Fine Art


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.
See also

Sato-Tate conjecture


Chapter V of Silverman, Joseph H. (1994), The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR1329092,

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World