On the number of rational points on curves over finite fields with many automorphisms (Q1931510)

Let \(C\) be a geometrically connected smooth curve of genus \(g\) over a finite field \(\mathbb F_q\) with \(q\) elements of characteristic \(p\). For any finite extension \(\mathbb F_{q^r}\) of \(\mathbb F_q\) of degree \(r\), let \(N_r\) be the number of \(\mathbb F_{q^r}\)-points of \(C\). By a theorem of Weil, we have \[ |N_r-q^r-1|\leq 2gq^{\frac{r}{2}}. \] In some special cases, this estimate can be improved. Consider the case where \(C\) is an Artin-Schreier curve, that is, \(C\) is given by the equation \[ y^q-y=f(x) \] for some polynomial \(f\in\mathbb F_q[x]\). We have \[ N_r=q\# \{x\in\mathbb F_{q^r}|\mathrm{Tr}_{\mathbb F_{q^r}/\mathbb F_q}(f(x))=0\}. \] Using Weil descent, the author constructs a hypersurface in \(\mathbb A_{\mathbb F_q}^r\) whose number of \(\mathbb F_q\)-points is precisely the number of \(x\in\mathbb F_{q^r}\) such that \(\mathrm{Tr}_{\mathbb F_{q^r}/\mathbb F_q}(f(x))=0\). Under certain conditions on the smoothness of the projective closure of this hypersurface, the author proves \[ |N_r-q^r|\leq (d-1)^rq^{\frac{r+1}{2}} \] using Deligne's theorem. Consider the case where \(C\) is a Kummer type curve \[ y^{\frac{q-1}{e}}=f(x) \] for some divisor \(e\) of \(q-1\). We have \[ N_r=\delta+\frac{q-1}{e}\sum_{\lambda^e=1}\#\{x\in\mathbb F_{q^r}|\mathrm{N}_{\mathbb F_{q^r}/\mathbb F_q}(f(x))=\lambda\}, \] where \(\delta\) is the number of roots of \(f\) in \(\mathbb F_{q^r}\). Using Weil descent, the author constructs a hypersurface in \(\mathbb A^r_{\mathbb F_q}\) whose number of \(\mathbb F_q\)-points is precisely the number of \(x\in\mathbb F_{q^r}\) such that \(\mathrm{N}_{\mathbb F_{q^r}/\mathbb F_q}(f(x))=\lambda\). By studying its cohomology, the author proves \[ |N_r-q^r-\delta+1|\leq r(d-1)^r(q-1)q^{\frac{r-1}{2}}. \]