1 Singular Cubic Curves
and so we can write the Taylor expansion of at as follows:
for some and (an
algebraic closure of ).
The singular point is a node if . In this case there are two different tangent lines to at , namely:
If then we say that is a cusp, and there is a unique tangent line at .
Note: See the entry for elliptic curve for examples of cusps and nodes.
Let be given by a Weierstrass equation, and let be the discriminant and as in the definition of . Then:
is singular if and only if ,
has a node if and only if and ,
has a cusp if and only if .
2 Reduction of Elliptic Curves
If has a cusp then we say that has additive reduction at .
From Proposition 1 we deduce the following:
Let be an elliptic curve with coefficients in . Let be a prime. If has bad reduction at then .
has good reduction at .
However has bad reduction at , and the reduction is additive (since modulo we can write the equation as and the slope is ).
The elliptic curve has bad multiplicative reduction at and . The reduction at is split, while the reduction at is non-split. Indeed, modulo we could write the equation as , being the slopes and . However, for the slopes are not in ( is not in ).
- 1 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
- 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.
|Date of creation||2013-03-22 13:49:21|
|Last modified on||2013-03-22 13:49:21|
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