minimal model for an elliptic curve
Let be a local field, complete with respect to a discrete valuation (for example, could be , the field of http://planetmath.org/node/PAdicIntegers-adic numbers, which is complete with respect to the http://planetmath.org/node/PAdicValuation-adic valuation).
Let be an elliptic curve defined over given by a Weierstrass equation
where are constants in . By a suitable change of variables, we may assume that . As it is pointed out in http://planetmath.org/node/WeierstrassEquationOfAnEllipticCurvethis entry, any other Weierstrass equation for is obtained by a change of variables of the form
with and . Moreover, by Proposition 2 in the same entry, the discriminants of both equations satisfy , so they only differ by a th power of a non-zero number in . Let us define a set:
Since is a discrete valuation, the set is a set of non-negative integers, therefore it has a minimum value . Moreover, by the remark above, satisfies and is the unique number with .
Definition.
Let be an elliptic curve over a local field , complete with respect to a discrete valuation . A Weierstrass equation for with discriminant is said to be a minimal model for (at ) if , the minimum of the set above.
It follows from the discussion above that every elliptic curve over a local field has a minimal model over .
Definition.
Let be a number field and let be an infinite or finite place (archimedean or non-archimedean prime) of . Let be an elliptic curve over . A given Weierstrass model for is said to be minimal at if the same model is minimal over , the completion of at . A Weierstrass equation for is said to be minimal if it is minimal at for all places of .
It can be shown that all elliptic curves over have a global minimal model. However, this is not true over general number fields. There exist elliptic curves over a number field which do not have a global minimal model (i.e. any given model is not minimal at for every ).
Title | minimal model for an elliptic curve |
---|---|
Canonical name | MinimalModelForAnEllipticCurve |
Date of creation | 2013-03-22 15:48:03 |
Last modified on | 2013-03-22 15:48:03 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 14H52 |
Classification | msc 11G05 |
Classification | msc 11G07 |
Synonym | minimal equation |
Defines | minimal model |