1998), 203–208. 10.1515/crll.1998.505.203Suche in Google Scholar

[2] F. Barroero, CM relations in fibered powers of elliptic families, J. Inst. Math. Jussieu 18 (2019), no. 5, 941–956. 10.1017/S1474748017000287Suche in Google Scholar

[3] F. Barroero and L. Capuano, Linear relations in families of powers of elliptic curves, Algebra Number Theory 10 (2016), no. 1, 195–214. 10.2140/ant.2016.10.195Suche in Google Scholar

[4] F. Barroero and L. Capuano, Unlikely intersections in products of families of elliptic curves and the multiplicative group, Q. J. Math. 68 (2017), no. 4, 1117–1138. 10.1093/qmath/hax014Suche in Google Scholar

[5] F. Barroero and L. Capuano, Unlikely intersections in families of abelian varieties and the polynomial Pell equation, Proc. Lond. Math. Soc. (3) 120 (2020), no. 2, 192–219. 10.1112/plms.12289Suche in Google Scholar

[6] D. Bertrand, Théories de Galois différentielles et transcendance, Ann. Inst Fourier 59 (2009), no. 7, 2773–2803. 10.5802/aif.2507Suche in Google Scholar

[7] Y. Bilu, D. Masser and U. Zannier, An effective “theorem of André” for CM-points on a plane curve, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 145–152. 10.1017/S0305004112000461Suche in Google Scholar

[8] E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Math. Monogr., Cambridge University, Cambridge, 2006. Suche in Google Scholar

[9] F. Breuer, Heights of CM points on complex affine curves, Ramanujan J. 5 (2001), 311–317. 10.1023/A:1012982812988Suche in Google Scholar

[10] F. Diamond and J. M. Shurman, A First Course in Modular Forms, Grad. Texts in Math. 228, Springer, New York, 2005. Suche in Google Scholar

[11] G. Dill, Unlikely intersections between isogeny orbits and curves, J. Eur. Math. Soc. 23 (2021), no. 7, 2405–2438. 10.4171/jems/1057Suche in Google Scholar

[12] L. R. Ford, Automorphic Functions, 2nd ed., American Mathematical Society, Providence, 1951. Suche in Google Scholar

[13] E. Gaudron and G. Rémond, Théorème des périodes et degrés minimaux d’isogénies, Comment. Math. Helv. 89 (2014), no. 2, 343–403. 10.4171/cmh/322Suche in Google Scholar

[14] P. Habegger, Weakly bounded height on modular curves, Acta Math. Vietnam 35 (2010), no. 1, 43–69. Suche in Google Scholar

[15] P. Habegger and J. Pila, Some unlikely intersections beyond André–Oort, Compos. Math. 148 (2012), no. 1, 1–27. 10.1112/S0010437X11005604Suche in Google Scholar

[16] P. Habegger and J. Pila, O-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 4, 813–858. 10.24033/asens.2296Suche in Google Scholar

[17] M. Hindry and J. H. Silverman, Diophantine Geometry: An Introduction, Grad. Texts in Math. 201, Springer, New York, 2013. Suche in Google Scholar

[18] S. Lang, Elliptic Functions, Grad. Texts in Math. 112, Springer, New York, 1987. 10.1007/978-1-4612-4752-4Suche in Google Scholar

[19] D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), no. 6, 1677–1691. 10.1353/ajm.2010.a404145Suche in Google Scholar

[20] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Ann. 352 (2012), no. 2, 453–484. 10.1007/s00208-011-0645-4Suche in Google Scholar

[21] D. W. Masser, Linear relations on algebraic groups, New Advances in Transcendence Theory, Cambridge University, Cambridge (1988), 248–262. 10.1017/CBO9780511897184.016Suche in Google Scholar

[22] D. W. Masser and G. Wüstholz, Estimating isogenies on elliptic curves, Invent. Math. 100 (1990), no. 1, 1–24. 10.1007/BF01231178Suche in Google Scholar

[23] M. Orr, Height bounds and the Siegel property, Algebra Number Theory 12 (2018), no. 2, 455–478. 10.2140/ant.2018.12.455Suche in Google Scholar

[24] Y. Peterzil and S. Starchenko, Uniform definability of the weierstrass ℘ functions and generalized tori of dimension one, Selecta Math. (N.S.) 10 (2005), no. 4, 525–550. 10.1007/s00029-005-0393-ySuche in Google Scholar

