On differential equations satisfied by modular forms. (English) Zbl 1108.11040

Summary: We use the theory of modular functions to give a new proof of a result of P. F. Stiller [Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Mem. Am. Math. Soc. 299 (1984; Zbl 0536.10023)], 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 \(\text{SL}(2,\mathbb R)\) commensurable with \(\text{SL}(2,\mathbb Z)\), 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.


11F11 Holomorphic modular forms of integral weight


Zbl 0536.10023
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