| Modular cusp forms are certain periodic functions with rich inner symmetries and growth conditions. The Fourier coefficients of these cusp forms are related to fundamental problems in arithmetic geometry. The famous Sato-Tate conjecture, for example, is essentially a prediction of how these coefficients are distributed. We will prove a "vertical" analogue of this conjecture using general, all-purpose variants of Weyl's criterion and Erdos-Turan inequality. We will also discuss several applications related to the arithmetic of elliptic curves and modular curves.
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