Hasse's theorem on elliptic curves
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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.
Setting N = 0 one finds
which cannot happen since q > 1. Therefore N > 0 and there is always at least one point.
[edit] See also
[edit] 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
