the arithmetic of elliptic curves (original) (raw)

The Arithmetic of Elliptic Curves

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

1.2 Elliptic Curves over Finite Fields

1. 1. See bad reduction (http://planetmath.org/BadReduction2).
2. 1. The criterion of Néron-Ogg-Shafarevich.
4. 1. Hasse’s bound for elliptic curves over finite fields.

1.3 The Mordell-Weil Group E⁢(K)

1.4 The Torsion Subgroup of E⁢(K)

2. 1. Mazur’s theorem on torsion of elliptic curves (a classification of all possible torsion subgroups).
4. 1. 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. 1. A bound for the rank of an elliptic curve.

1.6 Complex Multiplication

4. 1. 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 forms.

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. Springer-Verlag, New York, 1986.
• 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
• 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

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