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Modular integrals and indefinite binary quadratic forms. (English) Zbl 0795.11022
Knopp, Marvin (ed.) et al., A tribute to Emil Grosswald: number theory and related analysis. Providence, RI: American Mathematical Society. Contemp. Math. 143, 513-523 (1993).
A modular integral of weight \(2k\), \(k\in\mathbb{Z}\), with rational period function \(q(z)\) is a meromorphic function in the upper half-plane satisfying \(f(z+1)= f(z)\), \(z^{-2k} f(-{1\over z})= f(z)+ q(z)\), where \(k\in\mathbb{Z}\) and \(q(z)\) is a rational function. For a large class of rational period functions the corresponding modular integral is given by an infinite sum of powers of binary quadratic forms of positive discriminant [D. Zagier, Invent. Math. 30, 1-46 (1975; Zbl 0308.10014)]. In this case the Fourier coefficients of the modular integral and the action of the Hecke operators on the modular integral are determined.
For the entire collection see [Zbl 0773.00030].

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
11E16 General binary quadratic forms