Hasse's Theorem On Elliptic Curves - Hasse-Weil Bound

Hasse-Weil Bound

A generalization of the Hasse bound to higher genus algebraic curves is the Hasse-Weil bound. This provides a bound on the number of points on a curve over a finite field. If the number of points on the curve C of genus g over the finite field of order q is, then

This result is again equivalent to the determination of the absolute value of the roots of the local zeta-function of C, and is the analog of the Riemann hypothesis for the function field associated with the curve.

The Hasse-Weil bound reduces to the usual Hasse bound when applied to elliptic curves, which have genus g=1.

The Hasse-Weil bound is a consequence of the Weil conjectures, originally proposed by André Weil in 1949. The proof was provided by Pierre Deligne in 1974.