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Regularized inner products and errors of modularity. (English) Zbl 1405.11052
Summary: We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth [W. Duke et al., Ramanujan J. 41, No. 1–3, 13–29 (2016; Zbl 1418.11069)] on regularized inner products of weakly holomorphic modular forms of weights \(0\) and \(3/2\). These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by R. Bruggeman et al. [Mem. Am. Math. Soc. 1212, iii-viii, 172 p. (2018; Zbl 07000172)]. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.

MSC:
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F75 Cohomology of arithmetic groups
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