Hasse's theorem on elliptic curves

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.

If N is the number of points on the elliptic curve E over a finite field with q elements, then Hasse's result states that

${\displaystyle |N-(q+1)|\leq 2{\sqrt {q}}.}$

The reason is that N differs from q + 1, the number of points of the projective line over the same field, by an 'error term' that is the sum of two complex numbers, each of absolute value ${\displaystyle {\sqrt {q}}.}$

This result had originally been conjectured by Emil Artin in his thesis.^[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.