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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.

MSC:

11F11 Holomorphic modular forms of integral weight

Citations:

Zbl 0536.10023
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