A supplement to Manin's proof of the Hasse inequality

Abstract

In 1956 Yu.I.~Manin published an elementary proof of Helmut Hasse's 1933 result stating that the Riemann hypothesis holds in the case of an elliptic function field over a finite field. We briefly explain how Manin's proof relates to more modern proofs of the same result. This enables us to present an analogous elementary proof for the case of finite fields of characteristic two, which was excluded in the original argument.