Exercise 49, § 1

Let $k$ be a finite field of characteristic $p$ and of cardinal $q$. We endow $k$ with the counting measure and denote by $\widehat{f}$ the Fourier cotransformation of a function $f:k\rightarrow \mathbf{C}$.

a) Let $\chi$ be a nontrivial character of $k^*$. We extend $\chi$ to $k$ by setting $\chi (0) = 0$.

Prove that $|\widehat{\chi}(\psi )|=\overline{q}$ for every $\psi \in \widehat{k}$ such that $\psi \not= 1$. (“Gauss sums”.)

b) Let $d|q-1$ be an integer. Prove that for every character $\psi \not= 1$ of $k$, one has

with equality if $d= 2$.

c) Let $\chi_1$ and $\chi_2$ be nontrivial characters of $k^*$, extended by 0 to $k$ as in question a). We put

(“Jacobi sums”). If $\chi_1\chi_2\not= 1$, then for every character $\psi \not= 1$ of $k$, one has

d) Suppose that $p$ is a prime number congruent to 1 modulo 4. There exist integers $a$ and $b$ such that $a^2+b^2=p$ (“Fermat’s theorem on two squares”). (Consider a Jacobi sum for characters of suitably chosen order of $\mathbf{F}^*_p.$)

e) Suppose that $p$ is odd. Let $\lambda \in \widehat{k}^*$ be the unique character of order 2 (“Legendre character”). Let $\chi$ be a nontrivial character of $k^*$ and put $\eta =\chi^2$. Prove that for every character $\psi \not= 1$ of $k$, one has

(“Hasse–Davenport formula”).

f) Suppose that $p$ is odd. Let $N_p$ be the number of $(x, y)\in \mathbf{F}_p\times \mathbf{F}_p$ such that $y^2=x^4+ 1$. One has

(When $p$ is congruent to 3 modulo 4, now prove that $N_p=p$; otherwise, verify that $N_p-p=-\sum_x\lambda (x^4+ 1)$, where $\lambda$ is the Legendre character modulo $p$, and express this sum using Jacobi sums.)

g) Compare these formulas with those satisfied by the gamma function (FVR, VII).