the arithmetic of elliptic curves
The Arithmetic of Elliptic Curves
An elliptic curve^{} over a field $K$ is a projective nonsingular^{} curve $E$ defined over $K$ of genus $1$ together with a point $O\in E$ defined over $K$. In the simple case $K=\mathbb{Q}$ every elliptic curve is isomorphic^{} (over $\mathbb{Q}$) to a curve defined by an equation of the form:
$${y}^{2}={x}^{3}+Ax+B$$ 
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 structure^{} of a group.
The theory of elliptic curves is a very rich mix of algebraic geometry^{} and number theory^{} (arithmetic geometry). As in many other areas of number theory, the concepts^{} 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.
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 numbers^{}).

2.
Some basic objects attached to an elliptic curve: $j$invariant (http://planetmath.org/JInvariant), discriminant^{} and invariant differential. The $j$invariant classifies elliptic curves up to isomorphism^{} (http://planetmath.org/JInvariantClassifiesEllipticCurvesUpToIsomorphism).

3.
Isogeny^{}, the dual isogeny and the Frobenius morphism.

4.
Elliptic curves over finite fields^{}: good reduction, bad reduction, multiplicative reduction, additive reduction, cusp, node.

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.
The conductor^{} of an elliptic curve is an integer quantity that measures the arithmetic complexity of the curve (the entry contains examples).

7.
The Tate module of an elliptic curve (it is also defined in the entry inverse limit^{}).

8.
The canonical height^{} on an elliptic curve (over $\mathbb{Q}$).

9.
The height matrix and the elliptic regulator of an elliptic curve.
1.2 Elliptic Curves over Finite Fields

1.
See bad reduction (http://planetmath.org/BadReduction2).

2.
The criterion of NéronOggShafarevich.
 3.

4.
Hasse’s bound for elliptic curves over finite fields.
1.3 The MordellWeil Group $E(K)$

1.
The structure of $E(K)$ is given by the MordellWeil theorem^{} (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 socalled descent theorem.

2.
The free rank of the abelian group^{} $E(K)$ is called the rank of an elliptic curve (http://planetmath.org/RankOfAnEllipticCurve) (the entry contains examples).

3.
Together with the MordellWeil group, one defines two other rather important groups: the Selmer groups^{} and the TateShafarevich group. The TateShafarevich group (or “Sha”) measures the failure of the Hasse principle^{} on the elliptic curve.

4.
Some examples: Mordell curves^{}.
1.4 The Torsion Subgroup of $E(K)$

1.
The NagellLutz Theorem.

2.
Mazur’s theorem on torsion^{} of elliptic curves (a classification of all possible torsion subgroups).

3.
Examples of torsion subgroups of elliptic curves (includes examples of all possible subgroups^{}).

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

2.
A bound for the rank of an elliptic curve.
1.6 Complex Multiplication

1.
Definition of the endomorphism ring (http://planetmath.org/EndomorphismRing) and complex multiplication.
 2.

3.
A connection between complex multiplication and class field theory: abelian extensions of quadratic imaginary number fields.

4.
Definition of Grössencharacters, in general.
1.7 Famous Problems and Conjectures

1.
Fermat’s Last Theorem was finally solved using the theory of elliptic curves and modular forms^{}.

2.
The Birch and SwinnertonDyer conjecture (relating the $L$series of an elliptic curve with the algebraic^{} rank).

3.
The TaniyamaShimuraWeil Conjecture (now a theorem!).
1.8 Cryptography
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. SpringerVerlag, New York, 1986.
 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1994.
 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Note: If you want to contribute to this entry, please send an email to the author (alozano).
Title  the arithmetic of elliptic curves 
Canonical name  TheArithmeticOfEllipticCurves 
Date of creation  20130322 15:06:19 
Last modified on  20130322 15:06:19 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  15 
Author  alozano (2414) 
Entry type  Topic 
Classification  msc 14H52 
Classification  msc 11G05 
Synonym  concepts in the theory of elliptic curves 
Related topic  EllipticCurve 
Related topic  CriterionOfNeronOggShafarevich 
Related topic  HassesBoundForEllipticCurvesOverFiniteFields 
Related topic  BirchAndSwinnertonDyerConjecture2 
Related topic  RankOfAnEllipticCurve 
Related topic  MazursTheoremOnTorsionOfEllipticCurves 
Related topic  TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve 