bad reduction
1 Singular Cubic Curves
Let be a cubic curve over a field with Weierstrass equation , where:
which has a singular point . This is equivalent to:
and so we can write the Taylor expansion of at as follows:
for some and (an
algebraic closure of ).
Definition 1.
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.
There is a very simple criterion to know whether a cubic curve in Weierstrass form is singular and to differentiate nodes from cusps:
Proposition 1.
Let be given by a Weierstrass equation, and let be the discriminant and as in the definition of . Then:
-
1.
is singular if and only if ,
-
2.
has a node if and only if and ,
-
3.
has a cusp if and only if .
Proof.
See [2], chapter III, Proposition 1.4, page 50. ∎
2 Reduction of Elliptic Curves
Let be an elliptic curve (we could work over any number field , but we choose for simplicity in the exposition). Assume that has a minimal model with Weierstrass equation:
with coefficients in . Let be a prime in . By reducing each of the coefficients modulo we obtain the equation of a cubic curve over the finite field (the field with elements).
Definition 2.
-
1.
If is a non-singular curve then is an elliptic curve over and we say that has good reduction at . Otherwise, we say that has bad reduction at .
-
2.
If has a cusp then we say that has additive reduction at .
-
3.
If has a node then we say that has multiplicative reduction at . If the slopes of the tangent lines ( and as above) are in then the reduction is said to be split multiplicative (and non-split otherwise).
From Proposition 1 we deduce the following:
Corollary 1.
Let be an elliptic curve with coefficients in . Let be a prime. If has bad reduction at then .
Examples:
-
1.
has good reduction at .
-
2.
However has bad reduction at , and the reduction is additive (since modulo we can write the equation as and the slope is ).
-
3.
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 ).
References
- 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.
Title | bad reduction |
Canonical name | BadReduction |
Date of creation | 2013-03-22 13:49:21 |
Last modified on | 2013-03-22 13:49:21 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 12 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 14H52 |
Related topic | EllipticCurve |
Related topic | JInvariant |
Related topic | HassesBoundForEllipticCurvesOverFiniteFields |
Related topic | TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve |
Related topic | ArithmeticOfEllipticCurves |
Related topic | SingularPointsOfPlaneCurve |
Defines | bad reduction |
Defines | good reduction |
Defines | cusp |
Defines | node |
Defines | multiplicative reduction |
Defines | additive reduction |