Notes on \(G_{2}\), determinants, and equidistribution
The paper examines the distribution law for the one-parameter family of exponential sums over \({\mathbb F}_p\): \[ \sum_{x\pmod p, x\neq 0}\chi_2(x)\exp(2\pi i(x^7+tx)/p) \] where \(\chi_2(x)=(x/p)\) is the quadratic character and \(p\) is any prime other than 2 or 7. The first observation is that for large characteristic (\(p>15\)), after dividing the sums by a suitable normalising factor, they are distributed like the traces of random elements of the exceptional group \(G_2\). The author shows how to make the distribution law explicit in all characteristics and explains the exceptional cases of characteristic 3 and 13 where a different group intervenes. In fact, the approach deals with the families of sums in which 7 is replaced by an odd integer \(n\geq 3\), and then the group \(G_2\) is replaced by \(\text{SO}(n)\). The argument is a tour de force based on the author's work on exponential sums and monodromy groups.
scientific article; zbMATH DE number 2083586