Joint Sato-Tate Laws for Transformations of Hecke Eigenvalues: The Vertical Case

We introduce a framework within which a large class of joint equidistribution problems can be studied and resolved with effective error terms. This involves proving a higher dimensional and $\mu$-analogue of the Erd\"{o}s-Tur\'{a}n inequality, and utilizing the theory of the Hardy-Krause (H-K) variation from analysis, where, in particular, we formulate a technique to approximate a broad class of relevant functions by functions of bounded H-K variation. Our main focus will be on the vertical Sato-Tate problem for spaces of cusp forms and for families of elliptic curves over finite fields. In particular, we obtain novel results concerning the distribution of arithmetic relations, and, more generally, multi-dimensional functions of Fourier coefficients and Frobenius traces.