In mathematics, a Néron differential, named after
André Néron
André Néron (; November 30, 1922, La Clayette, France – April 6, 1985, Paris, France) was a French mathematician at the Université de Poitiers who worked on elliptic curves and abelian varieties. He discovered the Néron minimal model ...
, is an almost canonical choice of
1-form
In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...
on an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
or
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
defined over a
local field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
or
global field
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global functio ...
. The Néron differential behaves well on the
Néron minimal model Neron or Néron may refer to:
* Neron (DC Comics), a fictional character in the DC Comics' universe.
* An alternative name of the Roman Emperor Nero
* André Néron, a mathematician, who introduced:
** Néron minimal model
** Néron differentia ...
s.
For an elliptic curve of the form
:
the Néron differential is
:
References
*
*
Elliptic curves
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