A supplement to Manin's proof of the Hasse inequality
Let \(p\) be a prime number and \(q=p^{n}\) for some \(n\in\mathbb{N}_{+}\). Let \(E\) be an elliptic curve defined over the finite field \(\mathbb{F}_{q}\) with \(q\)-elements and by \(E(\mathbb{F}_{q})\) denote the set of \(\mathbb{F}_{q}\)-rational points on \(E\). A famous result of \textit{H. Hasse} is the inequality \(|\sharp E(\mathbb{F}_{q})-(q+1)|\leq 2\sqrt{q}\). This result was proved in [J. Reine Angew. Math. 175, 193--208 (1936; Zbl 0014.24902)]. A completely elementary proof of Hasse's inequality for elliptic curves defined over field of characteristic \(>2\) was given by \textit{Yu. I. Manin} in [Izv. Akad. Nauk SSSR, Ser. Mat. 20, 673--678 (1956; Zbl 0072.03202)]. In the paper the authors present an elementary approach to Hasse inequality in the case of fields of characteristic 2.
scientific article; zbMATH DE number 6386231
scientific article