Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.
[1] It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.
[2] Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of E. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve.
This provides a bound on the number of points on a curve over a finite field.
, 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 analogue of the Riemann hypothesis for the function field associated with the curve.