Kloosterman sums as algebraic integers
Let \(p\) be a prime, \(q\) a power of \(p\), \(\mathbb{F}_ q\) the field with \(q\) elements, tr the trace from \(\mathbb{F}_ q\) to \(\mathbb{F}_ p\), and \(\zeta_ p= e^{2\pi i/p}\). The Kloosterman sums \[ \text{Kl} (\mathbb{F}_ q, a)= \sum_{xy=a} \zeta_ p^{\text{tr} (x+y)} \qquad (a\in \mathbb{F}_ q^ \times) \] are integers in the number field \(\mathbb{Q} [\zeta_ p ]\cap \mathbb{R}= \mathbb{Q} [\zeta_ p+ \zeta_ p^{-1}]\), so can be considered as elements of a lattice in \(\mathbb{R}^{(p- 1)/2}\) (assuming \(p>2\)) via the Minkowski embedding. After normalization, it is shown that the \(q-1\) Kloosterman sums for \(\mathbb{F}_ q\) are ``approximately equidistributed'', with a known distribution, in \([-2, 2]^{(p- 1)/2}\). This refines a theorem of Katz, according to which these sums are approximately equidistributed along each axis, i.e., via any one complex embedding of \(\mathbb{Q}[ \zeta_ p+ \zeta_ p^{-1}]\). Analogous results hold for generalized Kloosterman sums in several variables.
generalized Kloosterman sums in several variables
scientific article; zbMATH DE number 759279