Equations involving the modular j -function and its derivatives

Vahagn Aslanyan; Sebastian Eterović; Vincenzo Mantova

doi:10.1515/crelle-2025-0067

Equations involving the modular j -function and its derivatives

Vahagn Aslanyan, Sebastian Eterović, Vincenzo Mantova

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that for any polynomial F(X, Y₀, Y₁, Y₂) ∈ C[X, Y₀, Y₁, Y₂], the equation
F(z, j(z), j′(z), j′′(z)) = 0 has a Zariski dense set of solutions in the hypersurface F(X, Y₀, Y₁, Y₂) = 0, unless F is in C[X] or it is divisible by Y₀, Y₀ − 1728, or Y₁.

Our methods establish criteria for finding solutions to more general equations

involving periodic functions. Furthermore, they produce a qualitative description of

the distribution of these solutions.

Original language	English
Journal	Journal fur die Reine und Angewandte Mathematik
DOIs	https://doi.org/10.1515/crelle-2025-0067
Publication status	Published - 11 Oct 2025

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AU - Aslanyan, Vahagn

AU - Eterović, Sebastian

AU - Mantova, Vincenzo

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N2 - We show that for any polynomial F(X, Y0, Y1, Y2) ∈ C[X, Y0, Y1, Y2], the equation F(z, j(z), j′(z), j′′(z)) = 0 has a Zariski dense set of solutions in the hypersurface F(X, Y0, Y1, Y2) = 0, unless F is in C[X] or it is divisible by Y0, Y0 − 1728, or Y1.Our methods establish criteria for finding solutions to more general equationsinvolving periodic functions. Furthermore, they produce a qualitative description ofthe distribution of these solutions.

AB - We show that for any polynomial F(X, Y0, Y1, Y2) ∈ C[X, Y0, Y1, Y2], the equation F(z, j(z), j′(z), j′′(z)) = 0 has a Zariski dense set of solutions in the hypersurface F(X, Y0, Y1, Y2) = 0, unless F is in C[X] or it is divisible by Y0, Y0 − 1728, or Y1.Our methods establish criteria for finding solutions to more general equationsinvolving periodic functions. Furthermore, they produce a qualitative description ofthe distribution of these solutions.

UR - https://www.scopus.com/pages/publications/105019399958

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DO - 10.1515/crelle-2025-0067

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JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

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Equations involving the modular j -function and its derivatives

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