Hasse's theorem on elliptic curves

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 √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.

Contents 1 Hasse-Weil Bound 2 See also 3 Notes 4 References 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 ^ 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 ^ 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 ^ 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 hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Elliptic curvesFinite fieldsTheorems in algebraic number theory

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