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 Helmut Hasse's result states that
That is, the interpretation 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 √q.
This result had originally been conjectured by Emil Artin in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.
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 analog of the Riemann hypothesis for the function field associated with the elliptic curve.
Read more about Hasse's Theorem On Elliptic Curves: Hasse-Weil Bound
Famous quotes containing the words theorem and/or curves:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“At the end of every diet, the path curves back toward the trough.”
—Mason Cooley (b. 1927)