Hasse's theorem on elliptic curves
- Hasse's theorem on elliptic curves
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
:.
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".
ee also
*Sato-Tate conjecture
References
*Chapter V of Silverman, Joseph H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, No. 106, Princeton University Press, 1992. ISBN 0-387-96203-4
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