Hasse's theorem on elliptic curves

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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 Helmut Hasse's result states that

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

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.

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[edit]

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 \mathbb{F}_q of order q is \#C(\mathbb{F}_q), then

|\#C(\Bbb{F}_q) - (q+1)| \le 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.[3] The proof was provided by Pierre Deligne in 1974.[4]

Notes[edit]

  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, MR 1544652, Zbl 51.0144.05 
  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, 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 
  4. ^ Deligne, Pierre (1974), La Conjecture de Weil: I, Publications Mathématiques de l'IHÉS 43: 273–307, doi:10.1007/BF02684373, ISSN 0073-8301, MR 340258, Zbl 0287.14001 

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