Startseite Mathematik Ramanujan systems of Rankin–Cohen type and hyperbolic triangles
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Ramanujan systems of Rankin–Cohen type and hyperbolic triangles

  • Gabriele Bogo ORCID logo EMAIL logo und Younes Nikdelan ORCID logo
Veröffentlicht/Copyright: 8. August 2023
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 35 Heft 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

References

[1] M. Alim, Algebraic structure of t ⁢ t * equations for Calabi–Yau sigma models, Comm. Math. Phys. 353 (2017), no. 3, 963–1009. 10.1007/s00220-017-2889-7Suche in Google Scholar

[2] M. Alim and M. Vogrin, Gauss–Manin Lie algebra of mirror elliptic K3 surfaces, Math. Res. Lett. 28 (2021), no. 3, 637–663. 10.4310/MRL.2021.v28.n3.a1Suche in Google Scholar

[3] B. C. Berndt and M. I. Knopp, Hecke’s Theory of Modular Forms and Dirichlet Series, Monogr. Number Theory 5, World Scientific, Hackensack, 2008. 10.1142/6438Suche in Google Scholar

[4] J. H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of Modular Forms, Universitext, Springer, Berlin, 2008. 10.1007/978-3-540-74119-0Suche in Google Scholar

[5] H. Cohen, Sums involving the values at negative integers of 𝐿-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. 10.1007/BF01436180Suche in Google Scholar

[6] P. Cohen and J. Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. 56 (1990), no. 2, 93–110. 10.4064/aa-56-2-93-110Suche in Google Scholar

[7] P. B. Cohen, Y. Manin and D. Zagier, Automorphic pseudodifferential operators, Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl. 26, Birkhäuser, Boston (1997), 17–47. 10.1007/978-1-4612-2434-1_2Suche in Google Scholar

[8] A. Connes and H. Moscovici, Rankin–Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (2004), no. 1, 111–130, 311. 10.17323/1609-4514-2004-4-1-111-130Suche in Google Scholar

[9] G. Darboux, Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux, Ann. Sci. Éc. Norm. Supér. (2) 7 (1878), 101–150. 10.24033/asens.159Suche in Google Scholar

[10] C. F. Doran, T. Gannon, H. Movasati and K. M. Shokri, Automorphic forms for triangle groups, Commun. Number Theory Phys. 7 (2013), no. 4, 689–737. 10.4310/CNTP.2013.v7.n4.a4Suche in Google Scholar

[11] A. M. El Gradechi, The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators, Adv. Math. 207 (2006), no. 2, 484–531. 10.1016/j.aim.2005.12.002Suche in Google Scholar

[12] G. Halphén, On a system of differential equations, C. R. Acad. Sci. Paris 92 (1881), 1101–1103. Suche in Google Scholar

[13] E. Hecke, Lectures on Dirichlet Series, Modular Functions and Quadratic Forms, Vandenhoeck & Ruprecht, Göttingen, 1983. Suche in Google Scholar

[14] M. Möller and D. Zagier, Modular embeddings of Teichmüller curves, Compos. Math. 152 (2016), no. 11, 2269–2349. 10.1112/S0010437X16007636Suche in Google Scholar

[15] H. Movasati, Quasi-modular forms attached to elliptic curves, I, Ann. Math. Blaise Pascal 19 (2012), no. 2, 307–377. 10.5802/ambp.316Suche in Google Scholar

[16] H. Movasati, Modular-type functions attached to mirror quintic Calabi–Yau varieties, Math. Z. 281 (2015), no. 3–4, 907–929. 10.1007/s00209-015-1513-9Suche in Google Scholar

[17] H. Movasati, Gauss–Manin Connection in Disguise: Calabi–Yau Modular Forms, Surv. Mod. Math. 13, International Press, Somerville, 2017. Suche in Google Scholar

[18] H. Movasati and Y. Nikdelan, Gauss–Manin connection in disguise: Dwork family, J. Differential Geom. 119 (2021), no. 1, 73–98. 10.4310/jdg/1631124264Suche in Google Scholar

[19] Z. Nehari, Conformal Mapping, McGraw-Hill, New York, 1952. Suche in Google Scholar

[20] Y. V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65–96. 10.4213/sm158Suche in Google Scholar

[21] Y. Nikdelan, Rankin–Cohen brackets for Calabi–Yau modular forms, preprint (2019), https://arxiv.org/abs/1912.12809. Suche in Google Scholar

[22] Y. Nikdelan, Modular vector fields attached to Dwork family: sl 2 ⁢ ( C ) Lie algebra, Mosc. Math. J. 20 (2020), no. 1, 127–151. 10.17323/1609-4514-2020-20-1-127-151Suche in Google Scholar

[23] Y. Nikdelan, About quasi-modular forms, differential operators and Rankin–Cohen algebras, Talk 2021. Suche in Google Scholar

[24] Y. Ohyama, Systems of nonlinear differential equations related to second order linear equations, Osaka J. Math. 33 (1996), no. 4, 927–949. Suche in Google Scholar

[25] K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. 10.2969/jmsj/02910091Suche in Google Scholar

[26] D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 57–75. 10.1007/BF02830874Suche in Google Scholar

[27] W. Zudilin, The hypergeometric equation and Ramanujan functions, Ramanujan J. 7 (2003), no. 4, 435–447. 10.1023/B:RAMA.0000012426.23921.24Suche in Google Scholar
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

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Modular forms; systems of nonlinear ODEs; Rankin–Cohen brackets; triangle groups; modular embeddings

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2026 De Gruyter Brill
Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0378/html?lang=de
Button zum nach oben scrollen