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Ramanujan systems of Rankin–Cohen type and hyperbolic triangles

  • Gabriele Bogo ORCID logo EMAIL logo and Younes Nikdelan ORCID logo
Published/Copyright: August 8, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 35 Issue 6

Abstract

In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ ( C ) -module. We prove that such systems, called Ramanujan systems of Rankin–Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin–Cohen structure. In the second part of the paper, we consider triangle groups Δ ⁢ ( n , m , ∞ ) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ ( n , m , ∞ ) are generated by the solutions of one such system (which is of Rankin–Cohen type). As a corollary, we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on Δ ⁢ ( 2 , 5 , ∞ ) from solutions of systems of nonlinear ODEs.

Keywords: Modular forms; systems of nonlinear ODEs; Rankin–Cohen brackets; triangle groups; modular embeddings
MSC 2010: 11F03; 34A34; 16W50; 11F55

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 444845124

Funding statement: The first author is supported by the LOEWE research unit USAG, and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”, project number 444845124.

Acknowledgements

The second author started and concluded the major part of his results during his one-year visit to Max Planck Institute for Mathematics (MPIM) Bonn. So he would like to thank MPIM and its staff for preparing such an excellent ambience for doing mathematical work. Finally, the authors wants to thank the anonymous referees for many valuable suggestions that helped to improve the paper.

  1. Communicated by: Jan Bruinier

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Received: 2022-12-12
Revised: 2023-05-19
Published Online: 2023-08-08
Published in Print: 2023-11-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields
  3. Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
  4. Minimal Kähler submanifolds in product of space forms
  5. On the number of rational points of certain algebraic varieties over finite fields
  6. On two conjectures of Sun concerning Apéry-like series
  7. A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth
  8. Time-step heat problem on the mesh: asymptotic behavior and decay rates
  9. Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras
  10. Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
  11. Skew-braces and 𝑞-braces
  12. Products of unipotent elements in certain algebras
  13. On the Fourier orthonormal bases of a class of self-similar measures on ℝ n
  14. The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
  15. Restriction estimates in a conical singular space: Schrödinger equation
  16. Fractional integrals associated with Radon transforms
https://doi.org/10.1515/forum-2022-0378
Keywords for this article
Modular forms; systems of nonlinear ODEs; Rankin–Cohen brackets; triangle groups; modular embeddings

Articles in the same Issue

  1. Frontmatter
  2. Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields
  3. Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation
  4. Minimal Kähler submanifolds in product of space forms
  5. On the number of rational points of certain algebraic varieties over finite fields
  6. On two conjectures of Sun concerning Apéry-like series
  7. A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth
  8. Time-step heat problem on the mesh: asymptotic behavior and decay rates
  9. Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras
  10. Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
  11. Skew-braces and 𝑞-braces
  12. Products of unipotent elements in certain algebras
  13. On the Fourier orthonormal bases of a class of self-similar measures on ℝ n
  14. The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma
  15. Restriction estimates in a conical singular space: Schrödinger equation
  16. Fractional integrals associated with Radon transforms
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