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单词 TheArithmeticOfEllipticCurves
释义

the arithmetic of elliptic curves


The Arithmetic of Elliptic Curves

An elliptic curveMathworldPlanetmath over a field K is a projective nonsingularPlanetmathPlanetmath curve E defined over K of genus 1 together with a point OE defined over K. In the simple case K= every elliptic curve is isomorphicPlanetmathPlanetmathPlanetmath (over ) to a curve defined by an equation of the form:

y2=x3+Ax+B

where A,B are integers. The most remarkable feature of an elliptic curve is the fact that the group of points can be given the structureMathworldPlanetmath of a group.

The theory of elliptic curves is a very rich mix of algebraic geometryMathworldPlanetmathPlanetmath and number theoryMathworldPlanetmath (arithmetic geometry). As in many other areas of number theory, the conceptsMathworldPlanetmath are simple to state but the theory is extremely deep and beautiful. The intrinsic arithmetic of the points on an elliptic curve is absolutely compelling. The most prominent mathematicians of our time have contributed in the development of the theory. The ultimate goal of the theory is to completely understand the structure of the points on the elliptic curve over any field F and being able to find them.

1.1 Basic Definitions

  1. 1.

    For a basic exposition of the subject the reader should start with the entry elliptic curve (http://planetmath.org/EllipticCurve) (defines elliptic curve, the group law and gives some examples with graphs, also treats elliptic curves over the complex numbersMathworldPlanetmathPlanetmath).

  2. 2.

    Some basic objects attached to an elliptic curve: j-invariant (http://planetmath.org/JInvariant), discriminantPlanetmathPlanetmathPlanetmathPlanetmath and invariant differential. The j-invariant classifies elliptic curves up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/JInvariantClassifiesEllipticCurvesUpToIsomorphism).

  3. 3.

    IsogenyMathworldPlanetmath, the dual isogeny and the Frobenius morphism.

  4. 4.

    Elliptic curves over finite fieldsMathworldPlanetmath: good reduction, bad reduction, multiplicative reduction, additive reduction, cusp, node.

  5. 5.

    One of the most important objects that one can associate to an elliptic curve is the L-series (http://planetmath.org/LSeriesOfAnEllipticCurve) (the entry defines the L-series of an elliptic curve (http://planetmath.org/LSeriesOfAnEllipticCurve) and also talks about analytic continuation).

  6. 6.

    The conductorPlanetmathPlanetmathPlanetmath of an elliptic curve is an integer quantity that measures the arithmetic complexity of the curve (the entry contains examples).

  7. 7.

    The Tate module of an elliptic curve (it is also defined in the entry inverse limitMathworldPlanetmathPlanetmath).

  8. 8.

    The canonical heightPlanetmathPlanetmath on an elliptic curve (over ).

  9. 9.

    The height matrix and the elliptic regulator of an elliptic curve.

1.2 Elliptic Curves over Finite Fields

  1. 1.

    See bad reduction (http://planetmath.org/BadReduction2).

  2. 2.

    The criterion of Néron-Ogg-Shafarevich.

  3. 3.

    Supersingular reductionPlanetmathPlanetmathPlanetmath.

  4. 4.

    Hasse’s bound for elliptic curves over finite fields.

1.3 The Mordell-Weil Group E(K)

  1. 1.

    The structure of E(K) is given by the Mordell-Weil theoremMathworldPlanetmath (see also this entry (http://planetmath.org/RankOfAnEllipticCurve)). The main two ingredients of the proof of the theorem are the concept of height function and the so-called descent theorem.

  2. 2.

    The free rank of the abelian groupMathworldPlanetmath E(K) is called the rank of an elliptic curve (http://planetmath.org/RankOfAnEllipticCurve) (the entry contains examples).

  3. 3.

    Together with the Mordell-Weil group, one defines two other rather important groups: the Selmer groupsMathworldPlanetmath and the Tate-Shafarevich group. The Tate-Shafarevich group (or “Sha”) measures the failure of the Hasse principleMathworldPlanetmath on the elliptic curve.

  4. 4.

    Some examples: Mordell curvesMathworldPlanetmath.

1.4 The Torsion Subgroup of E(K)

  1. 1.

    The Nagell-Lutz Theorem.

  2. 2.

    Mazur’s theorem on torsionPlanetmathPlanetmathPlanetmath of elliptic curves (a classification of all possible torsion subgroups).

  3. 3.

    Examples of torsion subgroups of elliptic curves (includes examples of all possible subgroupsMathworldPlanetmathPlanetmath).

  4. 4.

    A way to determine the torsion group: the torsion subgroup of an elliptic curve injects in the reduction of the curve.

1.5 Computing the Rank

  1. 1.

    Read about the rank (http://planetmath.org/RankOfAnEllipticCurve).

  2. 2.

    A bound for the rank of an elliptic curve.

1.6 Complex Multiplication

  1. 1.

    Definition of the endomorphism ring (http://planetmath.org/EndomorphismRing) and complex multiplication.

  2. 2.

    Examples of elliptic curves with complex multiplication.

  3. 3.

    A connection between complex multiplication and class field theory: abelian extensions of quadratic imaginary number fields.

  4. 4.

    Definition of Grössencharacters, in general.

1.7 Famous Problems and Conjectures

  1. 1.

    Fermat’s Last Theorem was finally solved using the theory of elliptic curves and modular formsMathworldPlanetmath.

  2. 2.

    The Birch and Swinnerton-Dyer conjecture (relating the L-series of an elliptic curve with the algebraicMathworldPlanetmath rank).

  3. 3.

    The Taniyama-Shimura-Weil Conjecture (now a theorem!).

1.8 Cryptography

  1. 1.

    Cryptography and Number Theory.

  2. 2.

    The elliptic curve discrete logarithm problem.

  3. 3.

    The Diffie-Hellman key exchange.

References

  • 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 3 Joseph H. Silverman, Advanced Topics inthe Arithmetic of Elliptic Curves. Springer-Verlag, New York,1994.
  • 4 Goro Shimura, Introduction to theArithmetic Theory of Automorphic Functions. Princeton UniversityPress, Princeton, New Jersey, 1971.

Note: If you want to contribute to this entry, please send an email to the author (alozano).

Titlethe arithmetic of elliptic curves
Canonical nameTheArithmeticOfEllipticCurves
Date of creation2013-03-22 15:06:19
Last modified on2013-03-22 15:06:19
Owneralozano (2414)
Last modified byalozano (2414)
Numerical id15
Authoralozano (2414)
Entry typeTopic
Classificationmsc 14H52
Classificationmsc 11G05
Synonymconcepts in the theory of elliptic curves
Related topicEllipticCurve
Related topicCriterionOfNeronOggShafarevich
Related topicHassesBoundForEllipticCurvesOverFiniteFields
Related topicBirchAndSwinnertonDyerConjecture2
Related topicRankOfAnEllipticCurve
Related topicMazursTheoremOnTorsionOfEllipticCurves
Related topicTorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve
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