On Modular Forms Arising from a Differential Equation of Hypergeometric Type
نویسندگان
چکیده
The parameter k always stands for a non-negative integer or half an integer throughout the paper. This differential equation originates in the work [1] where in some cases (k ≡ 0, 4 mod 6) solutions which are modular on SL2(Z) were found and studied in connection with liftings of supersingular j-invariants of elliptic curves. The purpose of this paper is to give an explicit description of (conjecturally) all modular solutions of (#)k when k is an integer or half an integer (Theorem 1 in Section 2), as well as to discuss positiveness of Fourier coefficients of some of those solutions (Theorem 3 in Section 4). We shall also discuss an intrinsic characterization of the equation (#)k by the property that if f(τ) is a solution, then (cτ +d)f((aτ+b)/(cτ +d)) is also a solution for all (
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