Cycle Integrals of Maass Cusp Forms and of Holomorphic Functions with Rational Periods
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2023
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Doctoral Thesis
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Abstract
Akin to the Fourier expansion of a modular form, the values of its cycle integrals determine the modular form completely. Given the importance of the Fourier coefficients of modular forms in arithmetic applications, it is thus not surprising that their cousins, the cycle integrals, are of crucial arithmetic importance.
We will extend the Katok-Sarnak formula (1991), which relates the Fourier coefficients of the Shintani lift of a Maass cusp from of weight zero for the full modular group to the cycle integrals of the Maass cusp form, to Maass cusp forms of non-zero weights and higher level.
In the second part, we investigate a formula of S. Katok (1985), which expresses the imaginary part of the cycle integral of the hyperbolic Poincar´ere series geometrically. We add a complementary counterpart to this interpretation by considering a suitable cycle integral of the Parson Poincar´er series, a non-modular analog of the hyperbolic Poincar´e series.
Finally, we will study an integer-valued analog of the classical Dedekind sums, the Hardy sums. Dedekind sums appear in the study of the cycle integrals of the non-modular Eisenstein series of weight 2. We prove distributional results of these Hardy sums, and give a geometrical interpretation of them.
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Examiner : Imamoğlu, Özlem
Examiner : Tóth, Árpád
Examiner : Kowalski, Emmanuel
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ETH Zurich
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08799 - Imamoglu, Oezlem (Tit.-Prof.)