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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

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

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,

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