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

TL;DR

This paper develops a new framework for studying joint equidistribution problems related to Hecke eigenvalues, using advanced inequalities and variation theory, with applications to the vertical Sato-Tate conjecture.

Contribution

It introduces a higher-dimensional Erdős-Turán inequality analogue and a technique to approximate functions by those of bounded Hardy-Krause variation, addressing vertical Sato-Tate problems.

Findings

Established effective error bounds for joint equidistribution in the vertical Sato-Tate setting.

Derived new results on the distribution of Fourier coefficients and Frobenius traces.

Provided a general framework applicable to a broad class of arithmetic distribution problems.

Abstract

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.