Sato-Tate equidistribution of Kurlberg-Rudnick sums
Let \(\psi\) and \(\chi\) be nontrivial additive and multiplicative complex valued characters on a finite field \(k\). The Kurlberg-Rudnick sums of the title are the complex functions on \(k\) defined by \(H(\psi,\chi;t)= \sum_{x\in k}\psi(x^2+tx) \chi(x)\). After a normalization, the author proves that they are traces of conjugacy classes is \(\text{SU}_2\) and therefore define angles \(\theta(\psi,\chi;t)\in [0,\pi]\). He further shows that when \((k,\psi,\chi)\) vary with \(|k|\to\infty\), these angles are approximately equidistributed with respect to the Sato-Tate measure, and moreover, that given any \(r\)-tuple of such sequences of (nonequal and noncontragredient) parameters, the corresponding measures are statistically independent. The proof goes through a reinterpretation of the sums in terms of \(\ell\)-adic sheaves on the affine line, the study of their Fourier transforms, and some (clever) group theory.
scientific article; zbMATH DE number 1656647
scientific article