Automorphic forms and the distribution of points on odd-dimensional spheres.
The theory of Hecke operators can be used to construct uniformly distributed sets of points on spheres. The degree of uniform distribution is measured by a \(L^{2}\) discrepancy function which has to be compared with the number of points of the set used. The general result here is that for any \(n \geq 2\) there is a number \(h\) so that any prime \(p \geq 13\) there is a set (Hecke operator) of \(h(p^{n}-1)/(p-1)\) points on \(S^{2n-1}\) where the discrepancy is bounded by \(hnp^{-(n-1)/2}\). In proving this the author makes use of the theory of automorphic forms on unitary groups (they study more generally groups \(G\) where \(G(\mathbb R)\) is compact). The author reduces the estimation of the discrepancy by the techniques of harmonic analysis to questions about automorphic forms. The central estimate follows from arguments of base change (due here mainly to the author and Labesse) and the association of certain automorphic forms with Galois representations due to [\textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. Ann. Math. Studies 151. Princeton Univ. Press (2002; Zbl 1036.11027)].
scientific article; zbMATH DE number 1874392
scientific article