Hasse's theorem on elliptic curves

Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.

If N is the number of points on the elliptic curve E over a finite field with q elements, then Hasse's result states that

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

The reason 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 ${\displaystyle {\sqrt {q}}.}$

This result had originally been conjectured by Emil Artin in his thesis.^[1] It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.^[2]

Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of E. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve.

Hasse–Weil Bound

A generalization of the Hasse bound to higher genus algebraic curves is the Hasse–Weil bound. This provides a bound on the number of points on a curve over a finite field. If the number of points on the curve C of genus g over the finite field ${\displaystyle \mathbb {F} _{q}}$ of order q is ${\displaystyle \#C(\mathbb {F} _{q})}$, then

${\displaystyle |\#C(\mathbb {F} _{q})-(q+1)|\leq 2g{\sqrt {q}}.}$

This result is again equivalent to the determination of the absolute value of the roots of the local zeta-function of C, and is the analogue of the Riemann hypothesis for the function field associated with the curve.

The Hasse–Weil bound reduces to the usual Hasse bound when applied to elliptic curves, which have genus g=1.

The Hasse–Weil bound is a consequence of the Weil conjectures, originally proposed by André Weil in 1949 and proved by André Weil in the case of curves.^[3]

See also

• Sato–Tate conjecture
• Schoof's algorithm
• Weil's bound

Notes

1. Artin, Emil (1924), "Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil", Mathematische Zeitschrift, 19 (1): 207–246, doi:10.1007/BF01181075, ISSN 0025-5874, JFM 51.0144.05, MR 1544652, S2CID 117936362
2. Hasse, Helmut (1936), "Zur Theorie der abstrakten elliptischen Funktionenkörper. I, II & III", Crelle's Journal, 1936 (175), doi:10.1515/crll.1936.175.193, ISSN 0075-4102, S2CID 118733025, Zbl 0014.14903
3. Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393

References

• Hurt, Norman E. (2003), Many Rational Points. Coding Theory and Algebraic Geometry, Mathematics and its Applications, vol. 564, Dordrecht: Kluwer/Springer-Verlag, ISBN 1-4020-1766-9, MR 2042828
• Niederreiter, Harald; Xing, Chaoping (2009), Algebraic Geometry in Coding Theory and Cryptography, Princeton: Princeton University Press, ISBN 978-0-6911-0288-7, MR 2573098
• Chapter V of Silverman, Joseph H. (1994), The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR 1329092
• Washington, Lawrence C. (2008), Elliptic Curves. Number Theory and Cryptography, 2nd Ed, Discrete Mathematics and its Applications, Boca Raton: Chapman & Hall/CRC Press, ISBN 978-1-4200-7146-7, MR 2404461