Hellenica World

# .

In mathematics, Hasse's theorem on elliptic curves bounds the number of points on an elliptic curve over a finite field, 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

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

This had been a conjecture of Emil Artin. It is equivalent to the determination of the absolute value of the roots of the local zeta-function of E.

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.