Exercise 50, § 1

Let $k$ be a finite field of characteristic $p\not= 2$ and of cardinal $q$. We equip $k$ with the counting measure and denote by $\widehat{f}$ the Fourier cotransformation of a function $f:k\rightarrow \mathbf{C}$. We equip $k^*$ with the counting measure and fix a nontrivial character $\psi$ of $k$. We denote by $\lambda$ the unique character of order 2 of $k^*$, extended to $k$ by setting $\lambda (0) = 0$.

For all $a\in k^*$, we set

(“Salié sums”). We denote by $\mathscr{F}(S)$ the Fourier transform of the function S on $k^*$.

a) Let $\chi \in \widehat{k}^*$ and set $\eta =\chi^2$. One has

(use the Hasse-Davenport formula).

b) Deduce that

$y^2=4a$ c) Analogy with the Bessel functions $J_{k+1/2}$ (exercise 20) for $k\in \mathbf{N}$.