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ON THE MODULARITY OF SOLUTIONS TO CERTAIN DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

Part of: Discontinuous groups and automorphic forms Differential equations in the complex domain

Published online by Cambridge University Press:  15 September 2021

HICHAM SABER Open the ORCID record for HICHAM SABER [Opens in a new window]  and
ABDELLAH SEBBAR Open the ORCID record for ABDELLAH SEBBAR [Opens in a new window]
HICHAM SABER*
Affiliation:
Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Kingdom of Saudi Arabia
ABDELLAH SEBBAR
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada e-mail: asebbar@uottawa.ca
*
e-mail: hicham.saber7@gmail.com
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Abstract

We answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J. 7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

Keywords

Schwarz derivativemodular formsEisenstein seriesequivariant functionsrepresentations of the modular groupFuchsian differential equations

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F11: Holomorphic modular forms of integral weight 34M05: Entire and meromorphic solutions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 385 - 391
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

Kaneko, M., ‘On modular forms of weight $\left(6n+1\right)/ 5$ satisfying a certain differential equation’, in: Number Theory, Developments in Mathematics, 15 (eds. Zhang, W. and Tanigawa, Y.) (Springer, New York, 2006), 97–102.CrossRefGoogle Scholar
Kaneko, M. and Koike, M., ‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J. 7(1–3) (2003), 145–164.CrossRefGoogle Scholar
Kaneko, M. and Zagier, D., ‘Supersingular $j$ -invariants, hypergeometric series and Atkin’s orthogonal polynomials’, in: Computational Perspectives on Number Theory, Studies in Advanced Mathematics, 7 (eds. Buell, D. A. and Teitelbaum, J. T.) (American Mathematical Society, Providence, RI, 1998), 97–126.Google Scholar
McKay, J. and Sebbar, A., ‘Fuchsian groups, automorphic functions and Schwarzians’, Math. Ann. 318(2) (2000), 255–275.CrossRefGoogle Scholar
Saber, H. and Sebbar, A., ‘On the existence of vector-valued automorphic forms’, Kyushu J. Math. 71(2) (2017), 271–285.CrossRefGoogle Scholar
Saber, H. and Sebbar, A., ‘Automorphic Schwarzian equations and integrals of weight 2 forms’, Ramanujan J., to appear.Google Scholar
Saber, H. and Sebbar, A., ‘Equivariant solutions to modular Schwarzian equations’, Preprint, arXiv:2106.06903, 2021.Google Scholar
Sebbar, A. and Saber, H., ‘Automorphic Schwarzian equations’, Forum Math. 12(6) (2020), 1621–1636.CrossRefGoogle Scholar