Hasse's theorem on elliptic curves

Hasse's theorem on elliptic curves

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".

ee also

*Sato-Tate conjecture

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Hasse's theorem — In mathematics, three different theorems of Helmut Hasse are sometimes called Hasse s theorem:* The Hasse norm theorem * Hasse s theorem on elliptic curves * The Hasse Minkowski theorem …   Wikipedia

  • Elliptic curve — In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O . An elliptic curve is in fact an abelian variety mdash; that is, it has a multiplication defined algebraically with… …   Wikipedia

  • Hasse–Witt matrix — In mathematics, the Hasse–Witt matrix H of a non singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the… …   Wikipedia

  • Hasse-Witt matrix — In mathematics, the Hasse Witt matrix H of a non singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping ( p th power mapping where F has q elements, q a power of the prime number p ) with respect to a basis for… …   Wikipedia

  • Counting points on elliptic curves — An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields… …   Wikipedia

  • Moduli of algebraic curves — In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on… …   Wikipedia

  • De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… …   Wikipedia

  • Helmut Hasse — Infobox Scientist name=Helmut Hasse birth date = August 25 1898 death date = December 26 1979 field = MathematicsHelmut Hasse (IPA2|ˈhasə) (25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known… …   Wikipedia

  • Modularity theorem — In mathematics the modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity …   Wikipedia

  • Lenstra elliptic curve factorization — The Lenstra elliptic curve factorization or the elliptic curve factorization method (ECM) is a fast, sub exponential running time algorithm for integer factorization which employs elliptic curves. Technically, the ECM is classified as a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”