the arithmetic of elliptic curves
The Arithmetic of Elliptic Curves
An elliptic curve over a field is a projective nonsingular curve defined over of genus together with a point defined over . In the simple case every elliptic curve is isomorphic (over ) to a curve defined by an equation of the form:
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 and being able to find them.
1.1 Basic Definitions
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).
One of the most important objects that one can associate to an elliptic curve is the -series (http://planetmath.org/LSeriesOfAnEllipticCurve) (the entry defines the -series of an elliptic curve (http://planetmath.org/LSeriesOfAnEllipticCurve) and also talks about analytic continuation).
The canonical height on an elliptic curve (over ).
The height matrix and the elliptic regulator of an elliptic curve.
1.2 Elliptic Curves over Finite Fields
1.3 The Mordell-Weil Group
1.4 The Torsion Subgroup of
1.5 Computing the Rank
Read about the rank (http://planetmath.org/RankOfAnEllipticCurve).
A bound for the rank of an elliptic curve.
1.6 Complex Multiplication
A connection between complex multiplication and class field theory: abelian extensions of quadratic imaginary number fields.
Definition of Grössencharacters, in general.
1.7 Famous Problems and Conjectures
- 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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|Title||the arithmetic of elliptic curves|
|Date of creation||2013-03-22 15:06:19|
|Last modified on||2013-03-22 15:06:19|
|Last modified by||alozano (2414)|
|Synonym||concepts in the theory of elliptic curves|