[25] J. Pila, On the algebraic points of a definable set, Selecta Math. (N.S.) 15 (2009), no. 1, 151–170. 10.1007/s00029-009-0527-8Suche in Google Scholar

[26] J. Pila, Rational points of definable sets and results of André–Oort–Manin–Mumford type, Int. Math. Res. Not. IMRN 2009 (2009), 2476–2507. 10.1093/imrn/rnp022Suche in Google Scholar

[27] J. Pila, Point-Counting and the Zilber–Pink Conjecture, Cambridge Tracts in Math., Cambridge University, Cambridge, 2022. 10.1017/9781009170314Suche in Google Scholar

[28] J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin–Mumford conjecture, Rend. Lincei Mat. Appl. 19 (2008), no. 2, 149–162. 10.4171/rlm/514Suche in Google Scholar

[29] J. H. Silverman, Heights and Elliptic Curves, Arithmetic Geometry, Springer, New York (1986), 253–265. 10.1007/978-1-4613-8655-1_10Suche in Google Scholar

[30] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts in Math. 151, Springer, New York, 1994. 10.1007/978-1-4612-0851-8Suche in Google Scholar

[31] J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math. 106, Springer, New York, 2009. 10.1007/978-0-387-09494-6Suche in Google Scholar

[32] L. van den Dries, Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge University, Cambridge, 1998. 10.1017/CBO9780511525919Suche in Google Scholar

[33] L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), 19–56. 10.1007/BF02758635Suche in Google Scholar

[34] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. 10.1215/S0012-7094-96-08416-1Suche in Google Scholar

[35] U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Ann. of Math. Stud. 181, Princeton University, Princeton, 2012. 10.23943/princeton/9780691153704.001.0001Suche in Google Scholar

[36] U. Zannier, Lecture Notes on Diophantine Analysis, Edizioni della Normale Pisa, Pisa, 2014. 10.1007/978-88-7642-517-2Suche in Google Scholar

[37] H. G. Zimmer, On the difference of the Weil height and the Néron–Tate height, Math. Z. 147 (1976), no. 1, 35–51. 10.1007/BF01214273Suche in Google Scholar
Received: 2025-03-12
Revised: 2025-08-24
Published Online: 2025-10-23
Published in Print: 2026-05-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Functional inequalities related to Sobolev, Besov and Morrey spaces in the Dunkl setting
  3. Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds
  4. Non-vanishing mod p of theta lifts for (O2n+1, Mp4n )
  5. On the abscissae of Weil representation zeta functions for procyclic groups
  6. On the Cauchy problem of spherical capillary water waves
  7. The limit of a nonlocal p-Laplacian obstacle problem with nonhomogeneous term as p → ∞
  8. Central limit theorem for Hecke eigenvalues
  9. Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on 𝕋3
  10. p-adic Rankin–Selberg L-functions in universal deformation families and functional equations
  11. Rigidity and Toeplitz systems
  12. Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality
  13. Equivariant K-theory of cellular toric bundles and related spaces
  14. Isogeny relations in products of families of elliptic curves
  15. On rough oscillatory singular integral operators
https://doi.org/10.1515/forum-2025-0128
Schlagwörter für diesen Artikel
Unlikely intersections; Zilber–Pink conjecture; elliptic curves; isogenies

Artikel in diesem Heft

  1. Frontmatter
  2. Functional inequalities related to Sobolev, Besov and Morrey spaces in the Dunkl setting
  3. Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds
  4. Non-vanishing mod p of theta lifts for (O2n+1, Mp4n )
  5. On the abscissae of Weil representation zeta functions for procyclic groups
  6. On the Cauchy problem of spherical capillary water waves
  7. The limit of a nonlocal p-Laplacian obstacle problem with nonhomogeneous term as p → ∞
  8. Central limit theorem for Hecke eigenvalues
  9. Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on 𝕋3
  10. p-adic Rankin–Selberg L-functions in universal deformation families and functional equations
  11. Rigidity and Toeplitz systems
  12. Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality
  13. Equivariant K-theory of cellular toric bundles and related spaces
  14. Isogeny relations in products of families of elliptic curves
  15. On rough oscillatory singular integral operators
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