On the number of solutions of some Kummer equations over finite fields (Q740307)

Let \(k=\mathbb F_q\) be a finite field of characteristic \(p\) and order \(q\). Let \(\ell\neq p\) be another prime. The author computes the number \(N\) of solutions \((x,y)\) in \(k\) of the Kummer equation \[ y^{\ell}=x(x^{\ell^n}-1). \] For characters \(\varphi_1\), \(\varphi_2\) of \(k^\ast\), the Jacobi sum is: \[ \iota (\varphi_1, \varphi_2)=\-\sum_{c\in k} \varphi_1(c)\varphi_2(1-c). \] Let \(\zeta_m=\exp (2\pi i/m)\). The main result is: {\parindent=0.6cm\begin{itemize}\item[(a)] If \(q\not\equiv 1\pmod{\ell^{n+1}}\) then \(N=q\). \item[(b)] If \(q\equiv 1\pmod{\ell^{n+1}}\) then \(N=q-\mathrm{Tr}_{\mathbb Q(\zeta_{\ell^{n+1}})/\mathbb Q} (\eta_n)\), where \(\eta_n=\iota (\psi^{\ell^n},\psi)\) for a character \(\psi\) of \(k^*\) of order \(\ell^{n+1}\). \end{itemize}} (The paper gives the hypothesis of (a) as \(q\not\equiv 1\pmod{\ell}\) but that is a typographical error. The statement given here is what is proven.) Now \(|\eta_n|=\sqrt{q}\) and a further result determines \(\eta_n\) via its prime ideal factorization and a congruence in the ring \(\mathbb Z[\zeta_{\ell^{n+1}}]\).