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On differential equations satisfied by modular forms

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Abstract.

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL 2 (ℝ) commensurable with SL 2 (ℤ), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined.

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  1. Department of Mathematics, National University of Singapore, Singapore

    Yifan Yang

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  1. Yifan Yang
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Correspondence to Yifan Yang.

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Mathematics Subject Classification (2000):11F03, 11F11.

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Yang, Y. On differential equations satisfied by modular forms. Math. Z. 246, 1–19 (2004). https://doi.org/10.1007/s00209-003-0573-4

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  • DOI: https://doi.org/10.1007/s00209-003-0573-4

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