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).
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
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 .
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 .
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|
|Date of creation||2013-03-22 15:48:03|
|Last modified on||2013-03-22 15:48:03|
|Last modified by||alozano (2414)|