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minimal model for an elliptic curve
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(Definition)
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Let be a local field, complete with respect to a discrete valuation (for example, could be
, the field of -adic numbers, which is complete with respect to the -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 this 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:
 is the discriminant of a Weierstrass eq. for $E$ and 
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
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Definition 1 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 .
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 ).
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"minimal model for an elliptic curve" is owned by alozano.
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(view preamble)
| Other names: |
minimal equation |
| Also defines: |
minimal model |
This object's parent.
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Cross-references: places, completion, minimal, Weierstrass model, prime, non-archimedean, archimedean, finite place, infinite, number field, integers, number, equations, discriminants, proposition, variables, Weierstrass equation, elliptic curve, field, discrete valuation, complete, local field
There are 2 references to this entry.
This is version 1 of minimal model for an elliptic curve, born on 2006-03-23.
Object id is 7763, canonical name is MinimalModelForAnEllipticCurve.
Accessed 1817 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) | | | 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields) | | | 11G07 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over local fields) |
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Pending Errata and Addenda
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