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Isogeny relations in products of families of elliptic curves

  • Luca Ferrigno ORCID logo EMAIL logo
Published/Copyright: October 23, 2025
Forum Mathematicum
From the journal Forum Mathematicum

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 ) ) .

MSC 2020: 11G05; 11G50; 11U09; 14K05

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

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Received: 2025-03-12
Revised: 2025-08-24
Published Online: 2025-10-23

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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